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

THE DISCRETE FOURIER SERIES

PART A

Prepared by Chao Xu 2

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

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

Introducing

• Define

• The DFS and IDFS pair

Prepared by Chao Xu 5

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

CONTINUED…

Prepared by Chao Xu 7

CONTINUED…

NOT Efficient for large

N!!!

Prepared by Chao Xu 8

An Matlab Example

Prepared by Chao Xu 9

CONTINUED…

Prepared by Chao Xu 10

CONTINUED…

Noting the sum of the geometric terms given as below:

Then, the magnitude is given by

Prepared by Chao Xu 11

CONTINUED…

Prepared by Chao Xu 12

CONTINUED…

Prepared by Chao Xu 13

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

Z-Transform & DFS

Prepared by Chao Xu 15

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

Time Sequence Reconstruction

Prepared by Chao Xu 17

Come close and touch each other

Peri. prolong. DFS

IDFS

An Matlab Example

Prepared by Chao Xu 18

Prepared by Chao Xu 19

CONTINUED…

Sample more points

in the frequency domain!

Mathematical Description

Prepared by Chao Xu 20

We sample $X(z)$ on

the unit circle, we

obtain a periodic

sequence in time

domain!

Prepared by Chao Xu 21

CONTINUED…

Prepared by Chao Xu 22

Interpolation 𝑋(𝑒𝑗𝜔) using 𝑋 (𝑘)

THE DISCRETE FOURIER TRANSFORM

PART B

Prepared by Chao Xu 23

Using DFS to Finite Duration Sequences – Discrete Fourier Transform/DFT

• Most signal in practice are NOT periodic

• Periodic prolongation!

Prepared by Chao Xu 24

Prepared by Chao Xu 25

CONTINUED…

Prepared by Chao Xu 26

Again, NOT Efficient for large

$N$!!!

Prepared by Chao Xu 27

An Example

Prepared by Chao Xu 28

CONTINUED…

Prepared by Chao Xu 29

CONTINUED…

Prepared by Chao Xu 30

CONTINUED…

Prepared by Chao Xu 31

CONTINUED…

Prepared by Chao Xu 32

CONTINUED…

Prepared by Chao Xu 33

CONTINUED…

Some DFT Properties

Prepared by Chao Xu 34

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)

Prepared by Chao Xu 36

Prepared by Chao Xu 37

Example:

Prepared by Chao Xu 38

CONTINUED…

Periodic prolongation Window Operation

Prepared by Chao Xu 39

Prepared by Chao Xu 40

Example:

Prepared by Chao Xu 41

CONTINUED…

Prepared by Chao Xu 42

CONTINUED…

Prepared by Chao Xu 43

Prepared by Chao Xu 44

Linear Convolution using DFT

Prepared by Chao Xu 45

Linear convolution:

Prepared by Chao Xu 46

Circular convolution:

Prepared by Chao Xu 47

Prepared by Chao Xu 48

Example:

Prepared by Chao Xu 49

CONTINUED…

Prepared by Chao Xu 50

Block convolution: