Transcript DSP Lecture 4 – DTFT

Digital Signal Processing
Lecture 4
DTFT
Dr. Shoab Khan
Complex Exp Input Signal
The Frequency Response
Delay and First Difference
More on the Ideal Delay
Discrete Time Fourier Transform
Fourier with DTFT Mag-Phase
Properties
Properties…(cont)
Symmetric Properties
DTFT Properties:
x[n]  x[n]
even
x[n]  X(e  j )
x * [n]  X * (e  j )
 X(e j )  X(e  j )
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 X(e j )  X *(e j )
then Re[X(e j )] is even
Im[X(e j )] is odd
X(e j ) is even
X(e j ) is odd
And
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 + e 2 j
 e 2 j (e 2 j + 2e j + 3 + 2e  j + e 2 j )
 e 2 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