Fourier transforms of discrete signals (DSP) 5
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The document discusses Fourier transforms of discrete signals. It covers topics like sampling, the discrete time Fourier transform (DTFT), the discrete Fourier transform (DFT), and fast Fourier transform (FFT) algorithms. The key points are: - Continuous signals are converted to discrete signals through sampling at discrete time intervals. The sampling rate must be high enough to avoid aliasing. - The DTFT analyzes the frequency spectrum of a discrete signal and is periodic in frequency space. The DFT provides the frequency spectrum of a finite discrete signal. - FFT algorithms like the radix-2 algorithm improve the efficiency of computing DFTs and DFTs by decomposing the computation into smaller subproblems.
![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](https://image.slidesharecdn.com/fouriertransformsofdiscretesignalsdsp5-170119085215/85/Fourier-transforms-of-discrete-signals-DSP-5-6-320.jpg)
![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](https://image.slidesharecdn.com/fouriertransformsofdiscretesignalsdsp5-170119085215/85/Fourier-transforms-of-discrete-signals-DSP-5-7-320.jpg)
![Finding DTFT For periodic signals
• Starting with xo[n]
• DTFT of xo[n]
Mr. HIMANSHU DIWAKAR JETGI 8](https://image.slidesharecdn.com/fouriertransformsofdiscretesignalsdsp5-170119085215/85/Fourier-transforms-of-discrete-signals-DSP-5-8-320.jpg)
![Example A & B
X[n]=a|n|, 0 < a < 1.
Mr. HIMANSHU DIWAKAR JETGI 9](https://image.slidesharecdn.com/fouriertransformsofdiscretesignalsdsp5-170119085215/85/Fourier-transforms-of-discrete-signals-DSP-5-9-320.jpg)
![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](https://image.slidesharecdn.com/fouriertransformsofdiscretesignalsdsp5-170119085215/85/Fourier-transforms-of-discrete-signals-DSP-5-12-320.jpg)
![Example of DFT
• Find X[k]
• We know k=1,.., 7; N=8
Mr. HIMANSHU DIWAKAR JETGI 16](https://image.slidesharecdn.com/fouriertransformsofdiscretesignalsdsp5-170119085215/85/Fourier-transforms-of-discrete-signals-DSP-5-16-320.jpg)
![Example of DFT
Summation for X[k]
Using the shift property!
Mr. HIMANSHU DIWAKAR JETGI 20](https://image.slidesharecdn.com/fouriertransformsofdiscretesignalsdsp5-170119085215/85/Fourier-transforms-of-discrete-signals-DSP-5-20-320.jpg)
![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], …
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![Example
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Let’s take a simple example where only two points are given n=0, n=1; N=2
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Fourier transforms of discrete signals (DSP) 5
- 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 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/ 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