discrete time fourier transform finite fourier series

Dr. Shoab Khan

Digital Signal Processing

Lecture 4

DTFT

Complex Exp Input Signal

The Frequency Response

Discrete Time Fourier Transform

Properties

Properties…(cont)

Symmetric Properties

DTFT Properties: 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 odd

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

real Hermitian symmetric

Consequences of Hermitian Symmetry

If

then

And

X(e j ) X*(e j )

Re[X(e j )] is even

Im[X(e j )] is odd

X(e j ) is even

X(e j ) is odd

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

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

symmetry

DTFT- Sinusoids

DTFT of Unit Impulse

Ideal Lowpass Filter

Example

Magnitude and Angle Form

Magnitude and Angle Plot

Example

Real and Even ( Zero Phase)

Consider an LTI system with an even unit sample response

DTFT is e2 j + 2e

j + 3+ 2e j + e

2 j

2cos(2 ) + 4cos( ) + 3

Real & Even (Zero Phase)

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

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

Linear Phase

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

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

e2 j (2cos(2 )+ 4cos( )+3)

symmetryLP

Useful DTFT Pairs

Convolution Theorem

Linear Phase… ( cont.)

freqfilter

Frequency Response of DE

Matlab

Example

Ideal Filters

Ideal Filters

Ideal Lowpass Filter

h[n] of ideal filter

Approximations

Freq Axis

Inverse System