Fourier transforms of discrete signals (DSP) 5
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JETGI 1
Fourier Transforms of Discrete Signals
Mr. HIMANSHU DIWAKARAssistant Professor
JETGI
Mr. HIMANSHU DIWAKAR
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JETGI 2Mr. HIMANSHU DIWAKAR
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JETGI 3
Sampling
• Continuous signals are digitized using digital computers
• When we sample, we calculate the value of the continuous signal at discrete points
• How fast do we sample • What is the value of each point
• Quantization determines the value of each samples value
Mr. HIMANSHU DIWAKAR
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JETGI 4
Sampling Periodic Functions
- Note that wb = Bandwidth, thus if then aliasing occurs (signal overlaps)-To avoid aliasing -According sampling theory:
To hear music up to 20KHz a CD should sample at the rate of 44.1 KHzMr. HIMANSHU DIWAKAR
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JETGI 5
Discrete Time Fourier Transform
• In likely we only have access to finite amount of data sequences (after sampling)
• Recall for continuous time Fourier transform, when the signal is sampled:
• Assuming• Discrete-Time Fourier Transform (DTFT):
Mr. HIMANSHU DIWAKAR
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JETGI 6
Discrete Time Fourier Transform
• Discrete-Time Fourier Transform (DTFT):
• A few points• DTFT is periodic in frequency with period of 2p
• X[n] is a discrete signal • DTFT allows us to find the spectrum of the discrete signal as viewed from a
window
Mr. HIMANSHU DIWAKAR
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JETGI 7
Example of Convolution
• Convolution• We can write x[n] (a periodic function) as an infinite sum of the function xo[n]
(a non-periodic function) shifted N units at a time
• This will result
• Thus
Mr. HIMANSHU DIWAKAR
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JETGI 8
Finding DTFT For periodic signals
• Starting with xo[n]
• DTFT of xo[n]
Mr. HIMANSHU DIWAKAR
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JETGI 9
Example A & B
X[n]=a|n|, 0 < a < 1.
Mr. HIMANSHU DIWAKAR
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JETGI 10
Table of Common Discrete Time Fourier Transform (DTFT) Pairs
Mr. HIMANSHU DIWAKAR
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JETGI 11
Table of Common Discrete Time Fourier Transform (DTFT) Pairs
Mr. HIMANSHU DIWAKAR
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JETGI 12
Discrete Fourier Transform
• We often do not have an infinite amount of data which is required by DTFT• For example in a computer we cannot calculate uncountable infinite (continuum) of
frequencies as required by DTFT• Thus, we use DTF to look at finite segment of data
• We only observe the data through a window
• In this case the xo[n] is just a sampled data between n=0, n=N-1 (N points)
Mr. HIMANSHU DIWAKAR
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JETGI 13
Discrete Fourier Transform
• It turns out that DFT can be defined as
• Note that in this case the points are spaced 2pi/N; thus the resolution of the samples of the frequency spectrum is 2pi/N.
• We can think of DFT as one period of discrete Fourier series
Mr. HIMANSHU DIWAKAR
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JETGI 14
A short hand notation
Remember:
Mr. HIMANSHU DIWAKAR
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JETGI 15
Inverse of DFT
• We can obtain the inverse of DFT
• Note that
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JETGI 16
Example of DFT
• Find X[k]
• We know k=1,.., 7; N=8
Mr. HIMANSHU DIWAKAR
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JETGI 17
Example of DFT
Mr. HIMANSHU DIWAKAR
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JETGI 18
Example of DFT
Time shift Property of DFT
Polar plot for
Mr. HIMANSHU DIWAKAR
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JETGI 19
Example of DFT
Mr. HIMANSHU DIWAKAR
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JETGI 20
Example of DFT
Summation for X[k]
Using the shift property!
Mr. HIMANSHU DIWAKAR
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JETGI 21
Example of IDFT
Remember:
Mr. HIMANSHU DIWAKAR
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JETGI 22
Example of IDFT
Remember:
Mr. HIMANSHU DIWAKAR
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JETGI 23
Fast Fourier Transform Algorithms
• Consider DTFT
• Basic idea is to split the sum into 2 subsequences of length N/2 and continue all the way down until you have N/2 subsequences of length 2
Log2(8)
N
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JETGI 24
Radix-2 FFT Algorithms - Two point FFT• We assume N=2^m
• This is called Radix-2 FFT Algorithms
• Let’s take a simple example where only two points are given n=0, n=1; N=2
y0
y1
Butterfly FFT
Advantage: Less computationally intensive: N/2.log(N)
Mr. HIMANSHU DIWAKAR
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25
General FFT Algorithm • First break x[n] into even and odd
• Let n=2m for even and n=2m+1 for odd • Even and odd parts are both DFT of a N/2
point sequence
• Break up the size N/2 subsequent in half by letting 2mm
• The first subsequence here is the term x[0], x[4], …
• The second subsequent is x[2], x[6], … 1
1)2sin()2cos(
)]12[(]2[
2/
2
2/2/
2/2/2/
2/
2/2
12/
02/
12/
02/
NN
jNN
mN
NN
mN
NmN
mkN
mkN
N
m
mkN
kN
N
m
mkN
W
jeW
WWWW
WW
mxWWmxW
Mr. HIMANSHU DIWAKAR JETGI
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JETGI 26
Example
1
1)2sin()2cos(
)]12[(]2[][
2/
2
2/2/
2/2/2/
2/
2/2
12/
02/
12/
02/
NN
jNN
mN
NN
mN
NmN
mkN
mkN
N
m
mkN
kN
N
m
mkN
W
jeW
WWWW
WW
mxWWmxWkX
Let’s take a simple example where only two points are given n=0, n=1; N=2
]1[]0[]1[]0[)]1[(]0[]1[
]1[]0[)]1[(]0[]0[
11
0
0
1.01
11
0
0
1.01
0
0
0.01
01
0
0
0.01
xxxWxxWWxWkX
xxxWWxWkX
mm
mm
Same result
Mr. HIMANSHU DIWAKAR
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JETGI 27
FFT Algorithms - Four point FFT
First find even and odd parts and then combine them:
The general form:
Mr. HIMANSHU DIWAKAR
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JETGI 28
FFT Algorithms - 8 point FFT
Mr. HIMANSHU DIWAKAR
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JETGI 29
THANK YOU
Mr. HIMANSHU DIWAKAR