Fourier transforms of discrete signals (DSP)

1. Fourier Transforms of Discrete
Signals
Mr. HIMANSHU DIWAKAR
Assistant Professor
JETGI
Mr. HIMANSHU DIWAKAR JETGI 1

2. Mr. HIMANSHU DIWAKAR JETGI 2

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 JETGI 3

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 JETGI 4

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 JETGI 5

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 JETGI 6

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 JETGI 7

8. Finding DTFT For periodic signals
• Starting with xo[n]
• DTFT of xo[n]
Mr. HIMANSHU DIWAKAR JETGI 8

9. Example A & B
X[n]=a|n|, 0 < a < 1.
Mr. HIMANSHU DIWAKAR JETGI 9

10. Table of Common Discrete Time Fourier Transform (DTFT) Pairs
Mr. HIMANSHU DIWAKAR JETGI 10

11. Table of Common Discrete Time Fourier Transform (DTFT) Pairs
Mr. HIMANSHU DIWAKAR JETGI 11

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 JETGI 12

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 JETGI 13

14. A short hand notation
Remember:
Mr. HIMANSHU DIWAKAR JETGI 14

15. Inverse of DFT
• We can obtain the inverse of DFT
• Note that
Mr. HIMANSHU DIWAKAR JETGI 15

16. Example of DFT
• Find X[k]
• We know k=1,.., 7; N=8
Mr. HIMANSHU DIWAKAR JETGI 16

17. Example of DFT
Mr. HIMANSHU DIWAKAR JETGI 17

18. Example of DFT
Time shift Property of DFT
Polar plot for
Mr. HIMANSHU DIWAKAR JETGI 18

19. Example of DFT
Mr. HIMANSHU DIWAKAR JETGI 19

20. Example of DFT
Summation for X[k]
Using the shift property!
Mr. HIMANSHU DIWAKAR JETGI 20

21. Example of IDFT
Remember:
Mr. HIMANSHU DIWAKAR JETGI 21

22. Example of IDFT
Remember:
Mr. HIMANSHU DIWAKAR JETGI 22

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
Mr. HIMANSHU DIWAKAR JETGI 23

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 JETGI 24

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 2mm
• 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/
0
2/
12/
0
2/












N
N
jN
N
m
N
N
N
m
N
Nm
N
mk
N
mk
N
N
m
mk
N
k
N
N
m
mk
N
W
jeW
WWWW
WW
mxWWmxW

Mr. HIMANSHU DIWAKAR JETGI 25

26. Example
1
1)2sin()2cos(
)]12[(]2[][
2/
2
2/
2/
2/2/
2/
2/
2/
2
12/
0
2/
12/
0
2/












N
N
jN
N
m
N
N
N
m
N
Nm
N
mk
N
mk
N
N
m
mk
N
k
N
N
m
mk
N
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[
1
1
0
0
1.0
1
1
1
0
0
1.0
1
0
0
0.0
1
0
1
0
0
0.0
1
xxxWxxWWxWkX
xxxWWxWkX
mm
mm






Same result
Mr. HIMANSHU DIWAKAR JETGI 26

27. FFT Algorithms - Four point FFT
First find even and odd parts and then combine them:
The general form:
Mr. HIMANSHU DIWAKAR JETGI 27

28. FFT Algorithms - 8 point FFT
Mr. HIMANSHU DIWAKAR JETGI 28

29. THANK YOU
Mr. HIMANSHU DIWAKAR JETGI 29