![List of Figures
1.1 A time-domain plot of x(t) = 5 cos(2πt) versus t. . . . . . . . . . . . . . . . . 3
1.2 A frequency-domain plot of x(t) = 5 cos(2πt). . . . . . . . . . . . . . . . . . . 4
1.3 Time-domain plots of x(t) and its components. . . . . . . . . . . . . . . . . . . 5
1.4 The time and frequency-domain plots of composite x(t). . . . . . . . . . . . . . 6
1.5 An example: the sum of 11 cosine and 11 sine components. . . . . . . . . . . . 7
1.6 Time plot and complex exponential-mode frequency plots. . . . . . . . . . . . . 8
1.7 Time plot and complex exponential-mode frequency plots. . . . . . . . . . . . . 10
2.1 Changing variable from t ∈ [0, T] to θ = 2πt/T ∈ [0, 2π]. . . . . . . . . . . . . 28
2.2 Equally-spaced samples and computed DFT coef cients. . . . . . . . . . . . . 29
2.3 Analog frequency grids and corresponding digital frequency grids. . . . . . . . 31
2.4 The function interpolating two samples is not unique. . . . . . . . . . . . . . . 33
2.5 Functions x(θ) and y(θ) have same values at 0 and π. . . . . . . . . . . . . . . 33
2.6 The aliasing of frequencies outside the Nyquist interval. . . . . . . . . . . . . . 34
2.7 Sampling rate and Nyquist frequency. . . . . . . . . . . . . . . . . . . . . . . . 35
2.8 Taking N = 2n+1 samples from a single period [0, T]. . . . . . . . . . . . . . 39
2.9 Rearranging N = 2n+1 samples on the time grid. . . . . . . . . . . . . . . . . 40
2.10 The placement of samples after changing variable t to θ = 2πt/T . . . . . . . . 40
2.11 Rearranging N = 2n+2 samples on the time grid. . . . . . . . . . . . . . . . . 43
2.12 The placement of samples after changing variable t to θ = 2πt/T . . . . . . . . 43
2.13 Taking N = 2n+2 samples from the period [0, 2π] or [−π, π]. . . . . . . . . . 44
3.1 Illustrating the convergence of the N-term Fourier series. . . . . . . . . . . . . 50
3.2 The behavior of the N-term Fourier series near a jump discontinuity. . . . . . . 50
3.3 The converging Fourier series of an even function. . . . . . . . . . . . . . . . . 55
3.4 The converging Fourier series of an odd function. . . . . . . . . . . . . . . . . . 55
3.5 De ning f(t) = t − t2
for the full range: −1 ≤ t ≤ 1. . . . . . . . . . . . . . . 56
3.6 The converging Fourier series of f(t) with jump discontinuities. . . . . . . . . . 57
3.7 The converging Fourier series of f(t) with jump discontinuities. . . . . . . . . . 58
3.8 The graphs of periodic (even) g1(t) and g
1(t). . . . . . . . . . . . . . . . . . . . 80
3.9 The graphs of periodic (odd) g2(t) and g
2(t). . . . . . . . . . . . . . . . . . . . 81
3.10 The graphs of three periods of g3(t). . . . . . . . . . . . . . . . . . . . . . . . . 82
3.11 Gibbs phenomenon and nite Fourier series of the square wave. . . . . . . . . . 90
3.12 The Dirichlet kernel Dn(λ) for n = 8, 12, 16, 20. . . . . . . . . . . . . . . . . 93
3.13 One period of the Dirichlet kernel Dn(λ) for n=8. . . . . . . . . . . . . . . . 93
3.14 One period of the Fejer kernel Fn(λ) for n = 8. . . . . . . . . . . . . . . . . . 98
xi](https://image.slidesharecdn.com/5901135-250621095806-6d5d24f2/85/Discrete-And-Continuous-Fourier-Transforms-Analysis-Applications-And-Fast-Algorithms-Chu-16-320.jpg)
![4 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS
one cycle. The same cycle repeats for each following time intervals: t ∈ [1, 2], t ∈ [2, 3], and
so on. The time it takes for a periodic function x(t) to complete one cycle is called the period,
and it is denoted by T . In this case, we have T = 1 unit of time (appropriate units may be used
to suit the application in hand), and x(t + T ) = x(t) for t ≥ 0.
While the function x(t) is fully speci ed in its analytical form, the graph of x(t) reveals
how the numerical function values change with time. Since a graph is plotted from a table
of pre-computed function values, the cont ents of the graph are the numbers in the table.
However, compared to reading a large table of data, reading the graph is a much more conve-
nient and effective way to s ee the trend or pattern represented by the data, the approximate
locations of minimum, maximum, or zero function values. With this understanding, the time-
domain (or time) content of x(t) (in this simple case) is the graph which plots x(t) versus
t.
For a single sinusoidal function like x(t) = 5 cos(2πt), one can easily tell from its time-
domain graph that it goes through one cycle (or 2π radians) per unit time, so its frequency is
f = 1. It is also apparent from the same graph that the amplitude of x(t) = 5 cos(2πt) is
A = 5. However, strictly for our future needs, let us formally represent the frequency-domain
(or frequency) content of x(t) in Figure 1.1 by a two-tuple (f, A) = (1, 5) in the amplitude-
versus-frequency stem plot given below. The usefulness of the frequency-domain plot will
be apparent in the next section.
Figure 1.2 A frequency-domain plot of x(t) = 5 cos(2πt).
0 0.5 1 1.5 2 2.5 3 3.5 4
−10
−8
−6
−4
−2
0
2
4
6
8
10
Frequency: f cycles per unit time
Amplitude
f = 1
A = 5
1.2 The Frequency-Domain Plots as Graphical Tools
We next consider a function synthesized from a linear combination of several cosine functions
each with a different amplitude as well as a different frequency. For example, let
x(t) = x1(t) + x2(t) + x3(t)
= A1 cos(2πf1t) − A2 cos(2πf2t) + A3 cos(2πf3t)
= 5 cos(2πt) − 7 cos(4πt) + 11.5 cos(6πt).
We see that the rst component function x1(t) = 5 cos(2πt) can be written as x1(t) =
A1 cos(2πf1t) with amplitude A1 = 5, and frequency f1 = 1. Similarly, the second compo-
nent function x2(t) = −7 cos(4πt) can be written as x2(t) = A2 cos(2πf2t) with amplitude](https://image.slidesharecdn.com/5901135-250621095806-6d5d24f2/85/Discrete-And-Continuous-Fourier-Transforms-Analysis-Applications-And-Fast-Algorithms-Chu-33-320.jpg)
![1.2. THE FREQUENCY-DOMAIN PLOTS AS GRAPHICAL TOOLS 5
A2 = −7 and frequency f2 = 2. For x3(t) = 11.5 cos(6πt), we have A3 = 11.5 and f3 = 3.
The function x1(t) was fully explained in the last section. In the case of x2(t), the cosine
function completes one cycle when its angle θ = 4πt goes from 0 radians to 2π radians, which
implies that t changes from 0 to 0.5 units. So the period of x2(t) is T2 = 0.5 units, and its
frequency is f2 = 1
T2
= 2 cycles per unit time. The expression in the form
xk(t) = Ak cos(2πfkt)
thus explicitly indicates that xk(t) repeats fk cycles per unit time. Now, we can see that the
time unit used to express fk will be canceled out when fk is multiplied by t units of time.
Therefore, θ = 2πfkt remains dimension-less, and the same holds regardless of whether the
time is measured in seconds, minutes, hours, days, months, or years. Note that the equivalent
expression xk(t) = Ak cos(ωkt) is also commonly used, where ωk ≡ 2πfk radians per unit
time is called theangular frequency.
In the time domain, a graph of the composite x(t) can be obtained by adding the three
graphs representing x1(t), x2(t), and x3(t) as shown below. The time-domain plot of x(t)
reveals a periodic composite function with a common period T = 1: the graph of x(t) for
t ∈ [0, 1] is seen to repeat four times in Figure 1.3.
Figure 1.3 Time-domain plots of x(t) and its components.
0 2 4
−20
−10
0
10
20
0 2 4
−20
−10
0
10
20
0 2 4
−20
−10
0
10
20
0 0.5 1 1.5 2 2.5 3 3.5 4
−20
−10
0
10
20
x1
(t) = 5cos(2πt) x2
(t) = −7cos(4πt) x
3
(t) = 11.5cos(6πt)
x(t) = 5cos(2πt) − 7cos(4πt) + 11.5cos(6πt)
In the frequency domain, suppose that the two-tuple (fk, Ak) represents the frequency
content of xk(t), the collection {(f1, A1), (f2, A2), (f3, A3)} de nes the frequency content of
x(t) = x1(t) + x2(t) + x3(t). Note that when x(t) is composite, we speak of the individual
frequencies and amplitudes of its components and they collectively represent the frequency](https://image.slidesharecdn.com/5901135-250621095806-6d5d24f2/85/Discrete-And-Continuous-Fourier-Transforms-Analysis-Applications-And-Fast-Algorithms-Chu-34-320.jpg)
![1.4. USING COMPLEX EXPONENTIAL MODES 7
range of frequencies in both modes, we obtain the following expression:
(1.1) y(t) =
n
k=1
Ak cos(2πfkt) + Bk sin(2πfkt).
The frequency content of y(t) can now be conveniently represented by a set of three-tuples
{ (f1, A1, B1), (f2, A2, B2), . . . , (fn, An, Bn) }, with the understanding that Ak is the ampli-
tude of a pure cosine mode at frequency fk, and Bk is the amplitude of a pure sine mode at
fk. We still need two separate stem plots: one plots Ak versus fk, and the other one plots
Bk versus fk. The time-domain and frequency-domain plots of the sum of eleven cosine and
eleven sine component functions are shown in Figure 1.5, where for 1 ≤ k ≤ 11, fk = k, with
amplitudes 0 Ak ≤ 2 and 0 Bk ≤ 3 randomly generated. The time-domain plot of x(t)
again reveals a periodic composite function with a common period T = 1; the graph of x(t)
for t ∈ [0, 1] is seen to repeat four times in Figure 1.5.
Figure 1.5 An example: the sum of 11 cosine and 11 sine components.
0 0.5 1 1.5 2 2.5 3 3.5 4
−10
0
10
20
0 1 2 3 4 5 6 7 8 9 10 11 12
−1
0
1
2
3
0 1 2 3 4 5 6 7 8 9 10 11 12
−1
0
1
2
3
Pure cosine modes
Pure sine modes
1.4 Using Complex Exponential Modes
By using complex arithmetics, Euler s formulaejθ
= cos θ + j sin θ, where j ≡
√
−1, and the
resulting identities
cos θ =
ejθ
+ e−jθ
2
, sin θ =
ejθ
− e−jθ
2j
,](https://image.slidesharecdn.com/5901135-250621095806-6d5d24f2/85/Discrete-And-Continuous-Fourier-Transforms-Analysis-Applications-And-Fast-Algorithms-Chu-36-320.jpg)
![1.10. FOURIER SERIES: A TOPIC TO COME 23
Example 1.14 The discrete-time sinusoid g = cos(0.7π + π/8) can be written as
g = cos(2πFα + π/8)
with Fα = 0.7/2 = 0.35 = 7/20, so the given sequence is periodic with period N = 20
(samples). In this case, we have K = 7, so the N equispaced samples span seven periods of its
envelope function.
Example 1.15 The discrete-time sinusoid y = cos(
√
3π) is not periodic, because when we
express
y = cos(2πFβ),
we have Fβ =
√
3/2, which is not a rational fraction.
Example 1.16 The discrete-time sinusoid z = cos(2 + π/6) is not periodic, because when
we express
z = cos(2πFγ + π/6),
we have Fγ = 1/π, which is not a rational fraction.
Sampling and reconstruction of signals will be formally treated in Chapters 2, 5 and 6.
1.10 Fourier Series: A Topic to Come
In this chapter we limit our discussion to functions consisting of explicitly given sines and
cosines, because their frequency contents are precisely de ned and easy to understand. To
extend the de n itions and results to an arbitrary function f(t), we must seek to represent
f(t) as a sum of sinusoidal modes this process is called Spectral Decomposition or Spectral
Analysis. The Fourier series refers to such a representation with frequencies speci ed at fk =
k/T cycles per unit time for k = 0, 1, 2, . . ., ∞. The unknowns to be determined are the
amplitudes (or coef cients) Ak and Bk so that
(1.43) f(t) =
∞
k=0
Ak cos
2πkt
T
+ Bk sin
2πkt
T
.
If we are successful, the Fourier series of f(t) is given by the commensurate sum in the right-
hand side, and we have f(t+T ) = f(t). That is, T is the common period of f(t) and f1 = 1/T
is the fundamental frequency of f(t). Note that f(t) completes one cycle over any interval of
length T , including the commonly used [−T/2, T/2].
Depending on the application context, the Fourier series of function f(t) may appear in
variants of the following forms:
1. Using pure cosine and sine modes with variable t,
(1.44) f(t) =
A0
2
+
∞
k=1
Ak cos
2πkt
T
+ Bk sin
2πkt
T
.
Note that f(t) has a nonzero DC term, namely, A0/2, for which we have the following
remarks:](https://image.slidesharecdn.com/5901135-250621095806-6d5d24f2/85/Discrete-And-Continuous-Fourier-Transforms-Analysis-Applications-And-Fast-Algorithms-Chu-52-320.jpg)
![24 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS
Remark 1. For k = 0, we have cos 0 = 1 and sin 0 = 0; hence, the constant term
in (1.43) is given by (A0 cos 0 + B0 sin 0) = A0.
Remark 2. By convention the constant (DC) term in the Fourier series (1.44) is denoted
by 1
2 A0 instead of A0 so that one mathematical formula de nes Ak for all k, in-
cluding k = 0. The analytical formulas which de ne Ak and Bk will be presented
when we study the theory of Fourier series in Chapter 3.
A common variant uses T = 2L with spatial variable x,
(1.45) f(x) =
A0
2
+
∞
k=1
Ak cos
πkx
L
+ Bk sin
πkx
L
.
Note that f(x + 2L) = f(x), and a commonly chosen interval of length 2L is [−L, L].
2. Using cosine modes with phase shifts,
(1.46) f(t) = D0 +
∞
k=1
Dk cos
2πkt
T
− φ̂k
.
The individual terms Dk cos(2πkt
T −φ̂k) a re called the harmonics of f(t). Note that
the spacing between the harmonic frequencies is f = fk+1 −fk = 1
T . Hence, periodic
analog signals are said to have discrete spectra, and the spacing in the frequency domain
is the reciprocal of the period in the time domain.
3. Using complex exponential modes with variable t,
(1.47) f(t) =
∞
k=−∞
Xkej2πkt/T
.
Note that X0 = A0/2 (see above).
4. Using pure cosine and sine modes with dimension-less variable θ = 2πt/T radians,
(1.48) g(θ) =
A0
2
+
∞
k=1
Ak cos kθ + Bk sin kθ.
Since t varies from 0 to T , θ = 2πt/T varies from 0 to 2π, we have g(θ + 2π) =
g(θ). Note that g(θ) completes one cycle over any interval of length 2π, including the
commonly used [−π, π].
5. Using complex exponential modes with dimension-less variable θ = 2πt/T radians,
(1.49) g(θ) =
∞
k=−∞
Xkejkθ
.
6. In Chapter 5, we will learn that the frequency contents of a nonperiodic function x(t)
are de ned by a continuous-frequency function X(f), and we will also encounter the
Fourier series representation of the periodically extended X(f), which appears in the
two forms given below. A full derivation of the continuous-frequency function X(f)
and its Fourier series (when it exists) will be given in Chapter 5.](https://image.slidesharecdn.com/5901135-250621095806-6d5d24f2/85/Discrete-And-Continuous-Fourier-Transforms-Analysis-Applications-And-Fast-Algorithms-Chu-53-320.jpg)
![1.11. TERMINOLOGY 25
Using pure cosine and sine modes with variable f (which represents the continuously
varying frequency) and bandwidth F, that is to say, f ∈ [−F/2, F/2],
(1.50) X(f) =
∞
k=0
ak cos
2πkf
F
+ bk sin
2πkf
F
.
Using complex exponential modes with variable f and bandwidth F,
(1.51) X(f) =
∞
k=−∞
ckej2πkf/F
.
Instead of using the variable f ∈ [−F/2, F/2], a dimension-less variable
θ = 2πf/F ∈ [−π, π] may also be used in the frequency domain. Corresponding
to the two forms of X(f) given above, we have
G(θ) =
∞
k=0
ak cos kθ + bk sin kθ,
(1.52)
and
G(θ) =
∞
k=−∞
ckejkθ
, where θ ∈ [−π, π].
(1.53)
Observe that because the Fourier series expression in θ may be used for both time-domain func-
tion x(t) and frequency-domain function X(f), the dimension-less variable θ is also known as
a neutral variable. Since the Fourier series expression is signi can tly simpli ed by using the
neutral variable θ, it is often the variable of choice in mathematical study of Fourier series.
