Dr. Shoab Khan Digital Signal Processing Lecture 4 DTFT
Jan 31, 2016
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