discrete time fourier transform finite fourier series
DESCRIPTION
discrete Time Fourier transform for finite Fourier series it is lecture for masters students in electrical engineeringTRANSCRIPT
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