The theory and techniques for deriving the Fourier series representation of a given function
will be covered in Chapter 3.
1.11 Terminology
Analog signals Signals continuous in time and amplitude are called analog signals.
Temporal and spatial variables The temporal variable t measures time in chosen units; the
spatial variable x measures distance in chosen units.
Period and wavelength The period T satis es f(t + T ) = f(t); the wavelength 2L satis es
g(x + 2L) = g(x).
Frequency and wave number The (rotational) frequency is de ned by
1
T
(cycles per unit
time); the wave number is de ned by
1
2L
(wave numbers per unit length).
Sine and cosine modes A pure sine wave with a xed frequency fk is called a sin e mode
and it is denoted by sin(2πfkt); similarly, a cosine mode is denoted by cos(2πfkt).
Phase or phase shift It refers to the phase angle φ̂k (expressed in radians) in the shifted cosine
mode cos(2πfkt − φ̂k) or cos(2πfkx − φ̂k).](https://image.slidesharecdn.com/5901135-250621095806-6d5d24f2/85/Discrete-And-Continuous-Fourier-Transforms-Analysis-Applications-And-Fast-Algorithms-Chu-54-320.jpg)
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- 6. DISCRETE AND CONTINUOUS FOURIER TRANSFORMS ANALYSIS,APPLICATIONS AND FAST ALGORITHMS
- 8. DISCRETE AND CONTINUOUS FOURIER TRANSFORMS ANALYSIS,APPLICATIONS AND FAST ALGORITHMS Eleanor Chu University of Guelph Guelph, Ontario, Canada
- 9. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-6363-9 (Hardcover) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The Authors and Publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For orga- nizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 10. Contents List of Figures xi List of Tables xv Preface xvii Acknowledgments xxi About the Author xxiii I Fundamentals, Analysis and Applications 1 1 Analytical and Graphical Representation of Function Contents 3 1.1 Time and Frequency Contents of a Function . . . . . . . . . . . . . . . . . . 3 1.2 The Frequency-Domain Plots as Graphical Tools . . . . . . . . . . . . . . . 4 1.3 Identifying the Cosine and Sine Modes . . . . . . . . . . . . . . . . . . . . . 6 1.4 Using Complex Exponential Modes . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Using Cosine Modes with Phase or Time Shifts . . . . . . . . . . . . . . . . 9 1.6 Periodicity and Commensurate Frequencies . . . . . . . . . . . . . . . . . . 12 1.7 Review of Results and Techniques . . . . . . . . . . . . . . . . . . . . . . . 13 1.7.1 Practicing the techniques . . . . . . . . . . . . . . . . . . . . . . . . 15 1.8 Expressing Single Component Signals . . . . . . . . . . . . . . . . . . . . . 19 1.9 General Form of a Sinusoid in Signal Application . . . . . . . . . . . . . . . 20 1.9.1 Expressing sequences of discrete-time samples . . . . . . . . . . . . 21 1.9.2 Periodicity of sinusoidal sequences . . . . . . . . . . . . . . . . . . 22 1.10 Fourier Series: A Topic to Come . . . . . . . . . . . . . . . . . . . . . . . . 23 1.11 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Sampling and Reconstruction of Functions–Part I 27 2.1 DFT and Band-Limited Periodic Signal . . . . . . . . . . . . . . . . . . . . 27 2.2 Frequencies Aliased by Sampling . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Connection: Anti-Aliasing Filter . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4 Alternate Notations and Formulas . . . . . . . . . . . . . . . . . . . . . . . 36 2.5 Sampling Period and Alternate Forms of DFT . . . . . . . . . . . . . . . . . 38 2.6 Sample Size and Alternate Forms of DFT . . . . . . . . . . . . . . . . . . . 41 v
- 11. vi CONTENTS 3 The Fourier Series 45 3.1 Formal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Time-Limited Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Even and Odd Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 Half-Range Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5 Fourier Series Using Complex Exponential Modes . . . . . . . . . . . . . . . 60 3.6 Complex-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.7 Fourier Series in Other Variables . . . . . . . . . . . . . . . . . . . . . . . . 61 3.8 Truncated Fourier Series and Least Squares . . . . . . . . . . . . . . . . . . 61 3.9 Orthogonal Projections and Fourier Series . . . . . . . . . . . . . . . . . . . 63 3.9.1 The Cauchy Schw arz inequality . . . . . . . . . . . . . . . . . . . . 68 3.9.2 The Minkowski inequality . . . . . . . . . . . . . . . . . . . . . . . 71 3.9.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.9.4 Least-squares approximation . . . . . . . . . . . . . . . . . . . . . . 74 3.9.5 Bessel s inequality and Riemann s lemma. . . . . . . . . . . . . . . 77 3.10 Convergence of the Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 79 3.10.1 Starting with a concrete example . . . . . . . . . . . . . . . . . . . . 79 3.10.2 Pointwise convergence a local property . . . . . . . . . . . . . . . 82 3.10.3 The rate of convergence a global property . . . . . . . . . . . . . . 87 3.10.4 The Gibbs phenomenon . . . . . . . . . . . . . . . . . . . . . . . . 89 3.10.5 The Dirichlet kernel perspective . . . . . . . . . . . . . . . . . . . . 91 3.10.6 Eliminating the Gibbs effect by the Cesaro sum . . . . . . . . . . . . 95 3.10.7 Reducing the Gibbs effect by Lanczos smoothing . . . . . . . . . . . 99 3.10.8 The modi cation of Fourier series coef cients . . . . . . . . . . . . . 100 3.11 Accounting for Aliased Frequencies in DFT . . . . . . . . . . . . . . . . . . 102 3.11.1 Sampling functions with jump discontinuities . . . . . . . . . . . . . 104 4 DFT and Sampled Signals 109 4.1 Deriving the DFT and IDFT Formulas . . . . . . . . . . . . . . . . . . . . . 109 4.2 Direct Conversion Between Alternate Forms . . . . . . . . . . . . . . . . . . 114 4.3 DFT of Concatenated Sample Sequences . . . . . . . . . . . . . . . . . . . . 116 4.4 DFT Coef c ients of a Commensurate Sum . . . . . . . . . . . . . . . . . . . 117 4.4.1 DFT coef cients of single-component signals . . . . . . . . . . . . . 117 4.4.2 Making direct use of the digital frequencies . . . . . . . . . . . . . . 121 4.4.3 Common period of sampled composite signals . . . . . . . . . . . . 123 4.5 Frequency Distortion by Leakage . . . . . . . . . . . . . . . . . . . . . . . . 126 4.5.1 Fourier series expansion of a nonharmonic component . . . . . . . . 128 4.5.2 Aliased DFT coef cients of a nonharmonic component . . . . . . . . 129 4.5.3 Demonstrating leakage by numerical experiments . . . . . . . . . . . 131 4.5.4 Mismatching periodic extensions . . . . . . . . . . . . . . . . . . . . 131 4.5.5 Minimizing leakage in practice . . . . . . . . . . . . . . . . . . . . . 134 4.6 The Effects of Zero Padding . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.6.1 Zero padding the signal . . . . . . . . . . . . . . . . . . . . . . . . . 134
- 12. CONTENTS vii 4.6.2 Zero padding the DFT . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.7 Computing DFT De n ing Formulas Per Se . . . . . . . . . . . . . . . . . . . 147 4.7.1 Programming DFT in MATLAB R . . . . . . . . . . . . . . . . . . . 147 5 Sampling and Reconstruction of Functions–Part II 157 5.1 Sampling Nonperiodic Band-Limited Functions . . . . . . . . . . . . . . . . 158 5.1.1 Fourier series of frequency-limited X(f) . . . . . . . . . . . . . . . 159 5.1.2 Inverse Fourier transform of frequency-limited X(f) . . . . . . . . . 159 5.1.3 Recovering the signal analytically . . . . . . . . . . . . . . . . . . . 160 5.1.4 Further discussion of the sampling theorem . . . . . . . . . . . . . . 161 5.2 Deriving the Fourier Transform Pair . . . . . . . . . . . . . . . . . . . . . . 162 5.3 The Sine and Cosine Frequency Contents . . . . . . . . . . . . . . . . . . . 164 5.4 Tabulating Two Sets of Fundamental Formulas . . . . . . . . . . . . . . . . . 165 5.5 Connections with Time/Frequency Restrictions . . . . . . . . . . . . . . . . 165 5.5.1 Examples of Fourier transform pair . . . . . . . . . . . . . . . . . . 167 5.6 Fourier Transform Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.6.1 Deriving the properties . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.6.2 Utilities of the properties . . . . . . . . . . . . . . . . . . . . . . . . 175 5.7 Alternate Form of the Fourier Transform . . . . . . . . . . . . . . . . . . . . 177 5.8 Computing the Fourier Transform from Discrete-Time Samples . . . . . . . . 178 5.8.1 Almost time-limited and band-limited functions . . . . . . . . . . . . 179 5.9 Computing the Fourier Coef cients from Discrete-Time Samples . . . . . . . 181 5.9.1 Periodic and almost band-limited function . . . . . . . . . . . . . . . 182 6 Sampling and Reconstruction of Functions–Part III 185 6.1 Impulse Functions and Their Properties . . . . . . . . . . . . . . . . . . . . 185 6.2 Generating the Fourier Transform Pairs . . . . . . . . . . . . . . . . . . . . 188 6.3 Convolution and Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 189 6.4 Periodic Convolution and Fourier Series . . . . . . . . . . . . . . . . . . . . 192 6.5 Convolution with the Impulse Function . . . . . . . . . . . . . . . . . . . . . 194 6.6 Impulse Train as a Generalized Function . . . . . . . . . . . . . . . . . . . . 195 6.7 Impulse Sampling of Continuous-Time Signals . . . . . . . . . . . . . . . . 202 6.8 Nyquist Sampling Rate Rediscovered . . . . . . . . . . . . . . . . . . . . . . 203 6.9 Sampling Theorem for Band-Limited Signal . . . . . . . . . . . . . . . . . . 207 6.10 Sampling of Band-Pass Signals . . . . . . . . . . . . . . . . . . . . . . . . . 209 7 Fourier Transform of a Sequence 211 7.1 Deriving the Fourier Transform of a Sequence . . . . . . . . . . . . . . . . . 211 7.2 Properties of the Fourier Transform of a Sequence . . . . . . . . . . . . . . . 215 7.3 Generating the Fourier Transform Pairs . . . . . . . . . . . . . . . . . . . . 217 7.3.1 The Kronecker delta sequence . . . . . . . . . . . . . . . . . . . . . 217 7.3.2 Representing signals by Kronecker delta . . . . . . . . . . . . . . . . 218 7.3.3 Fourier transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.4 Duality in Connection with the Fourier Series . . . . . . . . . . . . . . . . . 226
- 13. viii CONTENTS 7.4.1 Periodic convolution and discrete convolution . . . . . . . . . . . . . 227 7.5 The Fourier Transform of a Periodic Sequence . . . . . . . . . . . . . . . . . 229 7.6 The DFT Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.6.1 The interpreted DFT and the Fourier transform . . . . . . . . . . . . 234 7.6.2 Time-limited case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.6.3 Band-limited case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.6.4 Periodic and band-limited case . . . . . . . . . . . . . . . . . . . . . 237 8 The Discrete Fourier Transform of a Windowed Sequence 239 8.1 A Rectangular Window of In nite Width . . . . . . . . . . . . . . . . . . . . 239 8.2 A Rectangular Window of Appropriate Finite Width . . . . . . . . . . . . . . 241 8.3 Frequency Distortion by Improper Truncation . . . . . . . . . . . . . . . . . 243 8.4 Windowing a General Nonperiodic Sequence . . . . . . . . . . . . . . . . . 244 8.5 Frequency-Domain Properties of Windows . . . . . . . . . . . . . . . . . . . 245 8.5.1 The rectangular window . . . . . . . . . . . . . . . . . . . . . . . . 246 8.5.2 The triangular window . . . . . . . . . . . . . . . . . . . . . . . . . 247 8.5.3 The von Hann window . . . . . . . . . . . . . . . . . . . . . . . . . 248 8.5.4 The Hamming window . . . . . . . . . . . . . . . . . . . . . . . . . 250 8.5.5 The Blackman window . . . . . . . . . . . . . . . . . . . . . . . . . 251 8.6 Applications of the Windowed DFT . . . . . . . . . . . . . . . . . . . . . . 252 8.6.1 Several scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.6.2 Selecting the length of DFT in practice . . . . . . . . . . . . . . . . 263 9 Discrete Convolution and the DFT 267 9.1 Linear Discrete Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 267 9.1.1 Linear convolution of two n ite sequences . . . . . . . . . . . . . . . 267 9.1.2 Sectioning a long sequence for linear convolution . . . . . . . . . . . 273 9.2 Periodic Discrete Convolution . . . . . . . . . . . . . . . . . . . . . . . . . 273 9.2.1 De n ition based on two periodic sequences . . . . . . . . . . . . . . 273 9.2.2 Converting linear to periodic convolution . . . . . . . . . . . . . . . 275 9.2.3 De ning the equivalent cyclic convolution . . . . . . . . . . . . . . . 275 9.2.4 The cyclic convolution in matrix form . . . . . . . . . . . . . . . . . 278 9.2.5 Converting linear to cyclic convolution . . . . . . . . . . . . . . . . 280 9.2.6 Two cyclic convolution theorems . . . . . . . . . . . . . . . . . . . . 280 9.2.7 Implementing sectioned linear convolution . . . . . . . . . . . . . . 283 9.3 The Chirp Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 284 9.3.1 The scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 9.3.2 The equivalent partial linear convolution . . . . . . . . . . . . . . . . 285 9.3.3 The equivalent partial cyclic convolution . . . . . . . . . . . . . . . 286 10 Applications of the DFT in Digital Filtering and Filters 291 10.1 The Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.2 Application-Oriented Terminology . . . . . . . . . . . . . . . . . . . . . . . 292 10.3 Revisit Gibbs Phenomenon from the Filtering Viewpoint . . . . . . . . . . . 294
- 14. CONTENTS ix 10.4 Experimenting with Digital Filtering and Filter Design . . . . . . . . . . . . 296 II Fast Algorithms 303 11 Index Mapping and Mixed-Radix FFTs 305 11.1 Algebraic DFT versus FFT-Computed DFT . . . . . . . . . . . . . . . . . . 305 11.2 The Role of Index Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 11.2.1 The decoupling process Stage I . . . . . . . . . . . . . . . . . . . 307 11.2.2 The decoupling process Stage II . . . . . . . . . . . . . . . . . . . 309 11.2.3 The decoupling process Stage III . . . . . . . . . . . . . . . . . . . 311 11.3 The Recursive Equation Approach . . . . . . . . . . . . . . . . . . . . . . . 313 11.3.1 Counting short DFT or DFT-like transforms . . . . . . . . . . . . . . 313 11.3.2 The recursive equation for arbitrary composite N . . . . . . . . . . . 313 11.3.3 Specialization to the radix-2 DIT FFT for N = 2ν . . . . . . . . . . 315 11.4 Other Forms by Alternate Index Splitting . . . . . . . . . . . . . . . . . . . . 317 11.4.1 The recursive equation for arbitrary composite N . . . . . . . . . . . 318 11.4.2 Specialization to the radix-2 DIF FFT for N = 2ν . . . . . . . . . . . 319 12 Kronecker Product Factorization and FFTs 321 12.1 Reformulating the Two-Factor Mixed-Radix FFT . . . . . . . . . . . . . . . 322 12.2 From Two-Factor to Multi-Factor Mixed-Radix FFT . . . . . . . . . . . . . . 328 12.2.1 Selected properties and rules for Kronecker products . . . . . . . . . 329 12.2.2 Complete factorization of the DFT matrix . . . . . . . . . . . . . . . 331 12.3 Other Forms by Alternate Index Splitting . . . . . . . . . . . . . . . . . . . . 333 12.4 Factorization Results by Alternate Expansion . . . . . . . . . . . . . . . . . 335 12.4.1 Unordered mixed-radix DIT FFT . . . . . . . . . . . . . . . . . . . . 335 12.4.2 Unordered mixed-radix DIF FFT . . . . . . . . . . . . . . . . . . . . 337 12.5 Unordered FFT for Scrambled Input . . . . . . . . . . . . . . . . . . . . . . 337 12.6 Utilities of the Kronecker Product Factorization . . . . . . . . . . . . . . . . 339 13 The Family of Prime Factor FFT Algorithms 341 13.1 Connecting the Relevant Ideas . . . . . . . . . . . . . . . . . . . . . . . . . 342 13.2 Deriving the Two-Factor PFA . . . . . . . . . . . . . . . . . . . . . . . . . . 343 13.2.1 Stage I: Nonstandard index mapping schemes . . . . . . . . . . . . . 343 13.2.2 Stage II: Decoupling the DFT computation . . . . . . . . . . . . . . 345 13.2.3 Organizing the PFA computation P art 1 . . . . . . . . . . . . . . . . 346 13.3 Matrix Formulation of the Two-Factor PFA . . . . . . . . . . . . . . . . . . 348 13.3.1 Stage III: The Kronecker product factorization . . . . . . . . . . . . 348 13.3.2 Stage IV: De ning permutation matrices . . . . . . . . . . . . . . . . 348 13.3.3 Stage V: Completing the matrix factorization . . . . . . . . . . . . . 350 13.4 Matrix Formulation of the Multi-Factor PFA . . . . . . . . . . . . . . . . . . 350 13.4.1 Organizing the PFA computation Part 2 . . . . . . . . . . . . . . . 352 13.5 Number Theory and Index Mapping by Permutations . . . . . . . . . . . . . 353
- 15. x CONTENTS 13.5.1 Some fundamental properties of integers . . . . . . . . . . . . . . . . 354 13.5.2 A simple case of index mapping by permutation . . . . . . . . . . . . 363 13.5.3 The Chinese remainder theorem . . . . . . . . . . . . . . . . . . . . 364 13.5.4 The ν-dimensional CRT index map . . . . . . . . . . . . . . . . . . 365 13.5.5 The ν-dimensional Ruritanian index map . . . . . . . . . . . . . . . 366 13.5.6 Organizing the ν-factor PFA computation Part 3 . . . . . . . . . . . 368 13.6 The In-Place and In-Order PFA . . . . . . . . . . . . . . . . . . . . . . . . . 368 13.6.1 The implementation-related concepts . . . . . . . . . . . . . . . . . 368 13.6.2 The in-order algorithm based on Ruritanian map . . . . . . . . . . . 371 13.6.3 The in-order algorithm based on CRT map . . . . . . . . . . . . . . . 371 13.7 Ef cient Implementation of the PFA . . . . . . . . . . . . . . . . . . . . . . 372 14 Computing the DFT of Large Prime Length 375 14.1 Performance of FFT for Prime N . . . . . . . . . . . . . . . . . . . . . . . . 376 14.2 Fast Algorithm I: Approximating the FFT . . . . . . . . . . . . . . . . . . . 378 14.2.1 Array-smart implementation in MATLAB R . . . . . . . . . . . . . . 379 14.2.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 14.3 Fast Algorithm II: Using Bluestein s FFT . . . . . . . . . . . . . . . . . . . 382 14.3.1 Bluestein s FFT and the chirp Fourier transform . . . . . . . . . . . . 382 14.3.2 The equivalent partial linear convolution . . . . . . . . . . . . . . . . 383 14.3.3 The equivalent partial cyclic convolution . . . . . . . . . . . . . . . 384 14.3.4 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 14.3.5 Array-smart implementation in MATLAB R . . . . . . . . . . . . . . 386 14.3.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 Bibliography 389 Index 393
- 16. List of Figures 1.1 A time-domain plot of x(t) = 5 cos(2πt) versus t. . . . . . . . . . . . . . . . . 3 1.2 A frequency-domain plot of x(t) = 5 cos(2πt). . . . . . . . . . . . . . . . . . . 4 1.3 Time-domain plots of x(t) and its components. . . . . . . . . . . . . . . . . . . 5 1.4 The time and frequency-domain plots of composite x(t). . . . . . . . . . . . . . 6 1.5 An example: the sum of 11 cosine and 11 sine components. . . . . . . . . . . . 7 1.6 Time plot and complex exponential-mode frequency plots. . . . . . . . . . . . . 8 1.7 Time plot and complex exponential-mode frequency plots. . . . . . . . . . . . . 10 2.1 Changing variable from t ∈ [0, T] to θ = 2πt/T ∈ [0, 2π]. . . . . . . . . . . . . 28 2.2 Equally-spaced samples and computed DFT coef cients. . . . . . . . . . . . . 29 2.3 Analog frequency grids and corresponding digital frequency grids. . . . . . . . 31 2.4 The function interpolating two samples is not unique. . . . . . . . . . . . . . . 33 2.5 Functions x(θ) and y(θ) have same values at 0 and π. . . . . . . . . . . . . . . 33 2.6 The aliasing of frequencies outside the Nyquist interval. . . . . . . . . . . . . . 34 2.7 Sampling rate and Nyquist frequency. . . . . . . . . . . . . . . . . . . . . . . . 35 2.8 Taking N = 2n+1 samples from a single period [0, T]. . . . . . . . . . . . . . 39 2.9 Rearranging N = 2n+1 samples on the time grid. . . . . . . . . . . . . . . . . 40 2.10 The placement of samples after changing variable t to θ = 2πt/T . . . . . . . . 40 2.11 Rearranging N = 2n+2 samples on the time grid. . . . . . . . . . . . . . . . . 43 2.12 The placement of samples after changing variable t to θ = 2πt/T . . . . . . . . 43 2.13 Taking N = 2n+2 samples from the period [0, 2π] or [−π, π]. . . . . . . . . . 44 3.1 Illustrating the convergence of the N-term Fourier series. . . . . . . . . . . . . 50 3.2 The behavior of the N-term Fourier series near a jump discontinuity. . . . . . . 50 3.3 The converging Fourier series of an even function. . . . . . . . . . . . . . . . . 55 3.4 The converging Fourier series of an odd function. . . . . . . . . . . . . . . . . . 55 3.5 De ning f(t) = t − t2 for the full range: −1 ≤ t ≤ 1. . . . . . . . . . . . . . . 56 3.6 The converging Fourier series of f(t) with jump discontinuities. . . . . . . . . . 57 3.7 The converging Fourier series of f(t) with jump discontinuities. . . . . . . . . . 58 3.8 The graphs of periodic (even) g1(t) and g 1(t). . . . . . . . . . . . . . . . . . . . 80 3.9 The graphs of periodic (odd) g2(t) and g 2(t). . . . . . . . . . . . . . . . . . . . 81 3.10 The graphs of three periods of g3(t). . . . . . . . . . . . . . . . . . . . . . . . . 82 3.11 Gibbs phenomenon and nite Fourier series of the square wave. . . . . . . . . . 90 3.12 The Dirichlet kernel Dn(λ) for n = 8, 12, 16, 20. . . . . . . . . . . . . . . . . 93 3.13 One period of the Dirichlet kernel Dn(λ) for n=8. . . . . . . . . . . . . . . . 93 3.14 One period of the Fejer kernel Fn(λ) for n = 8. . . . . . . . . . . . . . . . . . 98 xi
- 17. xii LIST OF FIGURES 3.15 Illustrating the convergence of the Cesaro sums of the square wave. . . . . . . . 99 3.16 Fourier series with coef cients modi ed by the Lanzcos sigma factor. . . . . . . 101 3.17 The three N-point frequency-domain windows for N = 2n+1=11. . . . . . . 102 3.18 Graphs of ˜ f(t) reconstructed using N computed DFT coef cients. . . . . . . . 105 4.1 Mapping t ∈ [0, T) to θ = 2πt/T ∈ [0, 2π) for 0 ≤ ≤ 2n+1. . . . . . . . 110 4.2 Sampling y(t) at 2 Hz (for three periods) and 3 Hz (for one period). . . . . . . . 125 4.3 Signal reconstructed using computed DFT coef cients from Table 4.1. . . . . . 127 4.4 Sampling y(t) at 2 Hz for 1.5 periods. . . . . . . . . . . . . . . . . . . . . . . 127 4.5 Signal reconstructed using M =10 DFT coef cients from Table 4.2. . . . . . . 133 4.6 Signal reconstructed using M =20 DFT coef cients from Table 4.2. . . . . . . 133 4.7 The Gaussian function x(t) and its Fourier transform X(f). . . . . . . . . . . . 138 4.8 Computing ten DFT coef cients from ten signal samples. . . . . . . . . . . . . 139 4.9 Computing twenty DFT coef cients by zero padding ten signal samples. . . . . 139 4.10 The effect of zero padding the DFT as done in Table 4.4. . . . . . . . . . . . . . 146 5.1 The graphs of L(t) for = −3, 0, 1. . . . . . . . . . . . . . . . . . . . . . . . 162 5.2 Time-domain and frequency-domain plots of x(t) = e−at . . . . . . . . . . . . . 167 5.3 Gaussian function and its real-valued Fourier transform. . . . . . . . . . . . . . 169 5.4 Time-limited rectangular pulse and its Fourier transform. . . . . . . . . . . . . 169 5.5 Connecting Fourier series coef cients to Fourier transform. . . . . . . . . . . . . 170 5.6 A band-limited Fourier transform pair. . . . . . . . . . . . . . . . . . . . . . . 172 5.7 Illustrating the time-shift property. . . . . . . . . . . . . . . . . . . . . . . . . 176 5.8 Illustrating the derivative of the transform property. . . . . . . . . . . . . . . . 177 5.9 Illustrating the derivative of the transform property (n = 2). . . . . . . . . . . . 178 6.1 De ning the Dirac delta function. . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.2 Illustrating properties of the unit impulse function. . . . . . . . . . . . . . . . . 187 6.3 Fourier transform pairs involving the impulse function. . . . . . . . . . . . . . 188 6.4 Illustrating the steps in convolving x(t) with h(t). . . . . . . . . . . . . . . . . 190 6.5 The result of continuous convolution w(t) = x(t) ∗ h(t). . . . . . . . . . . . . . 190 6.6 The periodic signal resulted from convolving x(t) with an impulse train. . . . . 196 6.7 The relationship between impulse train and its Fourier transform. . . . . . . . . 199 6.8 Several more examples of z(t) = x(t) ∗ PT (t). . . . . . . . . . . . . . . . . . . 200 6.9 Fourier transform of the sequence sampled from x(t) = e−at . . . . . . . . . . . 206 6.10 Reducing the effect of aliasing by increasing sampling rate. . . . . . . . . . . . . 206 7.1 Discrete exponential function and its Fourier transform. . . . . . . . . . . . . . 220 7.2 Obtaining Fourier transform pair by derivative of transform property. . . . . . . 220 7.3 Obtaining Fourier transform pair by the property of linearity. . . . . . . . . . . 221 7.4 The Fourier transform of a bilateral exponential function. . . . . . . . . . . . . 222 7.5 Connecting previously obtained results to new tasks. . . . . . . . . . . . . . . . 223 8.1 The rectangular window and its magnitude spectrum. . . . . . . . . . . . . . . 247 8.2 The triangular window and its magnitude spectrum. . . . . . . . . . . . . . . . 249 8.3 The von Hann window and its magnitude spectrum. . . . . . . . . . . . . . . . 250 8.4 The Hamming window and its magnitude spectrum. . . . . . . . . . . . . . . . 252
- 18. LIST OF FIGURES xiii 8.5 The Blackman window and its magnitude spectrum. . . . . . . . . . . . . . . . 253 8.6 The one-sided spectrum of UI(f) = 1 N F{xI(t) · wrect(t)}. . . . . . . . . . . . 256 8.7 Non-overlapped mainlobes and separate local maxima. . . . . . . . . . . . . . . 257 8.8 The merging of local maxima due to overlapped mainlobes. . . . . . . . . . . . 258 8.9 A local maximum is smeared out by overlapped mainlobes. . . . . . . . . . . . 259 8.10 Values of UI(fk) obtainable by the DFT, where fk = k/T (T = 2.2T ). . . . . . 260 8.11 Fourier transforms of zI(t) weighted by four different windows. . . . . . . . . . 261 8.12 The computed DFT of zI(t) truncated by a rectangular window. . . . . . . . . . 261 8.13 The computed DFT of zI(t) weighted by a triangular window. . . . . . . . . . . 262 8.14 The computed DFT of zI(t) weighted by a von Hann window. . . . . . . . . . . 262 8.15 The computed DFT of zI(t) weighted by a Blackman window. . . . . . . . . . 263 8.16 The effects of zero padding a windowed sequence. . . . . . . . . . . . . . . . . 264 8.17 Improving UI(f) = 1 N F{zI(t)·wtri(t)} by changing window length. . . . . . . 265 8.18 The computed DFT of zI(t)·wtri(f) after doubling the window length. . . . . . 265 8.19 Improving frequency detection by doubling the sampling rate. . . . . . . . . . . 266 9.1 The steps in performing continuous convolution u(t) = g(t) ∗ h(t). . . . . . . . 268 9.2 The result of continuous convolution u(t) = g(t) ∗ h(t). . . . . . . . . . . . . . 269 9.3 The steps in performing linear discrete convolution {u} = {g} ∗ {h}. . . . . . 270 9.4 The result of discrete convolution {uk} = {gk} ∗ {hk}. . . . . . . . . . . . . . 271 9.5 The results of discrete convolution {uk} = {gk} ∗ {hk}. . . . . . . . . . . . . . 272 9.6 Performing linear convolution {uk} = {gk} ∗ {hk} in two sections. . . . . . . . 274 9.7 The steps in performing periodic discrete convolution. . . . . . . . . . . . . . . 276 9.8 Converting linear to periodic discrete convolution. . . . . . . . . . . . . . . . . 277 9.9 De ning the equivalent cyclic convolution. . . . . . . . . . . . . . . . . . . . . 279 9.10 Converting linear to cyclic convolution. . . . . . . . . . . . . . . . . . . . . . . 281 9.11 Interpreting chirp Fourier transform as a partial linear convolution. . . . . . . . . 287 9.12 Interpreting chirp Fourier transform as a partial cyclic convolution. . . . . . . . . 288 10.1 Sampling H(f) to obtain impulse response of a FIR lter. . . . . . . . . . . . . 297 10.2 Sampled noisy signal x(t) and its magnitude spectrum. . . . . . . . . . . . . . 298 10.3 Discrete linear convolution of {x} and FIR lter {h}. . . . . . . . . . . . . . 299 10.4 Discrete periodic convolution of {x} and FIR lter {h}. . . . . . . . . . . . . 300 10.5 Computed DFT coef cients of the ltered sample sequence. . . . . . . . . . . . 301
- 20. List of Tables 2.1 Alternate symbols and alternate de nitions/assumptions. . . . . . . . . . . . . 37 2.2 Constants resulting from assuming unit period or unit spacing. . . . . . . . . . 37 2.3 Using analog frequency versus digital frequency. . . . . . . . . . . . . . . . . 38 3.1 The DFT coef cients computed in Example 3.66 (N = 8, 16, 32). . . . . . . . 106 4.1 Numerical values of M DFT coef cients when TM = To and TM = 3To. . . . 126 4.2 Numerical values of M distorted DFT coef cients when TM =1.5To. . . . . . 132 4.3 Numerical values of the DFT coef cients plotted in Figures 4.8 and 4.9. . . . . 140 4.4 Zero pad the DFT coef cie nts computed in Example 3.66 (N = 8, 16). . . . . 145 4.5 Variable names in MATLAB code. . . . . . . . . . . . . . . . . . . . . . . . . 148 4.6 Testing function dft1 matrix.m using MATLAB 5.3 and 7.4. . . . . . . . . . . 149 4.7 Testing function dft2 matrix.m using MATLAB 5.3 and 7.4. . . . . . . . . . . 150 4.8 Testing function dft3 matrix.m using MATLAB 5.3 and 7.4. . . . . . . . . . . 152 4.9 Testing function dft.m using MATLAB 5.3 and 7.4. . . . . . . . . . . . . . . . 155 5.1 Two sets of fundamental formulas in Fourier analysis. . . . . . . . . . . . . . . 166 5.2 Connections with time/frequency restrictions. . . . . . . . . . . . . . . . . . . 166 5.3 Fourier transform properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.4 Fourier transform properties (expressed in ω = 2πf). . . . . . . . . . . . . . . 179 5.5 Connections with time-limited restriction. . . . . . . . . . . . . . . . . . . . . 182 7.1 Properties of the Fourier transform X̂I(F) of a sequence. . . . . . . . . . . . . 215 7.2 Properties of the Fourier transform X̃I(θ) of a sequence (θ=2πF). . . . . . . . 217 8.1 Spectral characteristics of ve windows (λ = Tf = (Nt)f). . . . . . . . . . 253 14.1 Performance of MATLAB 5.3 built-in FFT. . . . . . . . . . . . . . . . . . . . 376 14.2 Measuring error in computing ifft(fft(x)) in MATLAB 5.3. . . . . . . . . . . . 377 14.3 Performance of MATLAB 7.4 built-in FFT. . . . . . . . . . . . . . . . . . . . 377 14.4 Measuring error in computing ifft(fft(x)) in MATLAB 7.4. . . . . . . . . . . . 378 14.5 Evaluating function M- les Tfft.m and iTfft.m for large prime N. . . . . . . . 382 14.6 Performance of Bluestein s FFT for large primeN. . . . . . . . . . . . . . . . 388 xv
- 22. Preface The topics in this book were selected to build a solid foundation for the application of Fourier analysis in the many diverging and continuously evolving areas in the digital signal processing enterprise. While Fourier transforms have long been used systematically in electrical engi- neering, the wide variety of modern-day applications of the discrete Fourier transform (DFT) on digital computers (made feasible by the fast Fourier transform (FFT) algorithms) motivates people in all branches of the physical sciences, computational sciences and engineering to learn the DFT, the FFT algorithms, as well as the many applications that directly impact our life to- day. To understand how the DFT can be deployed in any application area, one needs to have the core knowledge of Fourier analysis, which connects the DFT to the continuous Fourier transform, the Fourier series, and the all important sampling theorem. The tools offered by Fourier analysis enable us to correctly deploy and interpret the DFT results. This book presents the fundamentals of Fourier analysis and their deployment in signal processing by way of the DFT and the FFT algorithms in a logically careful manner so that the text is self-contained and accessible to senior undergraduate students, graduate students, and researchers and professionals in mathematical science, numerical analysis, computer science, physics, and the various disciplines in engineering and applied science. The contents of this book are divided into two parts and fourteen chapters with the following features, and the cited topics can be selected and combined in a number of suggested ways to suit one s interest or the need of a related course: • From the very beginning of the text a large number of graphical illustrations and worked examples are provided to help explain the many concepts and relationships; a detailed table of contents makes explicit the logical arrangement of topics in each chapter, each section, and each subsection. • Readers of this book are not required to have prior knowledge of Fourier analysis or signal processing. To provide background, the basic concepts of signals and signal sampling together with a practical introduction to the DFT are presented in Chapters 1 and 2, while the mathematical derivation of the DFT is deferred to Chapter 4. • The coverage of the Fourier series in Chapter 3 (Sections 3.1 3.8) is self-contained, and its relationship to the DFT is explained in Section 3.11. Section 3.9 on orthogonal projections and Section 3.10 on the convergence of Fourier series (including a detailed study of the Gibbs phenomenon) are more mathematical, and they can be skipped in the rst reading. • The DFT is formally derived in Chapter 4, and a thorough discussion of the relationships between the DFT spectra and sampled signals under various circumstances is presented with supporting numerical results and graphical illustrations. In Section 4.7 I provide instructional MATLAB R 1 codes for computing the DFT formulas per se, while the fast algorithms for 1MATLAB is a registered trademark of The MathWorks, Inc. xvii
- 23. xviii PREFACE computing the DFT are deferred to Part II of the book. • The continuous Fourier transform is introduced in Chapter 5. The concepts and results from Chapters 1 through 3 are used here to derive the sampling theorem and the Fourier trans- form pair. Worked examples of the Fourier transform pair are then given and the properties of Fourier transform are derived. The computing of Fourier transform from discrete-time sam- ples is investigated, and the relationship between sampled Fourier transform and Fourier series coef cients is also established in this chapter. • Chapter 6 is built on the material previously developed in Chapters 3 and 5. The topics covered in Chapter 6 include the Dirac delta function, the convolution theorems concerning the Fourier transform, and the periodic and discrete convolution theorems concerning the Fourier series. I then show how these mathematical tools interplay to model the sampling process and develop the sampling theorem directly. • With the foundations laid in Chapters 1 through 6, the Fourier transform of an ideally sampled signal is now formally de n ed (in mathematical terms) in Chapter 7, which provides the theoretical basis for appropriately constructing and deploying digital signal processing tools and correctly interpreting the processed results in Chapters 8 through 10. • In Chapter 8 the data-weighting window functions are introduced, the analysis of the possibly distorted DFT spectra of windowed sequences is pursued, and the various scenarios and consequences related to frequency detection are demonstrated graphically using numerical examples. • Chapter 9 covers discrete convolution algorithms, including the linear convolution algo- rithm, the periodic (and the equivalent circular or cyclic) convolution algorithm, and their im- plementation via the DFT (computed by the FFT). The relationship between the chirp Fourier transform and the cyclic convolution is also established in this chapter. • The application of the DFT in digital ltering and lters is the topic of Chapter 10. The Gibbs phenomenon is also revisited in this chapter from a ltering viewpoint. • Since the FFTs are the fast algorithms for computing the DFT and the associated con- volution, the Fourier analysis and digital ltering of sampled signals in Part I of the book are based solely on the DFTs, and Part II of the book is devoted to covering the FFTs exclusively. While Part II of this book is self-contained, the material in Chapters 11 through 13 is more advanced than the previous book: Eleanor Chu and Alan George, Inside the FFT Black Box: Serial and Parallel Fast Fourier Transform Algorithms, CRC Press, 2000. • In Chapter 11 the many ways to organize the mixed-radix DFT computation through index mapping are explored. This approach allows one to study the large family of mixed- radix FFT algorithms in a systematic manner, including the radix-2 special case. While this chapter can be read on its own, it also paves the way for the more specialized prime factor FFT algorithms covered in Chapter 13. • In Chapter 12 a connection is established between the multi-factor mixed-radix FFT algorithms and the Kronecker product factorization of the DFT matrix. This process results in a sparse matrix formulation of the mixed-radix FFT algorithm. • In Chapter 13 the family of prime factor FFT algorithms is presented. To cover the mathematical theory behind the prime factor algorithm, the relevant concepts from elementary number theory concerning the properties of integers are introduced, and the Chinese Remainder Theorem (CRT) is proved, because CRT and CRT-related index maps are responsible for the number-theoretic splitting of the DFT matrix, which gives rise to the prime factor algorithm.
