discrete time filter design by windowing 3
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JETGI 1
Discrete-Time Filter Design by Windowing
Mr. HIMANSHU DIWAKARAssistant Professor
JETGI
Mr. HIMANSHU DIWAKAR
JETGI 2Mr. HIMANSHU DIWAKAR
JETGI 3
Filter Design by Windowing• Simplest way of designing FIR filters• Method is all discrete-time no continuous-time involved• Start with ideal frequency response
• Choose ideal frequency response as desired response• Most ideal impulse responses are of infinite length• The easiest way to obtain a causal FIR filter from ideal is
• More generally
n
njd
jd enheH
deeH21nh njj
dd
else0Mn0nhnh d
else0Mn01nw where nwnhnh d
Mr. HIMANSHU DIWAKAR
JETGI 4
Windowing in Frequency Domain• Windowed frequency response
• The windowed version is smeared version of desired response
• If w[n]=1 for all n, then W(ej) is pulse train with 2 period
deWeH21eH jj
dj
Mr. HIMANSHU DIWAKAR
JETGI 5
Properties of Windows• Prefer windows that concentrate around DC in frequency
• Less smearing, closer approximation• Prefer window that has minimal span in time
• Less coefficient in designed filter, computationally efficient• So we want concentration in time and in frequency
• Contradictory requirements• Example: Rectangular window
2/sin
2/1Msinee1e1eeW 2/Mj
j
1MjM
0n
njj
Mr. HIMANSHU DIWAKAR
JETGI 6
Rectangular Window
else0Mn01nw
• Narrowest main lob• 4/(M+1)• Sharpest transitions at
discontinuities in frequency
• Large side lobs• -13 dB• Large oscillation around
discontinuities
• Simplest window possible
Mr. HIMANSHU DIWAKAR
JETGI 7
Bartlett (Triangular) Window
else0Mn2/MM/n222/Mn0M/n2
nw
• Medium main lob• 8/M
• Side lobs• -25 dB
• Hamming window performs better
• Simple equation
Mr. HIMANSHU DIWAKAR
JETGI 8
Hanning Window
else0
Mn0Mn2cos12
1nw
• Medium main lob• 8/M
• Side lobs• -31 dB
• Hamming window performs better
• Same complexity as Hamming
Mr. HIMANSHU DIWAKAR
JETGI 9
Hamming Window
else0
Mn0Mn2cos46.054.0nw
• Medium main lob
– 8/M
• Good side lobs
– -41 dB
• Simpler than Blackman
Mr. HIMANSHU DIWAKAR
JETGI 10
Blackman Window
else0
Mn0Mn4cos08.0M
n2cos5.042.0nw
• Large main lob
– 12/M
• Very good side lobs
– -57 dB
• Complex equation
Mr. HIMANSHU DIWAKAR
JETGI 11
Incorporation of Generalized Linear Phase• Windows are designed with linear phase in mind
• Symmetric around M/2
• So their Fourier transform are of the form
• Will keep symmetry properties of the desired impulse response• Assume symmetric desired response
• With symmetric window
• Periodic convolution of real functions
else0Mn0nMwnw
even and real a is eW where eeWeW je
2/Mjje
j
2/Mjje
jd eeHeH
deWeH21eA jj
ej
e
Mr. HIMANSHU DIWAKAR
JETGI 12
Linear-Phase Low pass filter• Desired frequency response
• Corresponding impulse response
• Desired response is even symmetric, use symmetric window
c
c2/Mj
jlp 0
eeH
2/Mn
2/Mnsinnh clp
nw2/Mn
2/Mnsinnh c
Mr. HIMANSHU DIWAKAR
JETGI 13
Kaiser Window Filter Design Method
• Parameterized equation forming a set of windows• Parameter to change main-lob
width and side-lob area trade-off
• I0(.) represents zeroth-order modified Bessel function of 1st kind
else0Mn0I
2/M2/Mn1I
nw0
2
0
Mr. HIMANSHU DIWAKAR
JETGI 14
Determining Kaiser Window Parameters• Given filter specifications Kaiser developed empirical equations
• Given the peak approximation error or in dB as A=-20log10 • and transition band width
• The shape parameter should be
• The filter order M is determined approximately by
21A050A2121A07886.021A5842.0
50A7.8A1102.04.0
ps
285.28AM
Mr. HIMANSHU DIWAKAR
JETGI 15
Example: Kaiser Window Design of a Lowpass Filter
• Specifications• Window design methods assume• Determine cut-off frequency• Due to the symmetry we can choose it to be
• Compute • And Kaiser window parameters
• Then the impulse response is given as
001.0,01.0,6.0,4.0 21pp
001.021
5.0c
2.0ps 60log20A 10
653.5 37M
else0Mn0653.5I
5.185.18n1653.5I
5.18n5.18n5.0sinnh
0
2
0
Mr. HIMANSHU DIWAKAR
JETGI 16
Example Cont’d
Approximation Error
Mr. HIMANSHU DIWAKAR
JETGI 17
General Frequency Selective Filters
• A general multiband impulse response can be written as
• Window methods can be applied to multiband filters• Example multiband frequency response
• Special cases of• Bandpass• Highpass• Bandstop
mbN
1k
k1kkmb 2/Mn
2/MnsinGGnh
Mr. HIMANSHU DIWAKAR
JETGI 18
THANK YOU
Mr. HIMANSHU DIWAKAR