3. convolution fourier

A presentation on Convolution, Fourier Series, and the
Fourier Transform
By:-
Chandan Kumar (2013UEE1203)
Sunil Kumar Yadav (2013UEE1176)
Pawan Kumar Jangid (2013UEE1166)
Manish Kumar Bagara (2013UEE1180)
Subject:- Network, Signal & Systems

Convolution, Fourier Series, and
the Fourier Transform

Convolution
 A mathematical operator which computes the
“amount of overlap” between two functions.
Can be thought of as a general moving
average
 Discrete domain:
 Continuous domain:

Discrete domain
 Basic steps
1. Flip (reverse) one of the digital functions.
2. Shift it along the time axis by one sample.
3. Multiply the corresponding values of the two digital functions.
4. Summate the products from step 3 to get one point of the
digital convolution.
5. Repeat steps 1-4 to obtain the digital convolution at all times
that the functions overlap.
 Example

LTI (Linear Time-Invariant) Systems
 Convolution can describe the effect of an LTI
system on a signal
 Assume we have an LTI system H, and its
impulse response h[n]
 Then if the input signal is x[n], the output signal
is y[n] = x[n] * h[n]
Hx[n] y[n] = x[n]*h[n]

Mathematics of Waves
 Periodic phenomena occur everywhere
– vision, sound, electronic communication, etc.
 In order to manipulate physical world, we need
mathematical tools

Fourier Series
 Any reasonable function can be expressed as a
(infinite) linear combination of sines and cosines
F(t) = a0 + a1cos (ωt) + b1sin(ωt) +
a2cos (2ωt) + b2sin(2ωt) + …
=
F(t) is a periodic function with
))sin()cos((0∑
∞
=
+n
nn tnbtna ωω
T
π
ω 2
=

Reasonable?
 F(t) is a periodic function with
 must satisfy certain other conditions
– finite number of discontinuities within T
– finite average within T
– finite number of minima and maxima
T
π
ω 2
=

Calculate Coefficients
∫=
T
k dttktf
T
a
0
)cos()(
2
ω
T
dttf
a
T
∫
= 0
0
)(
∫=
T
k dttktf
T
b
0
)sin()(
2
ω
))sin()cos(()( 0∑
∞
=
+= n
nn tnbtnatF ωω

Example
 F(t) = square wave, with T=1.0s ( )
0.1
2π
ω =
0
1
0 1 2 3 4 5 6 7
Series1

Example
ttF ω
π
sin
4
)( ≈
0
1
0 1 2 3 4 5 6 7
Series1
Series2

Example
ttttF ω
π
ω
π
ω
π
5sin
5
4
3sin
3
4
sin
4
)( ++≈
0
1
0 1 2 3 4 5 6 7
Series1
Series3

Example
 What if
– T = .01s;
– T = .05s;
– T = 50s;
ttttF ω
π
ω
π
ω
π
5sin
5
4
3sin
3
4
sin
4
)( ++≈
ttttF ω
π
ω
π
ω
π
5sin
5
4
3sin
3
4
sin
4
)( ++≈
ttttF ω
π
ω
π
ω
π
5sin
5
4
3sin
3
4
sin
4
)( ++≈

 Time
 Frequency
Time vs. Frequency Domain
0
1
0 1 2 3 4 5 6 7
Series1
Series3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1w 3w 5w
ttttF ω
π
ω
π
ω
π
5sin
5
4
3sin
3
4
sin
4
)( ++≈

Why Frequency Domain?
 Allows efficient representation of a good approximation
to the original function
 Makes filtering easy
 Note that convolution in the time domain is equivalent
to multiplication in the frequency domain (and vice
versa)
– When you are faced with a difficult convolution (or
multiplication), you can switch domains and do the
complement operation

Discrete Fourier Transform
 Rarely have closed form equation for F(t)
 Must sample at periodic intervals
– generates a series of numeric values
– apply transform to a time window of values
– N = number of sample points
– x[ ] contains sample points

DFT Notation
 ck[i] and sk [i] are the cosine and sine waves,
each N points in length
– ck is cosine wave for amplitude in ReX[k]
– sk is sine wave for amplitude in ImX[k]
 k refers to the frequency of the wave
– usually between 0 and N/2
)/2sin(][ Nkiisk π=
)/2cos(][ Nkiick π=

Complex exponentials
 Alternatively, we can use complex exponentials
 Euler’s formula: eiw
= cos(w) + i*sin(w)

Calculate DFT
 Separate sinusoids
 Complex exponentials
)/2cos(][][Re
1
0
NkiixkX
N
i
π∑
−
=
=
)/2sin(][][Im
1
0
NkiixkX
N
i
π∑
−
=
−=

Calculate Inverse DFT
 Coefficients first need to be normalized
 With two special cases
2/
][Re
][Re
N
kX
kX =
N
X
X
]0[Re
]0[Re =
2/
][Im
][Im
N
kX
kX −=
N
NX
NX
]2/[Re
]2/[Re =

Calculate Inverse DFT
 Corresponding points within the basis function
contribute to each input value
– Equation uses the normalized coefficients
 Result is exactly equal to original data (within
rounding error), ie IDFT(DFT(x[n])) = x[n]
)/2(sin][Im)/2(cos][Re][
2/
0
2/
0
NkikXNkikXix
N
k
N
k
ππ ∑∑ ==
+=

DCT (Discrete Cosine Transform)
 DCT is very similar to DFT
– Sine wave + phase shift equals cosine wave
– N coefficients of cosine basis functions
– Advantage: Result is purely real
– Most common is “Type II”
– Used in JPEG and MPEG

FFT (Fast Fourier Transform)
 FFT is an efficient algorithm for computing the DFT
– Naïve method for DFT requires O(N2
) operations
– FFT uses divide and conquer to break up problem into many 2-
point DFT’s (which are easy to compute)
– 2-point DFT: X[0] = x[0] + x[1]
X[1] = x[0] – x[1]
– log2N stages, O(N) operations per stage => O(N log N) total
operations
– Ideally, want N to be a power of 2 or close to a power of 2

4-point FFT “butterfly” diagram

Fourier Transform
 Definitions:
 Can be difficult to compute =>
Often rely upon table of transforms

Delta function
 Definition:
 Often, the result of the Fourier Transform needs
to be expressed in terms of the delta function

Fourier Transform pairs
 
 
 
 There is a duality in all transform pairs

3. convolution fourier

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3. convolution fourier