- 24. PREFACE xix • Chapter 14 provides full details of the mathematics behind Bluestein s FFT, which is a (deceptively simple) fast algorithm for computing the DFT of arbitrary length and is partic- ularly useful when the length is a large prime number. The MATLAB R implementation of Bluestein s FFT is given, and numerical and timing results are reported.
- 26. Acknowledgments My interest in the subject area of this book has arisen out of my research activities conducted at the University of Guelph, and I thank the Natural Sciences and Engineering Research Council of Canada for continued research grant support. Writing a book of this scope demands one s dedication to research and commitment of time and effort over multiple years, and I thank my husband, Robert Hiscott, for his understanding, consistent encouragement, and unwavering support at all fronts. I thank the reviewers of my book proposal and draft manuscript for their helpful sugges- tions and insightful comments, which led to many improvements. I extend my sincere thanks and appreciation to Robert Stern (Executive Editor) and his staff at Chapman Hall/CRC Press for their ongoing enthusiastic support of my writing projects. Eleanor Chu Guelph, Ontario xxi
- 28. About the Author Eleanor Chu, Ph.D., received her B.Sc. from National Taiwan University in 1973, her B.Sc. and M.Sc. from Acadia University, Canada, in 1980 and 1981, respectively, and her M.Math and Ph.D. in Computer Science from the University of Waterloo, Canada, in 1984 and 1988, respectively. From 1988 to 1991 Dr. Chu was a research assistant professor of computer science at the University of Waterloo. In 1991 she joined the faculty at the University of Guelph, where she has been Professor of Mathematics since 2001. Dr. Chu is the principal author of the book Inside the FFT Black Box: Serial and Parallel Fast Fourier Transform Algorithms (CRC Press, 2000). She has published journal articles in the broad area of computational mathematics, including scienti c computing, matrix analysis and applications, parallel computing, linear algebra and its applications, supercomputing, and high-performance computing applications. xxiii
- 30. Part I Fundamentals, Analysis and Applications 1
- 32. Chapter 1 Analytical and Graphical Representation of Function Contents Our objective in this chapter is to introduce the fundamental concepts and graphical tools for analyzing time-domain and frequency-domain function contents. Our initial discussion will be restricted to linear combinations of explicitly given sine and cosine functions, and we will show how the various representations of their frequency contents are connected to the Fourier series representation of periodic functions in general. 1.1 Time and Frequency Contents of a Function Let us consider a familiar trigonometric function x(t) = 5 cos(2πt). By plotting x(t) versus t over the interval 0 ≤ t ≤ 4, one obtains the following diagram. Figure 1.1 A time-domain plot of x(t) = 5 cos(2πt) versus t. 0 0.5 1 1.5 2 2.5 3 3.5 4 −10 −8 −6 −4 −2 0 2 4 6 8 10 Time Variable t Function x(t) The graph is the time-domain representation of x(t). We observe that when t varies from 0 to 1, the angle θ = 2πt goes from 0 radians to 2π radians, and the cosine function completes 3
- 33. 4 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS one cycle. The same cycle repeats for each following time intervals: t ∈ [1, 2], t ∈ [2, 3], and so on. The time it takes for a periodic function x(t) to complete one cycle is called the period, and it is denoted by T . In this case, we have T = 1 unit of time (appropriate units may be used to suit the application in hand), and x(t + T ) = x(t) for t ≥ 0. While the function x(t) is fully speci ed in its analytical form, the graph of x(t) reveals how the numerical function values change with time. Since a graph is plotted from a table of pre-computed function values, the cont ents of the graph are the numbers in the table. However, compared to reading a large table of data, reading the graph is a much more conve- nient and effective way to s ee the trend or pattern represented by the data, the approximate locations of minimum, maximum, or zero function values. With this understanding, the time- domain (or time) content of x(t) (in this simple case) is the graph which plots x(t) versus t. For a single sinusoidal function like x(t) = 5 cos(2πt), one can easily tell from its time- domain graph that it goes through one cycle (or 2π radians) per unit time, so its frequency is f = 1. It is also apparent from the same graph that the amplitude of x(t) = 5 cos(2πt) is A = 5. However, strictly for our future needs, let us formally represent the frequency-domain (or frequency) content of x(t) in Figure 1.1 by a two-tuple (f, A) = (1, 5) in the amplitude- versus-frequency stem plot given below. The usefulness of the frequency-domain plot will be apparent in the next section. Figure 1.2 A frequency-domain plot of x(t) = 5 cos(2πt). 0 0.5 1 1.5 2 2.5 3 3.5 4 −10 −8 −6 −4 −2 0 2 4 6 8 10 Frequency: f cycles per unit time Amplitude f = 1 A = 5 1.2 The Frequency-Domain Plots as Graphical Tools We next consider a function synthesized from a linear combination of several cosine functions each with a different amplitude as well as a different frequency. For example, let x(t) = x1(t) + x2(t) + x3(t) = A1 cos(2πf1t) − A2 cos(2πf2t) + A3 cos(2πf3t) = 5 cos(2πt) − 7 cos(4πt) + 11.5 cos(6πt). We see that the rst component function x1(t) = 5 cos(2πt) can be written as x1(t) = A1 cos(2πf1t) with amplitude A1 = 5, and frequency f1 = 1. Similarly, the second compo- nent function x2(t) = −7 cos(4πt) can be written as x2(t) = A2 cos(2πf2t) with amplitude
- 34. 1.2. THE FREQUENCY-DOMAIN PLOTS AS GRAPHICAL TOOLS 5 A2 = −7 and frequency f2 = 2. For x3(t) = 11.5 cos(6πt), we have A3 = 11.5 and f3 = 3. The function x1(t) was fully explained in the last section. In the case of x2(t), the cosine function completes one cycle when its angle θ = 4πt goes from 0 radians to 2π radians, which implies that t changes from 0 to 0.5 units. So the period of x2(t) is T2 = 0.5 units, and its frequency is f2 = 1 T2 = 2 cycles per unit time. The expression in the form xk(t) = Ak cos(2πfkt) thus explicitly indicates that xk(t) repeats fk cycles per unit time. Now, we can see that the time unit used to express fk will be canceled out when fk is multiplied by t units of time. Therefore, θ = 2πfkt remains dimension-less, and the same holds regardless of whether the time is measured in seconds, minutes, hours, days, months, or years. Note that the equivalent expression xk(t) = Ak cos(ωkt) is also commonly used, where ωk ≡ 2πfk radians per unit time is called theangular frequency. In the time domain, a graph of the composite x(t) can be obtained by adding the three graphs representing x1(t), x2(t), and x3(t) as shown below. The time-domain plot of x(t) reveals a periodic composite function with a common period T = 1: the graph of x(t) for t ∈ [0, 1] is seen to repeat four times in Figure 1.3. Figure 1.3 Time-domain plots of x(t) and its components. 0 2 4 −20 −10 0 10 20 0 2 4 −20 −10 0 10 20 0 2 4 −20 −10 0 10 20 0 0.5 1 1.5 2 2.5 3 3.5 4 −20 −10 0 10 20 x1 (t) = 5cos(2πt) x2 (t) = −7cos(4πt) x 3 (t) = 11.5cos(6πt) x(t) = 5cos(2πt) − 7cos(4πt) + 11.5cos(6πt) In the frequency domain, suppose that the two-tuple (fk, Ak) represents the frequency content of xk(t), the collection {(f1, A1), (f2, A2), (f3, A3)} de nes the frequency content of x(t) = x1(t) + x2(t) + x3(t). Note that when x(t) is composite, we speak of the individual frequencies and amplitudes of its components and they collectively represent the frequency
- 35. 6 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS content of x(t). The frequency plot of x(t) is obtained by superimposing the three component stem plots as shown in Figure 1.4. Figure 1.4 The time and frequency-domain plots of composite x(t). 0 0.5 1 1.5 2 2.5 3 3.5 4 −20 −10 0 10 20 x(t) 0 1 2 3 4 −10 −5 0 5 10 15 Amplitude Time−Domain Plot Frequency−Domain Plot A1 = 5 A 2 = −7 A 3 = 11.5 Time variable t Frequency f = 1, 2, 3 cycles per unit time Now, with the time-domain plot and the frequency-domain plot of x(t) both available, we see that when x(t) is composite, the frequency content of x(t) can no longer be deciphered from the time-domain plot of x(t) versus t one cannot visually decompose the graph of x(t) into its component graphs. The reverse is also true: the time-domain plot shows the behavior of x(t), which cannot be inferred from the frequency plot alone. Therefore, the time-domain and the frequency-domain plots are both needed, and they carry different but complementary information about the function x(t). 1.3 Identifying the Cosine and Sine Modes In general, a function may have both sine and cosine components, and the two modes must be explicitly identi ed in expressing the frequency content. For the previous example, the function x(t) = n=3 k=1 Ak cos 2πfkt has three cosine components, so each two-tuple in its frequency content {(f1, A1), (f2, A2), (f3, A3)} implicitly represents the amplitude and the frequency of a pure cosine mode, and they are shown together in a single frequency plot. However, the function y(t) below consists of two cosine and three sine components, y(t) = 5.3 cos(4πt) − 3.2 sin(6πt) − 2.5 cos(14πt) − 2.1 sin(4πt) + 9.5 sin(8πt), so the subset of two-tuples {(2, 5.3), (7, −2.5)} and its stem plot represent its pure cosine mode, whereas the other subset of two-tuples{(2, −2.1), (3, −3.2), (4, 9.5)} and a separate stem plot represent its pure sine mode. When we allow zero amplitude and use the same
- 36. 1.4. USING COMPLEX EXPONENTIAL MODES 7 range of frequencies in both modes, we obtain the following expression: (1.1) y(t) = n k=1 Ak cos(2πfkt) + Bk sin(2πfkt). The frequency content of y(t) can now be conveniently represented by a set of three-tuples { (f1, A1, B1), (f2, A2, B2), . . . , (fn, An, Bn) }, with the understanding that Ak is the ampli- tude of a pure cosine mode at frequency fk, and Bk is the amplitude of a pure sine mode at fk. We still need two separate stem plots: one plots Ak versus fk, and the other one plots Bk versus fk. The time-domain and frequency-domain plots of the sum of eleven cosine and eleven sine component functions are shown in Figure 1.5, where for 1 ≤ k ≤ 11, fk = k, with amplitudes 0 Ak ≤ 2 and 0 Bk ≤ 3 randomly generated. The time-domain plot of x(t) again reveals a periodic composite function with a common period T = 1; the graph of x(t) for t ∈ [0, 1] is seen to repeat four times in Figure 1.5. Figure 1.5 An example: the sum of 11 cosine and 11 sine components. 0 0.5 1 1.5 2 2.5 3 3.5 4 −10 0 10 20 0 1 2 3 4 5 6 7 8 9 10 11 12 −1 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 12 −1 0 1 2 3 Pure cosine modes Pure sine modes 1.4 Using Complex Exponential Modes By using complex arithmetics, Euler s formulaejθ = cos θ + j sin θ, where j ≡ √ −1, and the resulting identities cos θ = ejθ + e−jθ 2 , sin θ = ejθ − e−jθ 2j ,
- 37. 8 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS we can express y(t) in terms of complex exponential modes as shown below. y(t) = n k=1 Ak cos(2πfkt) + Bk sin(2πfkt) (1.2) = n k=1 Ak ej2πfkt + e−j2πfkt 2 + Bk ej2πfkt − e−j2πfkt 2j = n k=1 Ak − jBk 2 ej2πfkt + Ak + jBk 2 e−j2πfkt = n k=1 Xkej2πfkt + X−kej2πf−kt , Note: X±k ≡ Ak ∓ jBk 2 , f−k ≡ −fk = X0 + n k=1 Xkej2πfkt + X−kej2πf−kt , (Note: the term X0 ≡ 0 is added) = n k=−n Xkej2πfkt . When the complex number X±k is expressed in rectangular coordinates as (Re (X±k) , Im (X±k)), the frequency contents of y(t) are commonly expressed by two sets of two-tuples: (f±k, Re(X±k)) and (f±k, Im(X±k)). The example in Figure 1.5 is shown again in Figure 1.6 using the exponential mode. When comparing the two gures, note that Re(X±k) = Ak/2 and Im(X±k) = ∓Bk/2. Figure 1.6 Time plot and complex exponential-mode frequency plots. 0 0.5 1 1.5 2 2.5 3 3.5 4 −10 0 10 20 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 −2 −1 0 1 2 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 −2 −1 0 1 2 Amplitude Re(X k ) and Re(X −k ) Amplitude Im(X k ) and Im(X −k )
- 38. 1.5. USING COSINE MODES WITH PHASE OR TIME SHIFTS 9 Note that in order to simplify the terms in the summation, we have added the term X0 ≡ 0, and for 1 ≤ k ≤ n, we have de ned (1.3) Xk = Ak − jBk 2 , X−k = Ak + jBk 2 , f−k = −fk, and ω−k = −ωk = −2πfk. In the present context, since the negative frequencies are simply the consequence of applying trigonometric identities in our derivation of an alternative mathematical formula, they do not change the original problem. For example, if one uses the identity cos(θk) = cos(−θk), when θk = 2πfkt, −θk = 2π(−fk)t occurs, and it causes the presence of negative frequency −fk. (Note that a nonzero X0 = X0ej2πf0t term at f0 = 0 models a DC (direct current) term in electrical circuit applications.) Alternatively we may express the complex amplitude X±k using polar coordinates, namely, Xk = |Xk|ejφk = |Xk| (cos φk + j sin φk) , X−k = |X−k|ejφ−k = |X−k| (cos φ−k + j sin φ−k) , (1.4) where |X±k| = A2 k + B2 k 2 , with each φ±k chosen to satisfy both cos φ±k = Ak A2 k + B2 k , sin φ±k = ∓Bk A2 k + B2 k . Note that each angle φ±k is unique in the quadrant determined by the rectangular coordinates (Ak, ∓Bk) of the complex number 2Xk. In Figure 1.7, the frequency plots show |X±k| and φ±k versus f±k. In the next section we show that φ±k may also be interpreted as the phase shift angle. 1.5 Using Cosine Modes with Phase or Time Shifts Instead of separating the pure cosine and pure sine modes, we may use a pure cosine mode combined with phase shift angles, which is represented by a single set of three-tuples (fk, φ̂k, Dk) as de ned below. y(t) = n k=1 Ak cos(2πfkt) + Bk sin(2πfkt) = n k=1 A2 k + B2 k Ak A2 k + B2 k cos(2πfkt) + Bk A2 k + B2 k sin(2πfkt) = n k=1 Dk cos φ̂k cos(2πfkt) + sin ˆ φk sin(2πfkt) = n k=1 Dk cos(2πfkt − φ̂k), (1.5) where Dk ≡ A2 k + B2 k, with φ̂k satisfying both cos φ̂k = Ak Dk and sin φ̂k = Bk Dk .
- 39. 10 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS Figure 1.7 Time plot and complex exponential-mode frequency plots. 0 0.5 1 1.5 2 2.5 3 3.5 4 −10 0 10 20 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 −1 0 1 2 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 −2 −1 0 1 2 Phase φ k and φ −k (in radians) Magnitude |Xk | and |X−k | Therefore, each component function yk(t) may always be interpreted as a pure cosine mode shifted by a phase angle of φ̂k radians. The phase shifts may be interpreted as t ime shifts by rewriting Equation (1.5) as y(t) = n k=1 Dk cos(2πfkt − φ̂k) = n k=1 Dk cos 2πfk t − φ̂k 2πfk = n k=1 Dk cos 2π 1 Tk t − φ̂k 2π 1 Tk . ∵ fk ≡ 1 Tk (1.6) When it is known that the fundamental frequency f1 = 1 T and that fk = kf1 = k T for 1 ≤ k ≤ n, Equation (1.6) is commonly presented with time shifts tk de ned below. (1.7) y(t) = n k=1 Dk cos 2π k T (t − tk) , where tk ≡ φ̂k 2π k T . Since 2|Xk| = A2 k + B2 k, which is equal to |Dk| in Equations (1.6) and (1.7), we imme-
- 40. 1.5. USING COSINE MODES WITH PHASE OR TIME SHIFTS 11 diately obtain the following relationship. y(t) = n k=1 Xkej2πfk t + X−j2πfkt −k , where X±k = Ak ∓ jBk 2 , j ≡ √ −1, = n k=1 2|Xk| cos(2πfkt − φ̂k) = n k=1 2|Xk| cos 2π k T (t − tk) , if fk = k T , and tk ≡ φ̂k 2π k T . (1.8) Remark 1 In the literature any function of the form (1.9) f(t) = Dk sin(2πfkt + φk), where Dk, fk and φk are real constants, is said to be sinusoidal. Using the trigonometric identity cos θ − 1 2 π = cos θ cos 1 2 π + sin θ sin 1 2 π = sin θ with θ = 2πfkt + φk, we can also express (1.9) as a cosine function: f(t) = Dk sin(2πfkt + φk) = Dk cos 2πfkt + φk − 1 2 π . Hence, a sinusoidal function can be written in two forms which differ by 1 2 π in the phase angle: (1.10) Dk sin(2πfkt + φk) = Dk cos(2πfkt + φ̂k), where φ̂k = φk − 1 2 π. In particular, both sin(2πfkt) and cos(2πfkt) are sinusoidal functions by this de nition. Remark 2 Any component function of the form (1.11) gk(t) = Ak sin(2πfkt) + Bk cos(2πfkt) is said to be a sinusoidal component, because we have shown at the beginning of this section that it can be expressed as gk(t) = Dk cos 2πfkt − φ̂k , with Dk and φ̂k determined by Ak and Bk. Remark 3 The easiest way to add two or more sinusoidal functions of the same frequency is provided by form (1.11). For example, given f(t) = 5 sin(1.2t)+2 cos(1.2t) and g(t) = sin(1.2t) + cos(1.2t), we obtain the sum by adding the corresponding coef cients: h(t) = f(t) + g(t) = 6 sin(1.2t) + 3 cos(1.2t). Therefore, the sum of two or more sinusoidal functions of frequency fk is again a sinu- soidal function of frequency fk. Remark 4 Be aware that sinusoidal functions may be given in disguised forms: e.g., f(t) = sin(1.1t) cos(1.1t) is the disguised form of the sinusoidal f(t) = 1 2 sin(2.2t); g(t) = 1 − 2 sin2 t is the disguised form of the sinusoidal g(t) = cos 2t.
- 41. 12 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS 1.6 Periodicity and Commensurate Frequencies Recall that when we present the frequency-domain plots for speci c examples of y(t) = n k=1 Ak cos(2πfkt) + Bk sin(2πfkt), we have let fk = k cycles per unit time, and we plot the amplitudes Ak and Bk versus k. In such examples we automatically have uniform spacing with f = fk+1 − fk = 1, and we have fk = kf1 with f1 = 1 being the fundamental frequency. Since the time period T of composite y(t) is the shortest duration over which each sine or cosine component completes an integer number of cycles, we determine T by the LCM (least common multiple) of the individual periods. From fk = kf1 and Tk = 1/fk, we obtain T1 = kTk, so T1 is the LCM of the individual periods. Accordingly, the time period T of the composite y(t) is the reciprocal of the fundamental frequency f1. Note that f1 is the GCD (greatest common divisor) of the individual frequencies. In general, fk = k, and we need to distinguish periodic y(t) from non-periodic y(t) by examining its frequency contents. The conditions and results are given below. 1. The function y(t) is said to be a commensurate sum if the ratio of any two individual periods (or frequencies) is a rational fraction ratio of integers with common factors canceled out. Example 1.1 The function y(t) = 4.5 cos(2πfαt) + 7.2 cos(2πfβt) = 4.5 cos(1.2πt) + 7.2 cos(1.8πt) is a commensurate sum, because fα = 0.6 Hz, fβ = 0.9 Hz, and the ratio fα/fβ = 2/3 is a rational fraction. 2. A commensurate y(t) is periodic with its fundamental frequency being the GCD of the individual frequencies and its common period being the LCM of the individual periods. Example 1.2 We continue with Example 1.1: the fundamental frequency of the function y(t) = 4.5 cos(1.2πt) + 7.2 cos(1.8πt) is fo = GCD(0.6, 0.9) = 0.3 Hz; and the fundamental period is To = 1/fo = 31 3 seconds. We get the same result from To = LCM 1 0.6 , 1 0.9 = LCM 5 3 , 10 9 = 31 3 . It can be easily veri ed that y(t + To) = y(t). Example 1.3 When fk = k/T , the fundamental frequency is f1 = 1/T , and the com- posite function y(t) = n k=1 Ak cos 2πkt T + Bk sin 2πkt T is commensurate and periodic with common period T , i.e., y(t + T ) = y(t). Since we have uniform spacing f = fk+1 − fk = 1/T , we may still plot Ak and Bk versus k with the understanding that k is the index of equispaced fk; of course, one may plot Ak and Bk versus the values of fk if that is desired. (Note that fk = k/T = k if T = 1.)
- 42. 1.7. REVIEW OF RESULTS AND TECHNIQUES 13 3. A non-commensurate y(t) is not periodic, although all its components are periodic. For example, the function y(t) = sin(2πt) + 5 sin(2 √ 3πt) is not periodic because f1 = 1 and f2 = √ 3 are not commensurate. 1.7 Review of Results and Techniques In the preceding sections we show that a sum of sinusoidal modes can be expressed in a num- ber of ways. While the various formulas are mathematically equivalent, one form could be more convenient than another depending on the manipulations required for a particular appli- cation. Also, it is not uncommon that while one form is more suitable for describing a physical problem, another form is more desirable for a computational purpose. These formulas are summarized below. Form 1 Using pure cosine and sine modes (1.12) y(t) = n k=1 Ak cos(2πfkt) + Bk sin(2πfkt). If the angular frequency ωk = 2πfk is used, we obtain (1.13) y(t) = n k=1 Ak cos(ωkt) + Bk sin(ωkt). A common case: when y(t) = y(t+T ) with fk = k/T , this fact is explicitly recognized by expressing (1.14) y(t) = n k=1 Ak cos 2πkt T + Bk sin 2πkt T . Form 2 Using complex exponential modes (1.15) y(t) = n k=−n Xkej2πfkt . Form 3 Using cosine modes with phase shifts (1.16) y(t) = n k=1 Dk cos(2πfkt − φ̂k). Form 4 Using cosine modes with time shifts (1.17) y(t) = n k=1 Dk cos 2πfk (t − tk) . Form 5 Using complex exponential modes with phases (1.18) y(t) = n k=−n |Xk|ejφk ej2πfkt .
- 43. 14 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS A reminder: The de nitions fk = 1 Tk and ωk = 2πfk may be used to express y(t) in terms of Tk (individual period) or ωk (individual angular frequency)in all forms. Also, when fk = k/T , this fact is commonly recognized wherever fk is used. To convert one form to another, one may use the relationship between the coef cients as summarized below. Relation 1 De n e X0 ≡ 0 when A0 and B0 are missing. For 1 ≤ k ≤ n, X±k = Ak ∓ jBk 2 , and f−k = −fk. Relation 2 |X±k| = A2 k + B2 k 2 , and the phase angle φ±k satis es both cos φ±k = Ak A2 k + B2 k and sin φ±k = ∓Bk A2 k + B2 k . A reminder: φk is unique in the quadrant determined by the rectangular coordinates (Ak, −Bk) of the complex number 2Xk; φ−k is unique in the quadrant determined by the rectangular coordinates (Ak, Bk) of the complex number 2X−k. Relation 3 For 1 ≤ k ≤ n, Dk = A2 k + B2 k = 2|X±k|, tk = φ̂k 2πfk , where φ̂k = φ−k. Relation 4 For 1 ≤ k ≤ n, Ak = Xk + X−k = 2 Re(Xk); Bk = j(Xk − X−k) = −2 Im(Xk). We also identify the mathematical techniques used in deriving the various results in this section: Technique 1 Euler s identity in three forms: ejθ = cos θ + j sin θ, cos θ = ejθ + e−jθ 2 , and sin θ = ejθ − e−jθ 2j . Examples of future use: • Prove n k=−n ejkθ = sin n + 1 2 θ sin θ 2 . (Chapter 3, Section 3.10.2, page 84) • Prove π −π n k=−n ejkθ dθ = 2π. (Chapter 3, Section 3.10.2, page 85) • Prove π −π sin n + 1 2 θ sin θ 2 dθ = 2π. (Chapter 3, Section 3.10.2, page 85) • Prove 1 2fc fc −fc ej2πft df = sin 2πfct 2πfct . (Chapter 5, Example 5.4, page 171)
- 44. 1.7. REVIEW OF RESULTS AND TECHNIQUES 15 Technique 2 Trigonometric identities and their alternate forms: cos(α ± β) = cos α cos β ∓ sin α sin β, sin(α ± β) = sin α cos β ± cos α sin β, cos α cos β = cos(α + β) + cos(α − β) 2 , sin α cos β = sin(α + β) + sin(α − β) 2 , sin α sin β = cos(α − β) − cos(α + β) 2 , cos α sin β = sin(α + β) − sin(α − β) 2 . Examples of future use: • Letting α = β, we immediately have the useful identities cos 2α = cos2 α − sin2 α, sin 2α = 2 sin α cos α; cos2 α = 1 + cos 2α 2 , sin2 α = 1 − cos 2α 2 . • Letting α = mθ and β = nθ, it is straightforward to apply the identities given above to prove the following results for future use. π −π cos mθ cos nθ dθ = 0, if m = n; π, if m = n = 0; 2π, if m = n = 0. (1.19) π −π sin mθ sin nθ dθ = 0, if m = n; π, if m = n = 0; 0, if m = n = 0. (1.20) π −π cos mθ sin nθ dθ = 0. (1.21) 1.7.1 Practicing the techniques To practice the techniques in nontrivial settings, we show how to manipulate some trigonomet- ric series encountered in Fourier analysis in the examples that follow. Example 1.4 Derive the following identity: (1.22) n =1 sin(2 − 1)θ = sin2 nθ sin θ , and show that this identity is valid at θ = 0 by the limit convention. (When this convention is used, the value of a function at a point where a denominator vanishes is understood to be the
- 45. 16 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS limit, provided this limit is n ite.) ∵ sin θ n =1 sin(2 − 1)θ = sin2 θ + sin θ sin 3θ + sin θ sin 5θ + · · · + sin θ sin(2n − 1)θ = 1 − cos 2θ 2 + cos 2θ − cos4θ 2 + cos 4θ − cos 6θ 2 + · · · + cos(2n − 2)θ − cos(2n)θ 2 = 1 2 − cos 2θ 2 + cos 2θ 2 − cos 4θ 2 + · · · − cos(2n − 2)θ 2 + cos(2n − 2)θ 2 − cos(2n)θ 2 = 1 − cos 2nθ 2 (only the rst term and the last term remain) = sin2 nθ. recall sin2 α = 1 2 (1 − cos 2α) ∴ n =1 sin(2 − 1)θ = sin2 nθ sin θ . When θ = 0, since the right side is in the indeterminate form 0/0, we apply L Hospital s rule to determine the limit: lim θ→0 sin2 nθ sin θ = lim θ→0 2n sin nθ cos nθ cos θ = 2n sin 0 = 0. Hence the two sides are equal at θ = 0 by the limit convention. Example 1.5 Using Euler s identityejθ = cos θ + j sin θ, the nite sum of a geometric series in z = ejθ = 1, i.e., (1.23) n =0 z = 1 − zn+1 1 − z , and the complex arithmetic identity (1.24) c + jd a + jb = (c + jd)(a − jb) (a + jb)(a − jb) = ac + bd a2 + b2 + j ad − bc a2 + b2 , determine the closed-form sums of the following cosine and sine series: (1.25a) n =0 cos θ = 1 + cos θ + · · · + cos nθ =? (1.25b) n =1 sin θ = sin θ + sin 2θ + · · · + sin nθ =? By letting z = ejθ in the left side of (1.23), we identify the cosine series (1.25a) and the sine series (1.25b) as the real and imaginary parts: n =0 z = n =0 ejθ = n =0 cos θ + j sin θ = n =0 cos θ + j n =1 sin θ . (∵ sin 0 = 0) By letting z = ejθ in the right side of (1.23), we express (1.26) 1 − zn+1 1 − z = 1 − ej(n+1)θ 1 − ejθ = 1 − cos(n + 1)θ − j sin(n + 1)θ (1 − cos θ) − j sin θ = U + jV.
