Fourier Analysis of DiscreteTime Signals

1
Fourier Analysis of Discrete-Time Signals

• Discrete Frequency Analysis of aperiodic signals
  (DFS)
• Chapter 10.6
• Lecture 10

2
Types of Fourier functions
3
DFT intro

• Title page said analysis of Aperiodic signals
• Chart places the DFT (FFT or DFS) into same
  bracket of DTFS (periodic signals)
• What gives?
• If we have a perfectly periodic signal, great!
• Use DTFS
• If we have an aperiodic signal, must use DTFT
• Problem! What is it?

4

• Clicker Q
• What major problem with the DTFT?
• What type of signals do we normally work with?
• Is this signal periodic?

5
DFT intro

• Title page said analysis of Aperiodic signals
• Chart places the DFT (FFT or DFS) into same
  bracket of DTFS (periodic signals)
• What gives?
• If we have a perfectly periodic signal, great!
• Use DTFS
• If we have an aperiodic signal, must use DTFT
• Problem! What is it?
• By using the DFT and assuming the aperiodic
  signal IS periodic we achieve a discrete output
  for aperiodic signals! Truly the DSP WORKHORSE

6
Lets see how the DFT works

• Lets assume we have a finite length signal, xn
• Now lets make this periodic and use DTFS

7
(No Transcript)
8
Example Lecture 9, slide 8

• From Lecture 9 the DTFT
• X(O) 1-e-jO
• X(O) v(1-cosO) sin2(O)
• plot
• Now, the DFT
• NDr NSn0 xn e-jpnr
• ND0
• ND1

9
DFT properties

• Time shift
• Differentiation

10
Circular Convolution

• Why?
• Consider x1nx2n Dr1Dr2
• x1n 1 2 3 4 x2n 2 -2 1 -1
• Dr1
• Dr2

11

• Dr1 2.5 -1/2 j1/2 -1/2 -1/2 -j1/2
• Dr2 0 1/4 j1/4 -1.5 1/2 j1/2
  
• Dr1 Dr2
• DFT-1Dr1 Dr2

12
Repeat using Circular Convolution
13
Turn CirConvolution to LinConvolution