Brief Review of Fourier Analysis

Brief Review of Fourier Analysis

Elena Punskaya
www-sigproc.eng.cam.ac.uk/~op205

Some material adapted from courses by
Prof. Simon Godsill, Dr. Arnaud Doucet,
Dr. Malcolm Macleod and Prof. Peter Rayner

22

Time domain

Example: speech recognition

difficult to differentiate
between different sounds
in time domain
tiny segment

sound /i/
as in see

sound /a/
as in father
23

How do we hear?

Cochlea – spiral of tissue with liquid
and thousands of tiny hairs that
Inner Ear gradually get smaller

Each hair is connected to the nerve

The longer hair resonate with lower
frequencies, the shorter hair
resonate with higher frequencies

Thus the time-domain air pressure
signal is transformed into frequency
spectrum, which is then processed
www.uptodate.com by the brain
Our ear is a Natural Fourier Transform Analyser! 24

Fourier’s Discovery

Jean Baptiste Fourier showed that any
signal could be made up by adding together
a series of pure tones (sine wave) of
appropriate amplitude and phase
(Recall from 1A Maths)

Fourier Series
for periodic
square wave

infinitely large number
of sine waves is
required
25

Fourier Transform

The Fourier transform is an equation to
calculate the frequency, amplitude and
phase of each sine wave needed to
make up any given signal :

(recall from 1B Signal
and Data Analysis)

26

Prism Analogy

Analogy:
a prism which splits
white light into a
spectrum of colors
White Spectrum
light of colours
white light consists of all
frequencies mixed
together

Signal Fourier Spectrum
the prism breaks them
apart so we can see the
Transform
separate frequencies
27

Signal Spectrum

Every signal has a frequency spectrum.
• the signal defines the spectrum
• the spectrum defines the signal

We can move back and forth between
the time domain and the frequency
domain without losing information
28

Time domain / Frequency domain

•  Some signals are easier to visualise in the
frequency domain
•  Some signals are easier to visualise in the
time domain
•  Some signals are easier to define in the time
domain (amount of information needed)
•  Some signals are easier to define in the
frequency domain (amount of information
needed)

Fourier Transform is most useful
tool for DSP
29

Fourier Transforms Examples

signal spectrum

cosine t ω

added higher
frequency t ω
component

Back to our sound recognition problem:

peaks correspond to
t the resonances of
sound /a/ ω
the vocal tract shape
as in father in logarithmis units of dB

they can be used to
sound /i/ t differentiate between
as in see ω
sounds
in logarithmis units of dB
30

Discrete Time Fourier Transform (DTFT)

What about sampled signal?

The DTFT is defined as the Fourier transform of the sampled
signal. Define the sampled signal in the usual way:

Take Fourier transform directly

using the “sifting property of the δ-function to reach the last line
31

Discrete Time Fourier Transform – Signal Samples

Note that this expression known as DTFT is a periodic function of
the frequency usually written as

The signal sample values may be expressed in terms of DTFT by
noting that the equation above has the form of Fourier series (as a
function of ω) and hence the sampled signal can be obtained directly
as

[You can show this for yourself by first noting that (*) is a complex Fourier
series with coefficients however it is also covered in one of Part IB
Examples Papers]
32

Computing DTFT on Digital Computer

The DTFT

expresses the spectrum of a sampled signal in terms of
the signal samples but is not computable on a digital
computer for two reasons:

1.  The frequency variable ω is continuous.
2.  The summation involves an infinite number of
samples.

33

Overcoming problems with computing DTFT

The problems with computing DTFT on a digital
computer can be overcome by:

Step 1. Evaluating the DTFT at a finite
collection of discrete frequencies.

no undesirable consequences, any
frequency of interest can always be
included in the collection

Step 2. Performing the summation over a
finite number of data points

does have consequences since
signals are generally not of finite duration
34

The Discrete Fourier Transform (DFT)

The discrete set of frequencies chosen is arbitrary. However, since the
DTFT is periodic we generally choose a uniformly spaced grid of N
frequencies covering the range ωT from 0 to 2π. If the summation is then
truncated to just N data points we get the DFT

The inverse DFT can be used to obtain the sampled signal values from the
DFT: multiply each side by and sum over p=0 to N-1

Orthogonality property of complex exponentials

is N if n=q and 0 otherwise

35

The Discrete Fourier Transform Pair

36

Properties of the Discrete Fourier Transform (DFT)

•  is periodic, for each p

•  is periodic, for each n

•  for real data

[You should check that you can show these results from first principles]

37

DTFT – Normalised Frequency

Please also note the DTFT and IDTFT pair is often written as:

The assumption here is that ω is a normalized frequency
ω=2πfΤ = 2π(f/fs) - normalized frequency
(rad/sample)

f - cycles per second
fs - samples per second
f/fs - cycles per sample

38
We will adopt this notation for majority of the slides.

Brief Review of Fourier Analysis

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