- 46. 1.7. REVIEW OF RESULTS AND TECHNIQUES 17 Accordingly, the real part U represents the cosine series, and the imaginary part V represents the sine series. To express U and V in (1.26), we use identity (1.24) with c = 1 − cos(n + 1)θ, d = − sin(n + 1)θ, a = 1 − cos θ, and b = − sin θ: (1.27) U = 1 − cos(n + 1)θ (1 − cos θ) + sin(n + 1)θ sin θ (1 − cos θ)2 + sin2 θ = 1 − cos(n + 1)θ − cos θ + cos(n + 1)θ cos θ + sin(n + 1)θ sin θ 1 − 2 cosθ + cos2 θ + sin2 θ = 1 − cos(n + 1)θ − cos θ + cos (n + 1)θ − θ 1 − 2 cosθ + 1 = 1 − cos θ + cos nθ − cos(n + 1)θ 2 − 2 cosθ ; (1.28) V = −(1 − cos θ) sin(n + 1)θ + 1 − cos(n + 1)θ sin θ (1 − cos θ)2 + sin2 θ = sin(n + 1)θ cos θ − cos(n + 1)θ sin θ − sin(n + 1)θ + sin θ 1 − 2 cosθ + cos2 θ + sin2 θ = sin (n + 1)θ − θ − sin(n + 1)θ + sin θ 1 − 2 cosθ + 1 = sin θ + sin nθ − sin(n + 1)θ 2 − 2 cos θ . We have thus obtained (1.29) n =0 cos θ = 1 − cos θ + cos nθ − cos(n + 1)θ 2 − 2 cos θ ; (1.30) n =1 sin θ = sin θ + sin nθ − sin(n + 1)θ 2 − 2 cos θ . Example 1.6 Derive the trigonometric identity (1.31) 1 2 + n =1 cos θ = sin n + 1 2 θ 2 sin 1 2 θ , and show that it is valid at θ = 0 by the limit convention. Beginning with the identity (1.29), we obtain 1 2 + n =1 cos θ = 1 − cos θ + cos nθ − cos(n + 1)θ 2 − 2 cosθ − 1 2 = 2 sin2 1 2 θ + cos n + 1 2 θ − 1 2 θ − cos n + 1 2 θ + 1 2 θ 4 sin2 1 2 θ − 1 2 = 2 sin2 1 2 θ + 2 sin n + 1 2 θ sin 1 2 θ 4 sin2 1 2 θ − 1 2 = sin 1 2 θ + sin n + 1 2 θ 2 sin 1 2 θ − 1 2 = sin n + 1 2 θ 2 sin 1 2 θ .
- 47. 18 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS At θ = 0, because cos θ = cos 0 = 1 for 1 ≤ ≤ n in the left side, the sum is n + 1 2 . Here again the right side is in the indeterminate form 0/0, we apply L Hospital s rule to determine the limit: lim θ→0 sin n + 1 2 θ 2 sin 1 2 θ = lim θ→0 n + 1 2 cos n + 1 2 θ cos 1 2 θ = n + 1 2 . Hence the two sides are equal at θ = 0 by the limit convention. Example 1.7 Show that (1.32) n =0 cos (2m + 1)π n + 1 = 1. If we let θ = 2m+1 n+1 π in the geometric series (1.23), the numerator in the right side can be further simpli ed : (1.33) n =0 ejθ = 1 − ej(n+1)θ 1 − ejθ = 2 1 − cos θ − j sin θ (∵ θ = 2m+1 n+1 π ∴ ej(n+1)θ = −1) = 2(1 − cos θ + j sin θ) (1 − cos θ)2 + sin2 θ = 2 − 2 cosθ 1 − 2 cosθ + 1 + j 2 sin θ 1 − 2 cosθ + 1 Recall from Example 1.5 that the real part of the series (1.33) represents the cosine series, we have thus proved the desired result: If θ = (2m + 1)π n + 1 , then n =0 cos θ = 2 − 2 cosθ 2 − 2 cosθ = 1. Example 1.8 Show that, if the nonzero integer m is not a multiple of n + 1, we have (1.34) n =0 cos (2m)π n + 1 = 0. We again let θ = 2m n+1 π in the geometric series (1.23), we have (1.35) n =0 ejθ = 1 − ej(n+1)θ 1 − ejθ = 0 1 − ejθ (∵ θ = 2m n+1 π ∴ ej(n+1)θ = 1) = 0. Example 1.9 Show that the following alternative expressions for the nite sum of the sine series can be obtained from identity (1.30) in Example 1.5. (1.36) n =1 sin θ = cos 1 2 θ − cos n + 1 2 θ 2 sin 1 2 θ ;
- 48. 1.8. EXPRESSING SINGLE COMPONENT SIGNALS 19 (1.37) n =1 sin θ = sin n+1 2 θ sin n 2 θ sin 1 2 θ . To derive the two mathematically equivalent results, we continue from (1.30): n =1 sin θ = sin θ + sin nθ − sin(n + 1)θ 2 − 2 cosθ = 2 sin 1 2 θ cos 1 2 θ + sin n + 1 2 θ − 1 2 θ − sin n + 1 2 θ + 1 2 θ 4 sin2 1 2 θ = 2 sin 1 2 θ cos 1 2 θ − 2 cos n + 1 2 θ sin 1 2 θ 4 sin2 1 2 θ = cos 1 2 θ − cos n + 1 2 θ 2 sin 1 2 θ this is the desired result (1.36) = cos n+1 2 − n 2 θ − cos n+1 2 + n 2 θ 2 sin 1 2 θ = sin n+1 2 θ sin n 2 θ sin 1 2 θ . this is the desired result (1.37) 1.8 Expressing Single Component Signals Since many puzzling phenomena we encounter in analyzing or processing composite signals can be easily investigated through single-mode signals, they are indispensable tools in our continued study of signal sampling and transformations, and it pays to be very familiar (and comfortable) with expressing a single-mode signal in its various forms. Although we can formally put such a signal in one of the standard forms (with a single nonzero coef cient) and apply the full-force conversion formulas, it is much easier to forgo the formalities and work with the given signal directly, as demonstrated by the following examples. Example 1.10 f(t) = cos(2πfat) = cos(80πt) is a 40-Hertz sinusoidal signal, its amplitude is A = 1.0, its period is T = 1/fa = 1/40 = 0.025 seconds, and it has zero phase. We express f(t) in the complex exponential modes by applying Euler s formula directly: f(t) = cos(80πt) = 1 2 ej80πt + e−j80πt = 0.5e−j80πt + 0.5ej80πt . The difference between f(t) given above and g(t) = sin(80πt) lies in the phase angle, because the latter can be rewritten as a shifted cosine wave, namely, g(t) = cos(80πt−π/2). The phase can also be recognized directly from expressing g(t) in the complex exponential modes: g(t) = sin(80πt) = 1 2j ej80πt − e−j80πt = (0.5j)e−j80πt + (−0.5j)ej80πt . = 0.5ejπ/2 e−j80πt + 0.5e−jπ/2 ej80πt . The coef cients ±0.5j each has nonzero imaginary part, which re ects a nonzero phase in the signal. The polar expression ±j = e±jπ/2 reveals the phase explicitly.
- 49. 20 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS Example 1.11 For h(t) = 4 cos(7πt + α), we have h(t) = 4 cos(7πt + α) = 4 2 ej(7πt+α) + e−j(7πt+α) = 2e−jα e−j7πt + 2ejα ej7πt . Observe that when the phase α = 0, π, the coef cients 2e±jα = 2(cos α ± j sin α) have nonzero imaginary part. For u(t) = 4 sin(7πt+β), we may apply Euler s formula directly to the given sine function to obtain u(t) = 4 sin(7πt + β) = 4 2j ej(7πt+β) − e−j(7πt+β) = 2je−jβ e−j7πt + −2jejβ ej7πt = 2e−j(β−π/2) e−j7πt + 2ej(β−π/2) ej7πt . The same expression can also be obtained if we use the result already available for u(t) = 4 cos(7π + α) with α = β − π/2. Example 1.12 For v(t) = 3 cos(15πt) cos(35πt), be aware that it hides two cosine modes. To bring them out, we use the trigonometric identity for cos α cos β (given under Technique 2 in the previous section) to obtain v(t) = 3 cos(15πt) cos(35πt) = 1.5 cos(15 + 35)πt + cos(15 − 35)πt = 1.5 cos 50πt + cos 20πt = 1.5e−j50πt + 1.5e−j20πt + 1.5ej20πt + 1.5ej50πt . The two cosine modes may also be disguised as s(t) = 3 sin(15πt) sin(35πt), and they can again be obtained using the trigonometric identity for sin α sin β (given under Technique 2 in the previous section): s(t) = 3 sin(15πt) sin(35πt) = 1.5 cos(15 − 35)πt − cos(15 + 35)πt = 1.5(cos20πt − cos 50πt = −1.5e−j50πt + 1.5e−j20πt + 1.5ej20πt − 1.5ej50πt . 1.9 General Form of a Sinusoid in Signal Application When a cyclic physical phenomenon is described by a cosine curve, the general form used in many applications is the cosine mode with phase shift angle (or phase in short) (1.38) x(t) = Dα cos(2πfαt − φα), where the amplitude Dα, frequency fα, and phase φα (in radians) provide useful information about the physical problem at hand. For example, suppose that it is justi able to model the variation of monthly precipitation in each appropriately identi ed geographic region by a co- sine curve with period Tα = 1/fα = 12 months, then the amplitude of each tted cosine curve predicts the maximum precipitation for each region, and the phase (converted to time shift) predicts the date of maximum precipitation for each region. Graphically, the time shift tα (computed from the phase φα) is the actual distance between the origin and the crest of the
- 50. 1.9. GENERAL FORM OF A SINUSOID IN SIGNAL APPLICATION 21 cosine curve when the horizontal axis is time, because x(t) = Dα when 2πfαt − φα = 0 is satis ed by t = tα = φα/2πfα. Note that when a negative frequency fα 0 appears in the general form, it is interpreted as the result of phase reversal as shown below. x(t) = Dα cos(2πfαt − φα) = Dα cos(−2π ˆ fαt − φα) (∵ ˆ fα = −fα 0) = Dα cos −(2π ˆ fαt + φα) = Dα cos(2π ˆ fαt + φα) (∵ cos(−θ) = cos θ) = Dα cos 2π ˆ fαt − (−φα) . For example, to obtain the time-domain plot of x(t) = 2.5 cos(−40πt − π/6), we simply plot x(t) = 2.5 cos(40πt − φ) with φ = −π/6 (reversed from π/6) in the usual manner. 1.9.1 Expressing sequences of discrete-time samples When the sinusoid x(t) = Dα cos(2πfαt − φα) is sampled at intervals of t (measured in chosen time units), we obtain the discrete-time sinusoid (1.39) x ≡ x(t) = Dα cos(2πfαt − φα), = 0, 1, 2, . . . Observe that the sequence of discrete-time samples {x0, x1, x2, . . . } can also be represented by the three-tuple {fαt, φα, Dα}, where the product of the analog frequency fα (cycles per unit time) and the sampling interval t (elapsed time between consecutive samples) de nes the digital (or discrete) frequency Fα ≡ fαt (cycles per sample). Therefore, a discrete-time sinusoid has the general form (1.40) x = Dα cos(2πFα − φα), = 0, 1, 2, . . . Since fα = Fα/t, the digital frequencycan always be convertedback to the analog frequency as desired. Furthermore, because Fα ≡ fαt = 1 m fα mt = mfα 1 m t , an m-fold increase (or decrease) in t amounts to an m-fold decrease (or increase) in the analog frequency, i.e., fβ = Fα mt = 1 m Fα t ; fγ = Fα 1 m t = m Fα t . Consequently, by simply adjusting t at the time of output, the same set of digital samples may be converted to analog signals with different frequencies. This will provide further e xibility in the sampling and processing of signals. Corresponding to the (analog) angular frequency ωα = 2πfα (radians per second), we have the digital (or discrete) angular frequency Wα = 2πFα (radians per sample); hence, we may also express the two general forms as (1.41) x(t) = Dα cos(ωαt − φα),
- 51. 22 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS and (1.42) x = Dα cos(Wα − φα), = 0, 1, 2, . . . 1.9.2 Periodicity of sinusoidal sequences While the period of the sinusoid x(t) = Dα cos(2πfαt − φα) is always T = 1/fα, we cannot say the same for its sampled sequence for two reasons: 1. The discrete-time sample sequence may or may not be periodic depending on the sam- pling interval t; 2. If the discrete-time sample sequence is periodic, its period varies with the sampling interval t. To nd out whether a discrete-time sinusoid is periodic and to determine the period (measured by the number of samples), we make use of the mathematical expression for the th sample, namely, x = Dα cos(2πFα − φα), = 0, 1, 2, . . ., and we recall that Fα = fαt. We now relate the discrete-time samples represented by the sequence {x} to the period of its envelope function x(t) = Dα cos(2πfαt − φα) through the digital frequency Fα: 1. If we can express Fα = fαt = K N , where K and N are integers (with no common factor), then we have xN = Dα cos(2πK − φα), and xN is positioned exactly at the point where its envelope function x(t) completes K cycles, and we may conclude that the discrete-time sample sequence {x} is periodic with period T = N samples m eaning that x+N = x for 0 ≤ ≤ N − 1, and xN = x0 is the rst sample of the next period. 2. The sequence {x} is not periodic if we cannot express its digital frequency F as a rational fraction. We demonstrate the different cases by several examples below. Example 1.13 The discrete-time sinusoid x = cos(0.025π − π/6) can be written as x = cos(2πFα − π/6) with Fα = 0.025/2 = 0.0125 = 1/80, so the given sequence is periodic with period N = 80 (samples). In this case, we have K = 1, so the N samples are equally spaced over a single period of its envelope function.
- 52. 1.10. FOURIER SERIES: A TOPIC TO COME 23 Example 1.14 The discrete-time sinusoid g = cos(0.7π + π/8) can be written as g = cos(2πFα + π/8) with Fα = 0.7/2 = 0.35 = 7/20, so the given sequence is periodic with period N = 20 (samples). In this case, we have K = 7, so the N equispaced samples span seven periods of its envelope function. Example 1.15 The discrete-time sinusoid y = cos( √ 3π) is not periodic, because when we express y = cos(2πFβ), we have Fβ = √ 3/2, which is not a rational fraction. Example 1.16 The discrete-time sinusoid z = cos(2 + π/6) is not periodic, because when we express z = cos(2πFγ + π/6), we have Fγ = 1/π, which is not a rational fraction. Sampling and reconstruction of signals will be formally treated in Chapters 2, 5 and 6. 1.10 Fourier Series: A Topic to Come In this chapter we limit our discussion to functions consisting of explicitly given sines and cosines, because their frequency contents are precisely de ned and easy to understand. To extend the de n itions and results to an arbitrary function f(t), we must seek to represent f(t) as a sum of sinusoidal modes this process is called Spectral Decomposition or Spectral Analysis. The Fourier series refers to such a representation with frequencies speci ed at fk = k/T cycles per unit time for k = 0, 1, 2, . . ., ∞. The unknowns to be determined are the amplitudes (or coef cients) Ak and Bk so that (1.43) f(t) = ∞ k=0 Ak cos 2πkt T + Bk sin 2πkt T . If we are successful, the Fourier series of f(t) is given by the commensurate sum in the right- hand side, and we have f(t+T ) = f(t). That is, T is the common period of f(t) and f1 = 1/T is the fundamental frequency of f(t). Note that f(t) completes one cycle over any interval of length T , including the commonly used [−T/2, T/2]. Depending on the application context, the Fourier series of function f(t) may appear in variants of the following forms: 1. Using pure cosine and sine modes with variable t, (1.44) f(t) = A0 2 + ∞ k=1 Ak cos 2πkt T + Bk sin 2πkt T . Note that f(t) has a nonzero DC term, namely, A0/2, for which we have the following remarks:
- 53. 24 CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS Remark 1. For k = 0, we have cos 0 = 1 and sin 0 = 0; hence, the constant term in (1.43) is given by (A0 cos 0 + B0 sin 0) = A0. Remark 2. By convention the constant (DC) term in the Fourier series (1.44) is denoted by 1 2 A0 instead of A0 so that one mathematical formula de nes Ak for all k, in- cluding k = 0. The analytical formulas which de ne Ak and Bk will be presented when we study the theory of Fourier series in Chapter 3. A common variant uses T = 2L with spatial variable x, (1.45) f(x) = A0 2 + ∞ k=1 Ak cos πkx L + Bk sin πkx L . Note that f(x + 2L) = f(x), and a commonly chosen interval of length 2L is [−L, L]. 2. Using cosine modes with phase shifts, (1.46) f(t) = D0 + ∞ k=1 Dk cos 2πkt T − φ̂k . The individual terms Dk cos(2πkt T −φ̂k) a re called the harmonics of f(t). Note that the spacing between the harmonic frequencies is f = fk+1 −fk = 1 T . Hence, periodic analog signals are said to have discrete spectra, and the spacing in the frequency domain is the reciprocal of the period in the time domain. 3. Using complex exponential modes with variable t, (1.47) f(t) = ∞ k=−∞ Xkej2πkt/T . Note that X0 = A0/2 (see above). 4. Using pure cosine and sine modes with dimension-less variable θ = 2πt/T radians, (1.48) g(θ) = A0 2 + ∞ k=1 Ak cos kθ + Bk sin kθ. Since t varies from 0 to T , θ = 2πt/T varies from 0 to 2π, we have g(θ + 2π) = g(θ). Note that g(θ) completes one cycle over any interval of length 2π, including the commonly used [−π, π]. 5. Using complex exponential modes with dimension-less variable θ = 2πt/T radians, (1.49) g(θ) = ∞ k=−∞ Xkejkθ . 6. In Chapter 5, we will learn that the frequency contents of a nonperiodic function x(t) are de ned by a continuous-frequency function X(f), and we will also encounter the Fourier series representation of the periodically extended X(f), which appears in the two forms given below. A full derivation of the continuous-frequency function X(f) and its Fourier series (when it exists) will be given in Chapter 5.
