Lec 6 Discrete Time Fourier Transform

Discrete Time Fourier Transform

Discrete Time Fourier TransformDiscrete Time Fourier Transform

1

Compare Fourier Series and DTFTCompare Fourier Series and DTFT

Existence of the DTFTExistence of the DTFT

2

DTFT PropertiesDTFT Properties

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DTFT Properties

DTFT Properties1

Symmetric Properties Recall

from DTFT properties: If

then and ...

x[n] X(e j)x[ n] X(e j)x*[n] X*(e j)

x[n]x[ n] X(e j )X(e j)even even

x[n] x[ n] X(e j ) X(e j ) odd oddx[n]x*[n] X(e j )X*(e j )

real Hermitian symmetric

Symmetric Properties

4

Consequences of Hermitian Symmetry

If Then And

X(e j)X *(e j )Re[X(e j )] is evenIm[X(e j )] is oddX(e j ) is even

X(e j ) is odd

If x[n] is real and even, X(e j ) will be real and evenand if x[n] is real and odd, X(e j ) will be imaginary and odd

Consequences of Hermitian Symmetry

Real and Even: Zero Phasen even unit sample response:

Frequency response is real, so system has “zero” phase shift

This is to be expected since unit sample response is real and even

Real and Even: Zero Phase

e 2 j + 2 e j + 3 + 2 e j + e 2 j 2 cos( 2 ) + 4 cos( ) + 3

5

Linear PhaseNow delay the system’s sample response to make it

causal:

DTFT is now Comment:

Frequency response now exhibits linear phase shift

H(z)e2 j +2e j +3+3e j + e 2j

e 2j(e2j +2e j +3+2e j +e 2j )e 2j(2cos(2)+4cos()+3)

Linear Phase

6

Example

7

Frequency Response of MA System

otherwise

MnMMM

0

11

2121

++][nh

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Moving Average System

Ex1: Moving Average System

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Inverse Fourier Transform & evaluation of Impulse Response from Difference Equation