discrete time filter design by windowing 3

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Discrete-Time Filter Design by Windowing

Mr. HIMANSHU DIWAKARAssistant Professor

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Mr. HIMANSHU DIWAKAR

JETGI 2Mr. HIMANSHU DIWAKAR

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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


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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


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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


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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


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Bartlett (Triangular) Window

else0Mn2/MM/n222/Mn0M/n2

nw

• Medium main lob• 8/M

• Side lobs• -25 dB

• Hamming window performs better

• Simple equation


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Hanning Window

else0

Mn0Mn2cos12

1nw

• Medium main lob• 8/M

• Side lobs• -31 dB

• Hamming window performs better

• Same complexity as Hamming


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Hamming Window

else0

Mn0Mn2cos46.054.0nw

• Medium main lob

– 8/M

• Good side lobs

– -41 dB

• Simpler than Blackman


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Blackman Window

else0

Mn0Mn4cos08.0M

n2cos5.042.0nw

• Large main lob

– 12/M

• Very good side lobs

– -57 dB

• Complex equation


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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


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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


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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


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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


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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


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Example Cont’d

Approximation Error


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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


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THANK YOU


discrete time filter design by windowing 3

Engineering