- 54. 1.11. TERMINOLOGY 25 Using pure cosine and sine modes with variable f (which represents the continuously varying frequency) and bandwidth F, that is to say, f ∈ [−F/2, F/2], (1.50) X(f) = ∞ k=0 ak cos 2πkf F + bk sin 2πkf F . Using complex exponential modes with variable f and bandwidth F, (1.51) X(f) = ∞ k=−∞ ckej2πkf/F . Instead of using the variable f ∈ [−F/2, F/2], a dimension-less variable θ = 2πf/F ∈ [−π, π] may also be used in the frequency domain. Corresponding to the two forms of X(f) given above, we have G(θ) = ∞ k=0 ak cos kθ + bk sin kθ, (1.52) and G(θ) = ∞ k=−∞ ckejkθ , where θ ∈ [−π, π]. (1.53) Observe that because the Fourier series expression in θ may be used for both time-domain func- tion x(t) and frequency-domain function X(f), the dimension-less variable θ is also known as a neutral variable. Since the Fourier series expression is signi can tly simpli ed by using the neutral variable θ, it is often the variable of choice in mathematical study of Fourier series. The theory and techniques for deriving the Fourier series representation of a given function will be covered in Chapter 3. 1.11 Terminology Analog signals Signals continuous in time and amplitude are called analog signals. Temporal and spatial variables The temporal variable t measures time in chosen units; the spatial variable x measures distance in chosen units. Period and wavelength The period T satis es f(t + T ) = f(t); the wavelength 2L satis es g(x + 2L) = g(x). Frequency and wave number The (rotational) frequency is de ned by 1 T (cycles per unit time); the wave number is de ned by 1 2L (wave numbers per unit length). Sine and cosine modes A pure sine wave with a xed frequency fk is called a sin e mode and it is denoted by sin(2πfkt); similarly, a cosine mode is denoted by cos(2πfkt). Phase or phase shift It refers to the phase angle φ̂k (expressed in radians) in the shifted cosine mode cos(2πfkt − φ̂k) or cos(2πfkx − φ̂k).
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- 56. having the centre corresponding with the centre of gravity. A good billiard-player should, therefore, always try the ball before he engages to play for any large sum. The toy called the tombola reminds us of the egg-experiment, as there is usually a lump of lead inserted in the lower part of the hemisphere, and when the toy is pushed down it rapidly assumes the upright position because the centre of gravity is not in the lowest place to which it can descend; the latter position being only attained when the figure is upright. Fig. 47. No. 1. c. Centre of gravity in the lowest place, figure upright. No. 2. c. Centre of gravity raised as the figure is inclined on either side, but falling again into the lowest place as the figure gradually comes to rest. There is a popular paradox in mechanics—viz., a body having a tendency to fall by its own weight, may be prevented from falling by adding to it a weight on the same side on which it tends to fall, and the paradox is demonstrated by another well-known child's toy as depicted in the next cut.
- 57. Fig. 48. The line of direction falling beyond the base; the bent wire and lead weight throwing the centre of gravity under the table and near the leaden weight; the hind legs become the point of support, and the toy is perfectly balanced. Fig. 49. No. 1. Sword balanced on handle: the arc from c to d is very small, and if the centre, c, falls out of the line of direction it is not easily restored to the upright position. No 2. Sword balanced on the point: the arc from c to d much larger, and therefore the sword is more easily balanced.
- 58. After what has been explained regarding the improvement of the stability of the egg by lowering the situation of the centre of gravity, it may at first appear singular that a stick loaded with a weight at its upper extremity can be balanced perpendicularly with greater ease and precision than when the weight is lower down and nearer the hand; and that a sword can be balanced best when the hilt is uppermost; but this is easily explained when it is understood that with the handle downwards a much smaller arc is described as it falls than when reversed, so that in the former case the balancer has not time to re-adjust the centre, whilst in the latter position the arc described is so large that before the sword falls the centre of gravity may be restored within the line of direction of the base. For the same reason, a child tripping against a stone will fall quickly; whereas, a man can recover himself; this fact can be very nicely shown by fixing two square pieces of mahogany of different lengths, by hinges on a flat base or board, then if the board be pushed rapidly forward and struck against a lead weight or a nail put in the table, the short piece is seen to fall first and the long one afterwards; the difference of time occupied in the fall of each piece of wood (which may be carved to represent the human figure) being clearly denoted by the sounds produced as they strike the board. Fig. 50. No. 1. The two pieces of mahogany, carved to represent a man and a boy, one being 10 and the other 5 inches long, attached to board by hinges at h h.
- 59. Fig. 51. No. 2. The board pushed forward, striking against a nail, when the short piece falls first, and the long one second. Boat-accidents frequently arise in consequence of ignorance on the subject of the centre of gravity, and when persons are alarmed whilst sitting in a boat, they generally rise suddenly, raise the centre of gravity, which falling, by the oscillation of the frail bark, outside the line of direction of the base, cannot be restored, and the boat is upset; if the boat were fixed by the keel, raising the centre of gravity would be of little consequence, but as the boat is perfectly free to move and roll to one side or the other, the elevation of the centre of gravity is fatal, and it operates just as the removal of the lead would do, if changed from the base to the head of the tombola toy. A very striking experiment, exhibiting the danger of rising in a boat, maybe shown by the following model, as depicted at Nos. 1 and 2, figs. 52 and 53.
- 60. Fig. 52. No. 1. Sections of a toy-boat floating in water. b b b. Three brass wires placed at regular distances and screwed into the bottom of the boat, with cuts or slits at the top so that when the leaden bullets, l l l, which are perforated and slide upon them like beads, are raised to the top, they are retained by the brass cuts springing out; when the bullets are at the bottom of the lines they represent persons sitting in a boat, as shown in the lower cuts, and the centre of gravity will be within the vessel. We thus perceive that the stability of a body placed on a base depends upon the position of the line of direction and the height of the centre of gravity. Security results when the line of direction falls within the base. Instability when just at the edge. Incapability of standing when falling without the base. Fig. 53. No. 2. The leaden bullets raised to the top now show the result of persons suddenly rising, when the boat immediately turns over, and either sinks or floats on the surface with the keel upwards. The leaning-tower of Pisa is one hundred and eighty-two feet in height, and is swayed thirteen and a half feet from the perpendicular, but yet remains perfectly firm and secure, as the line of direction falls considerably within the base. If it was of a greater altitude it could no longer stand, because the centre of gravity would be so elevated that the line of direction would fall outside the base. This fact may be illustrated by taking a board several feet in length, and having cut it out to represent the architecture of the leaning-tower of Pisa, it may then be painted in distemper, and fixed at the right angle with a hinge to another board representing the ground, whilst a plumb-line may be dropped from the centre of gravity; and it may be shown that as long as the plummet falls within the base, the tower is safe; but directly the model tower is brought a little further forward by a wedge so that the plummet hangs outside, then, on removing the support, which may be a piece of string to be cut at the right moment, the model falls, and the fact is at once comprehended.
- 61. Fig. 54. f. Board cut and painted to represent the leaning-tower of Pisa. g. The centre of gravity and plummet line suspended from it. h. The hinge which attaches it to the base board. i. The string, sufficiently long to unwind and allow the plummet to hang outside the base, so that, when cut, the model falls in the direction of the arrow. The leaning-towers of Bologna are likewise celebrated for their great inclination; so also (in England) is the hanging-tower, or, more correctly, the massive wall which has formed part of a tower at Bridgenorth, Salop; it deviates from the perpendicular, but the centre of gravity and the line of direction fall within the base, and it remains secure; indeed, so little fears are entertained of its tumbling down, that a stable has been erected beneath it. Fig. 55. No. 1. Two billiard-cues arranged for the experiment and fixed to a board: the ball is rolling up. No. 2. Sections showing that the centre of gravity, c, is higher at a than at b, which represents the thick end of the cues; it therefore, in effect, rolls down hill. One of the most curious paradoxes is displayed in the ascent of a billiard-ball from the thin to the thick ends of two billiard-cues placed at an angle, as in our drawing above; here the
- 62. centre of gravity is raised at starting, and the ball moves in consequence of its actually falling from the high to the low level. Much of the stability of a body depends on the height through which the centre of gravity must be elevated before the body can be overthrown. The greater this height, the greater will be the immovability of the mass. One of the grandest examples of this fact is shown in the ancient Pyramids; and whilst gigantic palaces, with vast columns, and all the solid grandeur belonging to Egyptian architecture, have succumbed to time and lie more or less prostrate upon the earth, the Pyramids, in their simple form and solidity, remain almost as they were built, and it will be noticed, in the accompanying sketch, how difficult, if not impossible, it would be to attempt to overthrow bodily one of these great monuments of ancient times. Fig. 56. c. Centre of gravity, which must be raised to d before it can be overthrown. The principles already explained are directly applicable to the construction or secure loading of vehicles; and in proportion as the centre of gravity is elevated above the point of support (that is, the wheels), so is the insecurity of the carriage increased, and the contrary takes place if the centre of gravity is lowered. Again, if a waggon be loaded with a very heavy substance which does not occupy much space, such as iron, lead, or copper, or bricks, it will be in much less danger of an overthrow than if it carries an equal weight of a lighter body, such as pockets of hops, or bags of wool or bales of rags.
- 63. Fig. 57. No. 1. The centre of gravity is near the ground, and falls within the wheels. No. 2. The centre of gravity is much elevated, and the line of direction is outside the wheels. In the one instance, the centre of gravity is near the ground, and falls well within the base, as at No. 1, fig. 57. In the other, the centre of gravity is considerably elevated above the ground, and having met with an obstruction which has raised one side higher than the other, the line of direction has fallen outside the wheels, and the waggon is overturning as at No. 2. The various postures of the human body may be regarded as so many experiments upon the position of the centre of gravity which we are every moment unconsciously performing. To maintain an erect position, a man must so place his body as to cause the line of direction of his weight to fall within the base formed by his feet.
- 64. Fig. 58. The more the toes are turned outwards, the more contracted will be the base, and the body will be more liable to fall backwards or forwards; and the closer the feet are drawn together, the more likely is the body to fall on either side. The acrobats, and so-called India-Rubber Brothers, dancing dogs, c., unconsciously acquire the habit of accurately balancing themselves in all kinds of strange positions; but as these accomplishments are not to be recommended to young people, some other marvels (such as balancing a pail of water on a stick laid upon a table) may be adduced, as illustrated in fig. 59. Fig. 59. Let a b represent an ordinary table, upon which place a broomstick, c d, so that one-half shall lay upon the table and the other extend from it; place over the stick the handle of an empty pail (which may possibly require to be elongated for the experiment) so that the handle touches or falls into a notch at h; and in order to bring the pail well under the table, another stick is placed in the notch e, and is arranged in the line g f e, one end resting at g and the other at e. Having made these preparations, the pail may now be filled with water; and
- 65. although it appears to be a most marvellous result, to see the pail apparently balanced on the end of a stick which may easily tilt up, the principles already explained will enable the observer to understand that the centre of gravity of the pail falls within the line of direction shown by the dotted line; and it amounts in effect to nothing more than carrying a pail on the centre of a stick, one end of which is supported at e, and the other through the medium of the table, a b. This illustration may be modified by using a heavy weight, rope, and stick, as shown in our sketch below. Fig. 60. Before we dismiss this subject it is advisable to explain a term referring to a very useful truth, called the centre of percussion; a knowledge of which, gained instinctively or otherwise, enables the workman to wield his tools with increased power, and gives greater force to the cut of the swordsman, so that, with some physical strength, he may perform the feat of cutting a sheep in half, cleaving a bar of lead, or neatly dividing, à la Saladin, in ancient Saracen fashion, a silk handkerchief floating in the air. There is a feat, however, which does not require any very great strength, but is sufficiently startling to excite much surprise and some inquiry—viz., the one of cutting in half a broomstick supported at the ends on tumblers of water without spilling the water or cracking or otherwise damaging the glass supports.
- 66. Fig. 61. These and other feats are partly explained by reference to time: the force is so quickly applied and expended on the centre of the stick that it is not communicated to the supports; just as a bullet from a pistol may be sent through a pane of glass without shattering the whole square, but making a clean hole through it, or a candle may be sent through a plank, or a cannon-ball pass through a half opened door without causing it to move on its hinges. But the success of the several feats depends in a great measure on the attention that is paid to the delivery of the blows at the centre of percussion of the weapon; this is a point in a moving body where the percussion is the greatest, and about which the impetus or force of all parts is balanced on every side. It may be better understood by reference to our drawing below. Applying this principle to a model sword made of wood, cut in half in the centre of the blade, and then united with an elbow-joint, the handle being fixed to a board by a wire passed through it and the two upright pieces of wood, the fact is at once apparent, and is well shown in Nos. 1, 2, 3, fig. 62.
- 67. Fig. 62. No. 1, is the wooden sword, with an elbow-joint at c. No. 2. Sword attached to board at k, and being allowed to fall from any angle shown by dotted-line, it strikes the block, w, outside the centre of percussion, p, and as there is unequal motion in the parts of the sword it bends down (or, as it were, breaks) at the elbow-joint, c. No. 3 displays the same model; but here the blow has fallen on the block, w, precisely at the centre of percussion of the sword, p, and the elbow-joint remains perfectly firm. When a blow is not delivered with a stick or sword at the centre of percussion, a peculiar jar, or what is familiarly spoken of as a stinging sensation, is apparent in the hand; and the cause of this disagreeable result is further elucidated by fig. 63, in which the post, a, corresponds with the handle of the sword.
- 68. Fig. 63. a. The post to which a rope is attached. b and c are two horses running round in a circle, and it is plain that b will not move so quick as c, and that the latter will have the greatest moving force; consequently, if the rope was suddenly checked by striking against an object at the centre of gravity, the horse c would proceed faster than b, and would impart to b a backward motion, and thus make a great strain on the rope at a. But if the obstacle were placed so as to be struck at a certain point nearer c, viz., at or about the little star, the tendency of each horse to move on would balance and neutralize the other, so that there would be no strain at a. The little star indicates the centre of percussion. All military men, and especially those young gentlemen who are intended for the army, should bear in mind this important truth during their sword-practice; and with one of Mr. Wilkinson's swords, made only of the very best steel, they may conquer in a chance combat which might otherwise have proved fatal to them. To Mr. Wilkinson, of Pall Mall, the eminent sword-cutler, is due the great merit of improving the quality of the steel employed in the manufacture of officers' swords; and with one of his weapons, the author has repeatedly thrust through an iron plate about one-eighth of an inch in thickness without injuring the point, and has also bent one nearly double without fracturing it, the perfect elasticity of the steel bringing the sword straight again. These, and other severe tests applied to Wilkinson's swords, show that there is no reason why an officer should not possess a weapon that will bear comparison with, nay, surpass, the far-famed Toledo weapon, instead of submitting to mere army-tailor swords, which are often little better than hoops of beer barrels; and, in dire combat with Hindoo or Mussulman fanatics' Tulwah, may show too late the folly of the owner.
- 69. Fig. 64.
