applied digital signal processingmypage.zju.edu.cn/attachments/2011-05/07-1305245234...applied...
TRANSCRIPT
![Page 1: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/1.jpg)
APPLIED DIGITAL SIGNAL PROCESSING
Instructors: Chao Xu and Zaiyue Yang
ZHEJIANG UNIVERSITY
LECTURE 5-A: DISCRETE FOURIER TRANSFORM/May 13, 2011
Prepared by Chao Xu 1
![Page 2: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/2.jpg)
THE DISCRETE FOURIER SERIES
PART A
Prepared by Chao Xu 2
![Page 3: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/3.jpg)
The Discrete Fourier Transformation
• From the computation point of view, we should avoid computing infinite sums at uncountably infinite frequencies
• We truncate sequences and then evaluate the expressions at finitely many points
• Periodic sequences Fourier series representation using linear combination of harmonically complex exponentials DFS
Prepared by Chao Xu 3
![Page 4: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/4.jpg)
Review on DFS & IDFS
• Periodic sequences
• Fourier analysis tells us there exists FS
• are called the discrete Fourier series coefficients
Prepared by Chao Xu 4
![Page 5: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/5.jpg)
Introducing
• Define
• The DFS and IDFS pair
Prepared by Chao Xu 5
![Page 6: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/6.jpg)
An Example
• Find DFS representation of the periodic sequence
• Solution: The period is N=4, then
Now follow the DFS definition
Prepared by Chao Xu 6
![Page 7: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/7.jpg)
CONTINUED…
Prepared by Chao Xu 7
![Page 8: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/8.jpg)
CONTINUED…
NOT Efficient for large
N!!!
Prepared by Chao Xu 8
![Page 9: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/9.jpg)
An Matlab Example
Prepared by Chao Xu 9
![Page 10: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/10.jpg)
CONTINUED…
Prepared by Chao Xu 10
![Page 11: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/11.jpg)
CONTINUED…
Noting the sum of the geometric terms given as below:
Then, the magnitude is given by
Prepared by Chao Xu 11
![Page 12: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/12.jpg)
CONTINUED…
Prepared by Chao Xu 12
![Page 13: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/13.jpg)
CONTINUED…
Prepared by Chao Xu 13
![Page 14: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/14.jpg)
CONTINUED…
c. • The envelopes of the DFS coefficients of square waves look like “sinc” functions • The amplitude at $k=0$ is equal to $L$ • The zeros take place at multiples of $N/L$
Prepared by Chao Xu 14
![Page 15: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/15.jpg)
Z-Transform & DFS
Prepared by Chao Xu 15
![Page 16: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/16.jpg)
Thinking Further…
• How about if we only use the data within a period of $\tilde x(n)$?
Prepared by Chao Xu 16
the previous example
![Page 17: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/17.jpg)
Time Sequence Reconstruction
Prepared by Chao Xu 17
Come close and touch each other
Peri. prolong. DFS
IDFS
![Page 18: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/18.jpg)
An Matlab Example
Prepared by Chao Xu 18
![Page 19: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/19.jpg)
Prepared by Chao Xu 19
CONTINUED…
Sample more points
in the frequency domain!
![Page 20: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/20.jpg)
Mathematical Description
Prepared by Chao Xu 20
We sample $X(z)$ on
the unit circle, we
obtain a periodic
sequence in time
domain!
![Page 21: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/21.jpg)
Prepared by Chao Xu 21
CONTINUED…
![Page 22: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/22.jpg)
Prepared by Chao Xu 22
Interpolation 𝑋(𝑒𝑗𝜔) using 𝑋 (𝑘)
![Page 23: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/23.jpg)
THE DISCRETE FOURIER TRANSFORM
PART B
Prepared by Chao Xu 23
![Page 24: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/24.jpg)
Using DFS to Finite Duration Sequences – Discrete Fourier Transform/DFT
• Most signal in practice are NOT periodic
• Periodic prolongation!
Prepared by Chao Xu 24
![Page 25: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/25.jpg)
Prepared by Chao Xu 25
CONTINUED…
![Page 26: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/26.jpg)
Prepared by Chao Xu 26
Again, NOT Efficient for large
$N$!!!
![Page 27: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/27.jpg)
Prepared by Chao Xu 27
An Example
![Page 28: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/28.jpg)
Prepared by Chao Xu 28
CONTINUED…
![Page 29: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/29.jpg)
Prepared by Chao Xu 29
CONTINUED…
![Page 30: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/30.jpg)
Prepared by Chao Xu 30
CONTINUED…
![Page 31: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/31.jpg)
Prepared by Chao Xu 31
CONTINUED…
![Page 32: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/32.jpg)
Prepared by Chao Xu 32
CONTINUED…
![Page 33: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/33.jpg)
Prepared by Chao Xu 33
CONTINUED…
![Page 34: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/34.jpg)
Some DFT Properties
Prepared by Chao Xu 34
![Page 35: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/35.jpg)
Prepared by Chao Xu 35
CONTINUED…
M = mod(X,Y) if Y ~= 0, returns X - n.*Y where n = floor(X./Y). If Y is not an integer and the quotient X./Y is within roundoff error of an integer, then n is that integer.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 0 mod(-10:0,11)
![Page 36: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/36.jpg)
Prepared by Chao Xu 36
![Page 37: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/37.jpg)
Prepared by Chao Xu 37
Example:
![Page 38: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/38.jpg)
Prepared by Chao Xu 38
CONTINUED…
Periodic prolongation Window Operation
![Page 39: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/39.jpg)
Prepared by Chao Xu 39
![Page 40: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/40.jpg)
Prepared by Chao Xu 40
Example:
![Page 41: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/41.jpg)
Prepared by Chao Xu 41
CONTINUED…
![Page 42: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/42.jpg)
Prepared by Chao Xu 42
CONTINUED…
![Page 43: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/43.jpg)
Prepared by Chao Xu 43
![Page 44: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/44.jpg)
Prepared by Chao Xu 44
![Page 45: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/45.jpg)
Linear Convolution using DFT
Prepared by Chao Xu 45
Linear convolution:
![Page 46: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/46.jpg)
Prepared by Chao Xu 46
Circular convolution:
![Page 47: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/47.jpg)
Prepared by Chao Xu 47
![Page 48: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/48.jpg)
Prepared by Chao Xu 48
Example:
![Page 49: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/49.jpg)
Prepared by Chao Xu 49
CONTINUED…
![Page 50: APPLIED DIGITAL SIGNAL PROCESSINGmypage.zju.edu.cn/attachments/2011-05/07-1305245234...APPLIED DIGITAL SIGNAL PROCESSING Instructors: Chao Xu and Zaiyue Yang ZHEJIANG UNIVERSITY LECTURE](https://reader034.vdocument.in/reader034/viewer/2022051800/5acdb9327f8b9a93268deb9c/html5/thumbnails/50.jpg)
Prepared by Chao Xu 50
Block convolution: