1. Fourier Transforms of Discrete Signals By Dr. Varsha Shah
RCOE Dr. Varsha Shah DSPP

2. RCOE Dr. Varsha Shah DSPP
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
RCOE Dr. Varsha Shah DSPP

3. Sampling Periodic Functions
To hear music up to 20KHz a CD
should sample at the rate of 44.1 KHz
RCOE Dr. Varsha Shah DSPP

4. 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):
RCOE Dr. Varsha Shah DSPP

5. 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
RCOE Dr. Varsha Shah DSPP

6. Example of 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
RCOE Dr. Varsha Shah DSPP

7. Finding DTFT For periodic signals
Starting with Xo [n]
DTFT of Xo [n]
RCOE Dr. Varsha Shah DSPP

8. RCOE Dr. Varsha Shah DSPP
DT Fourier Transforms
W is in radian and it is between 0 and 2p in each discrete time interval
This is different from w where it was between – INF and + INF
Note that X(W) is periodic
RCOE Dr. Varsha Shah DSPP

9. RCOE Dr. Varsha Shah DSPP
Properties of DTFT
Remember:
For time scaling note that m>1  Signal spreading
RCOE Dr. Varsha Shah DSPP

10. 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)
RCOE Dr. Varsha Shah DSPP

11. 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
RCOE Dr. Varsha Shah DSPP

12. RCOE Dr. Varsha Shah DSPP
A short hand notation
remember:
RCOE Dr. Varsha Shah DSPP

13. RCOE Dr. Varsha Shah DSPP
Inverse of DFT
We can obtain the inverse of DFT
Note that
RCOE Dr. Varsha Shah DSPP

14. Using MATLAB to Calculate DFT
Example:
Assume N=4
x[n]=[1,2,3,4]
n=0,…,3
Find X[k]; k=0,…,3 or
RCOE Dr. Varsha Shah DSPP

15. RCOE Dr. Varsha Shah DSPP
Example of DFT
Find X[k]
We know k=1,.., 7; N=8
RCOE Dr. Varsha Shah DSPP

16. RCOE Dr. Varsha Shah DSPP
Example of DFT
RCOE Dr. Varsha Shah DSPP

17. RCOE Dr. Varsha Shah DSPP
Example of DFT
Polar plot for
Time shift Property of DFT
RCOE Dr. Varsha Shah DSPP

18. RCOE Dr. Varsha Shah DSPP
Example of DFT
RCOE Dr. Varsha Shah DSPP

19. Example of DFT Summation for X[k] Using the shift property!
RCOE Dr. Varsha Shah DSPP

20. RCOE Dr. Varsha Shah DSPP
Example of IDFT
Remember:
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21. RCOE Dr. Varsha Shah DSPP
Example of IDFT
Remember:
RCOE Dr. Varsha Shah DSPP

22. 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
RCOE Dr. Varsha Shah DSPP

23. 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
Butterfly FFT y0 y0 y1
Advantage: Less computationally intensive: N/2.log(N)
RCOE Dr. Varsha Shah DSPP

24. RCOE Dr. Varsha Shah DSPP
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 2mm
The first subsequence here is the term x[0], x[4], …
The second subsequent is x[2], x[6], …
RCOE Dr. Varsha Shah DSPP

25. RCOE Dr. Varsha Shah DSPP
Example
Let’s take a simple example where only two points are given n=0, n=1; N=2
Same result
RCOE Dr. Varsha Shah DSPP

26. FFT Algorithms - Four point FFT
First find even and odd parts and then combine them:
The general form:
RCOE Dr. Varsha Shah DSPP

27. FFT Algorithms - 8 point FFT
RCOE Dr. Varsha Shah DSPP

28. A Simple Application for FFT
t = 0:0.001:0.6; % 600 points
x = sin(2*pi*50*t)+sin(2*pi*120*t);
y = x + 2*randn(size(t));
plot(1000*t(1:50),y(1:50))
title('Signal Corrupted with Zero-Mean Random Noise')
xlabel('time (milliseconds)')
Taking the 512-point fast Fourier transform (FFT): Y = fft(y,512)
The power spectrum, a measurement of the power at various frequencies, is Pyy = Y.* conj(Y) / 512;
Graph the first 257 points (the other 255 points are redundant) on a meaningful frequency axis: f = 1000*(0:256)/512;
plot(f,Pyy(1:257))
title('Frequency content of y')
xlabel('frequency (Hz)')
This helps us to design an effective filter!
RCOE Dr. Varsha Shah DSPP