- 70. CHAPTER V. SPECIFIC GRAVITY. It is recorded of the great Dr. Wollaston, that when Sir Humphry Davy placed in his hand, what was then considered to be the scientific wonder of the day—viz., a small bit of the metal potassium, he exclaimed at once, How heavy it is, and was greatly surprised, when Sir Humphry threw the metal on water, to see it not only take fire, but actually float upon the surface; here, then, was a philosopher possessing the deepest learning, unable, by the sense of touch and by ordinary handling, to state correctly whether the new substance (and that a metal), was heavy or light; hence it is apparent that the property of specific gravity is one of importance, and being derived from the Latin, means species, a particular sort or kind; and gravis, heavy or weight—i.e., the particular weight of every substance compared with a fixed standard of water. Fig. 65. a. A large cylindrical vessel containing water, in which the egg sinks till it reaches the bottom of the glass. b. A similar glass vessel containing half brine and half water, in which the egg floats in the centre—viz., just at the point where the brine and water touch. We are so constantly in the habit of referring to a standard of perfection in music and the arts of painting and sculpture, that the youngest will comprehend the office of water when told that it is the philosopher's unit or starting-point for the estimation of the relative weights
- 71. Fig. 66. A vessel half full of water, and as the brine is poured down the tube the egg gradually rises. of solids and liquids. A good idea of the scope and meaning of the term specific gravity, is acquired by a few simple experiments, thus: if a cylindrical glass, say eighteen inches long, and two and a half wide, is filled with water, and another of the same size is also filled, one half with water and the other half with a saturated solution of common salt, or what is commonly termed brine, a most amusing comparison of the relative weights of equal bulks of water and brine, can be made with the help of two eggs; when one of the eggs is placed in the glass containing water, it immediately sinks to the bottom, showing that it has a greater specific gravity than water; but when the other egg is placed in the second glass containing the brine, it sinks through the water till it reaches the strong solution of salt, where it is suspended, and presents a most curious and pretty appearance; seeming to float like a balloon in air, and apparently suspended upon nothing, it provokes the inquiry, whether magnetism has anything to do with it? The answer, of course, is in the negative, it merely floats in the centre, in obedience to the common principle, that all bodies float in others which are heavier than themselves; the brine has, therefore, a greater weight than an equal bulk of water, and is also heavier than the egg. A pleasing sequel to this experiment may be shown by demonstrating how the brine is placed in the vessel without mixing with the water above it; this is done by using a glass tube and funnel, and after pouring away half the water contained in the vessel (Fig. 65), the egg can be floated from the bottom to the centre of the glass, by pouring the brine down the funnel and tube. The saturated solution of salt remains in the lower part of the vessel and displaces the water, which floats upon its surface like oil on water, carrying the egg with it. The water of the Dead Sea is said to contain about twenty-six per cent. of saline matter, which chiefly consists of common salt. It is perfectly clear and bright, and in consequence of the great density, a person may easily float on its surface, like the egg on the brine, so that if a ship could be heavily laden whilst floating on the water of the Dead Sea, it would most likely sink if transported to the Thames. This illustration of specific gravity is also shown by a model ship, which being first floated on the brine, will afterwards sink if conveyed to another vessel containing water. One of the tin model ships sold as a magnetic toy answers nicely for this experiment, but it must be weighted or adjusted so that it just floats in the brine, a; then it will sink, when placed, in another vessel containing only water.
- 72. Fig. 67. a. Vessel containing brine, upon which the little model floats. b. Vessel containing water, in which the ship sinks. Another amusing illustration of the same kind is displayed with goldfish, which swim easily in water, floating on brine, but cannot dive to the bottom of the vessel, owing to the density of the saturated solution of salt. If the fish are taken out immediately after the experiment, and placed in fresh water, they will not be hurt by contact with the strong salt water. These examples of the relative weights of equal bulks, enable the youthful mind to grasp the more difficult problem of ascertaining the specific gravity of any solid or liquid substance; and here the strict meaning of terms should not be passed by. Specific weight must not be confounded with Absolute weight; the latter means the entire amount of ponderable matter in any body: thus, twenty-four cubic feet of sand weigh about one ton, whilst specific weight means the relation that subsists between the absolute weight and the volume or space which that weight occupies. Thus a cubic foot of water weighs sixty-two and a half pounds, or 1000 ounces avoirdupois, but changed to gold, the cubic foot weighs more than half a ton, and would be equal to about 19,300 ounces—hence the relation between the cubic foot of water and that of gold is nearly as 1 to 19.3; the latter is therefore called the specific gravity of gold. Such a mode of taking the specific gravity of different substances—viz., by the weight of equal bulks, whether cubic feet or inches, could not be employed in consequence of the difficulty of procuring exact cubic inches or feet of the various substances which by their peculiar properties of brittleness or hardness would present insuperable obstacles to any
- 73. attempt to fashion or shape them into exact volumes. It is therefore necessary to adopt the method first devised by Archimedes, 600 b.c., when he discovered the admixture of another metal with the gold of King Hiero's crown. This amusing story, ending in the discovery of a philosophical truth, may be thus described: —King Hiero gave out from the royal treasury a certain quantity of gold, which he required to be fashioned into a crown; when, however, the emblem of power was produced by the goldsmith, it was not found deficient in weight, but had that appearance which indicated to the monarch that a surreptitious addition of some other metal must have been made. It may be assumed that King Hiero consulted his friend and philosopher Archimedes, and he might have said, Tell me, Archimedes, without pulling my crown to pieces, if it has been adulterated with any other metal? The philosopher asked time to solve the problem, and going to take his accustomed bath, discovered then specially what he had never particularly remarked before—that, as he entered the vessel of water, the liquid rose on each side of him —that he, in fact, displaced a certain quantity of liquid. Thus, supposing the bath to have been full of water, directly Archimedes stepped in, it would overflow. Let it be assumed that the water displaced was collected, and weighed 90 pounds, whilst the philosopher had weighed, say 200 pounds. Now, the train of reasoning in his mind might be of this kind: —My body displaces 90 pounds of water; if I had an exact cast of it in lead, the same bulk and weight of liquid would overflow; but the weight of my body was, say 200 pounds, the cast in lead 1000 pounds; these two sums divided by 90 would give very different results, and they would be the specific gravities, because the rule is thus stated:—'Divide the gross weight by the loss of weight in water, the water displaced, and the quotient gives the specific gravity.' The rule is soon tested with the help of an ordinary pair of scales, and the experiment made more interesting by taking a model crown of some metal, which may be nicely gilt and burnished by Messrs. Elkington, the celebrated electro-platers of Birmingham. For convenience, the pan of one scale is suspended by shorter chains than the other, and should have a hook inserted in the middle; upon this is placed the crown, supported by very thin copper wire. For the sake of argument, let it be supposed that the crown weighs 17½ ounces avoirdupois, which are duly placed in the other scale-pan, and without touching these weights, the crown is now placed in a vessel of water. It might be supposed that directly the crown enters the water, it would gain weight, in consequence of being wetted, but the contrary is the case, and by thrusting the crown into the water, it may be seen to rise with great buoyancy so long as the 17½ ounces are retained in the other scale-pan; and it will be found necessary to place at least two ounces in the scale-pan to which the crown is attached before the latter sinks in the water; and thus it is distinctly shown that the crown weighs only about 15½ ounces in the water, and has therefore lost instead of gaining weight whilst immersed in the liquid. The rule may now be worked out: Ounces. Weight of crown in air 17½ Ditto in water 15½ ——— Less in water 2 ——— 17½ / 2 = 8·75
- 74. The quotient 8¾ demonstrates that the crown is manufactured of copper, because it would have been about 19¼ if made of pure gold. Fig. 68. a. Ordinary pair of scales. b. Scale-pan, containing 17½ ounces, being the weight of the crown in air. c. Pan, with hook and crown attached, which is sunk in the water contained in the vessel d; this pan contains the two ounces, which must be placed there to make the crown sink and exactly balance b. Table of the Specific Gravities of the Metals in common use. Platinum20.98 Gold 19.26 to 19.3 and 19.64 Mercury 13.57 Lead 11.35 Silver 10.47 to 10.5 Bismuth 9.82 Copper 8.89 Iron 7.79 Tin 7.29 Zinc 6.5 to 7.4 The simple rule already explained may be applied to all metals of any size or weight, and when the mass is of an irregular shape, having various cavities on the surface, there may be some difficulty in taking the specific gravity, in consequence of the adhesion of air-bubbles; but this may be obviated either by brushing them away with a feather, or, what is frequently much better, by dipping the metal or mineral first into alcohol, and then into water, before placing it in the vessel of water, by which the actual specific gravity is to be taken.
- 75. The mode of taking the specific gravity of liquids is very simple, and is usually performed in the laboratory by means of a thin globular bottle which holds exactly 1000 grains of pure distilled water at 60° Fahrenheit. A little counterpoise of lead is made of the exact weight of the dry globular bottle, and the liquid under examination is poured into the bottle and up to the graduated mark in the neck; the bottle is then placed in one scale-pan, the counterpoise and the 1000-grain weight in the other; if the liquid (such as oil of vitriol) is heavier than water, then more weight will be required—viz., 845 grains—and these figures added to the 1000 would indicate at once that the specific gravity of oil of vitriol was 1.845 as compared with water, which is 1.000. When the liquid, such as alcohol, is lighter than water, the 1000- grain weight will be found too much, and grain weights must be added to the same scale- pan in which the bottle is standing, until the two are exactly balanced. If ordinary alcohol is being examined, it will be found necessary to place 180 grains with the bottle, and these figures deducted from the 1000 grains in the other scale-pan, leave 820, which, marked with a dot before the first figure (sic .820), indicates the specific gravity of alcohol to be less than that of water. The difference in the gravities of various liquids is displayed in a very pleasing manner by an experiment devised by Professor Griffiths, to whom chemical lecturers are especially indebted for some of the most ingenious and beautiful illustrations which have ever been devised. The experiment consists in the arrangement of five distinct liquids of various densities and colours, the one resting on the other, and distinguished not only by the optical line of demarcation, but by little balls of wax, which are adjusted by leaden shot inside, so as to sink through the upper strata of liquids, and rest only upon the one that it is intended to indicate. The manipulation for this experiment is somewhat troublesome, and is commenced by procuring some pure bright quicksilver, upon which an iron bullet (black-leaded, or painted of any colour) is placed, or one of those pretty glass balls which are sold in such quantities at the Crystal Palace. Secondly. Put as much white vitriol (sulphate of zinc) into a half pint of boiling water as it will dissolve, and, when cold, pour off the clear liquid, make up a ball of coloured wax (say red), and adjust it by placing little shot inside, until it sinks in a solution of sulphate of copper and floats on that of the white vitriol. Thirdly. Make a solution of sulphate of copper in precisely the same manner, and adjust another wax ball to sink in water, and float on this solution. Fourthly. Some clear distilled water must be provided. Fifthly. A little cochineal is to be dissolved in some common spirits of wine (alcohol), and a ball of cork painted white provided. Finally. A long cylindrical glass, at least eighteen inches high, and two and a half or three inches diameter, must be made to receive these five liquids, which are arranged in their proper order of specific gravity by means of a long tube and funnel. The four balls—viz., the iron, the two wax, and the cork balls, are allowed to slide down the long glass, which is inclined at an angle; and then, by means of the tube and funnel, pour in the tincture of cochineal, and all the balls will remain at the bottom of the glass. The water is poured down next, and now the cork ball floats up on the water, and marks the boundary
- 76. line of the alcohol and water. Then the solution of blue vitriol, when a wax ball floats upon it. Thirdly, the solution of white vitriol, upon which the second wax ball takes its place; and lastly, the quicksilver is poured down the tube, and upon this heavy metallic fluid the iron or glass ball floats like a cork on water.
- 77. Fig. 69. Long cylindrical glass, 18 × 3 inches, containing the five liquids. The tube may now be carefully removed, pausing at each liquid, so that no mixture take place between them; and the result is the arrangement of five liquids, giving the appearance of a cylindrical glass painted with bands of crimson, blue, and silver; and the liquids will not mingle with each other for many days. A more permanent arrangement can be devised by using liquids which have no affinity, or will not mix with each other—such as mercury, water, and turpentine. The specific weight or weights of an equal measure of air and other gases is determined on the same principle as liquids, although a different apparatus is required. A light capped glass globe, with stop-cock, from 50 to 100 cubic inches capacity, is weighed full of air, then exhausted by an air-pump, and weighed empty, the loss being taken as the weight of its volume of air; these figures are carefully noted, because air instead of water is the standard of comparison for all gases. When the specific gravity of any other gas is to be taken, the glass globe is again exhausted, and screwed on to a gas jar provided with a proper stop-cock, in which the gas is contained; and when perfect accuracy is required, the gas must be dried by passing it over some asbestos moistened with oil of vitriol, and contained in a glass tube, and the gas jar should stand in a mercurial trough. (Fig. 70.) The stop-cocks are gradually turned, and the gas admitted to the exhausted globe from the gas jar; when full, the cocks are turned off, the globe unscrewed, and again weighed, and by the common rule of proportion, as the weight of the air first found is to the weight of the gas, so is unity (1.000, the density of air) to a number which expresses the density of the gas required. If oxygen had been the gas tried, the number would be 1.111, being the specific gravity of that gaseous element. If chlorine, 2.470. Carbonic acid, 1.500. Hydrogen being much less than air, the number would only be 69, or decimally 0.069. Fig. 70.
- 78. a. Glass globe to contain the gas. b. Gas jar standing in the mercurial trough, d. c. Tube containing asbestos moistened with oil of vitriol. A very good approximation to the correct specific gravity (particularly where a number of trials have to be made with the same gas, such as ordinary coal gas) is obtained by suspending a light paper box, with holes at one end, on one arm of a balance, and a counterpoise on the other. The box can be made carefully, and should have a capacity equal to a half or quarter cubic foot; it is suspended with the holes downward, and is filled by blowing in the coal gas until it issues from the apertures, and can be recognised by the smell. The rule in this case would be equally simple: as the known weight of the half or quarter cubic foot of common air is to the weight of the coal gas, so is 1.000 to the number required. (Fig. 71.) Fig. 71. a. The balance. b. The paper box, of a known capacity. c. Gas-pipe blowing in coal-gas, the arrows showing entrance of gas and exit of the air. As an illustration of the different specific weights of the gases, a small balloon, containing a mixture of hydrogen and air, may be so adjusted that it will just sink in a tall glass shade inverted and supported on a pad made of a piece of oilcloth shaped round and bound with list. On passing in quickly a large quantity of carbonic acid, the little balloon will float on its surface; and if another balloon, containing only hydrogen, is held in the top part of the open shade, and a sheet of glass carefully slid over the open end, the density of the gases (although they are perfectly invisible) is perfectly indicated; and, as a climax to the experiment, a third balloon can be filled with laughing gas, and may be placed in the glass shade, taking care that the one full of pure hydrogen does not escape; the last balloon will sink to the bottom of the jar, because laughing gas is almost as heavy as carbonic acid, and the weight of the balloon will determine its descent. (Fig. 72.)
- 79. Fig. 72. Inverted large glass shade, containing half carbonic acid and half common air. Fig. 73. a. Inverted glass shade, containing the material, b, for generating carbonic acid gas. c. The soap-bubble. d d. The glass tube for blowing the bubbles. e. Small lantern, to throw a bright beam of light from the oxy-hydrogen jet upon the thin soap-bubble, which then displays the most beautiful iridescent colours. A soap-bubble will rest most perfectly on a surface of carbonic acid gas, and the aerial and elastic cushion supports the bubble till it bursts. The experiment is best performed by taking a glass shade twelve inches broad and deep in proportion, and resting it on a pad; half a pound of sesquicarbonate of soda is then placed in the vessel, and upon this is poured a mixture of half a pint of oil of vitriol and half a pint of water, the latter being previously mixed and allowed to cool before use. An enormous quantity of carbonic acid gas is suddenly generated, and rising to the edge, overflows at the top of the glass shade. A well-formed soap-bubble, detached neatly from the end of a glass-tube, oscillates gently on the surface of the heavy gas, and presents a most curious and pleasing appearance. The soapy water is prepared by cutting a few pieces of yellow soap, and placing them in a two-ounce bottle containing distilled water. (Fig. 73.) The specific gravity of the gases, may therefore be either greater, or less than atmospheric air, which has been already mentioned as the standard of comparison, and examined by this test the vapours of some of the compounds of carbon and hydrogen are found to possess a remarkably high gravity; in proof of which, the vapour of ether may be adduced as an example, although it does not consist only of the two elements mentioned, but contains a certain quantity of oxygen. In a cylindrical tin vessel, two feet high and one foot in diameter, place an ordinary hot-water plate, of course full of boiling water; upon this warm surface
- 80. pour about half an ounce of the best ether; and, after waiting a few minutes until the whole is converted into vapour, take a syphon made of half-inch pewter tube, and warm it by pouring through it a little hot water, taking care to allow the water to drain away from it before use. After placing the syphon in the tin vessel, a light may be applied to the extremity of the long leg outside the tin vessel, to show that no ether is passing over until the air is sucked out as with the water-syphon; and after this has been done, several warm glass vessels may be filled with this heavy vapour of ether, which burns on the application of flame. Finally, the remainder of the vapour may be burnt at the end of the syphon tube, demonstrating in the most satisfactory manner that the vapour is flowing through the syphon just as spirit is removed by the distillers from the casks into cellars of the public-houses. (Fig. 74.) Fig. 74. a. Tin vessel containing the hot-water plate, b, upon which the ether is poured. c. The syphon. d. Glass to receive the vapour. e. Combustion of the ether vapour in another vessel. Before dismissing the important subject of specific gravity (or, as it is termed by the French savants, density), it may be as well to state that astronomers have been enabled, by taking the density of the earth and by astronomical observations, to calculate the gravity of the planets belonging to our solar system; and it is interesting to observe that the density of the planet Venus is the only one approaching the gravity of the earth:— The Earth 1.000 The Sun .254 The Moon .742 Mercury 2.583 Venus 1.037 Mars .650 Jupiter .258 Saturn .104
- 81. Herschel .220
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