# Discrete-Time Filter Design by Windowing Quote of the

Discrete-Time Filter Design by Windowing Quote of the Day In mathematics you don't understand things. You just get used to them. Johann von Neumann Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice Hall Inc. Filter Design by Windowing Simplest way of designing FIR filters Method is all discrete-time no continuous-time involved Start with ideal frequency response 1 j j n j j n Hd e hd n e hd n H e e d d 2 n 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 hd n 0 n M hn else 0 More generally hn hd n wn Copyright (C) 2005 Gner Arslan where 1 0 n M wn else 0 351M Digital Signal Processing 2 Windowing in Frequency Domain Windowed frequency response 1 j

j H ej H e W e d d 2 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 Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 3 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 M 1 e j M1 j j n j M / 2 sin M 1 / 2 W e e e sin / 2 1 e j n 0 Demo Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 4 Rectangular Window Narrowest main lob 4/(M+1)

Sharpest transitions at discontinuities in frequency Large side lobs -13 dB Large oscillation around discontinuities Simplest window possible 1 0 n M wn else 0 Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 5 Bartlett (Triangular) Window Medium main lob 8/M Side lobs -25 dB Hamming window performs better Simple equation 0 n M / 2

2n / M wn 2 2n / M M / 2 n M 0 else Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 6 Hanning Window Medium main lob 8/M Side lobs -31 dB Hamming window performs better Same complexity as Hamming 1 2n 1 cos wn 2 M

0 Copyright (C) 2005 Gner Arslan 0 n M else 351M Digital Signal Processing 7 Hamming Window Medium main lob 8/M Good side lobs -41 dB Simpler than Blackman 2n 0. 54 0. 46 cos 0 n M wn M 0 else Copyright (C) 2005 Gner Arslan

351M Digital Signal Processing 8 Blackman Window Large main lob 12/M Very good side lobs -57 dB Complex equation 2n 4n 0. 42 0. 5 cos 0. 08 cos 0 n M wn M M 0 else Windows Demo Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing

9 Incorporation of Generalized Linear Phase Windows are designed with linear phase in mind Symmetric around M/2 wM n 0 n M wn else 0 So their Fourier transform are of the form W e j We e j e j M / 2 where We e j is a real and even Will keep symmetry properties of the desired impulse response Assume symmetric desired response Hd e j He e j e j M / 2 With symmetric window

Ae e j 1 j j H e W e d e 2 Periodic convolution of real functions Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 10 Linear-Phase Lowpass filter

Desired frequency response e j M / 2 c j Hlp e c 0 Corresponding impulse response sin c n M / 2 hlp n n M / 2 Desired response is even symmetric, use symmetric window hn sin c n M / 2 wn n M / 2 Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 11 Kaiser Window Filter Design Method Parameterized equation

forming a set of windows Parameter to change mainlob width and side-lob area trade-off 2 n M / 2 I 0 1 M/ 2 wn I 0 0 0 n M else I0(.) represents zeroth-order modified Bessel function of 1st kind Copyright (C) 2005 Gner Arslan

351M Digital Signal Processing 12 Determining Kaiser Window Parameters Given filter specifications Kaiser developed empirical equations Given the peak approximation error s or p in dB as A=-20log10 and transition band width The shape parameter should be 0. 1102 A 8. 7 A 50 0. 4 0. 5842 A 21 0. 07886 A 21 21 A 50 0 A 21 The filter order M is determined approximately by A 8 M 2. 285 Copyright (C) 2005 Gner Arslan

351M Digital Signal Processing 13 Example: Kaiser Window Design of a Lowpass Filter Specifications p 0. 4, p 0. 6, 1 0. 01, 2 0. 001 Window design methods assume 1 2 0. 001 Determine cut-off frequency Due to the symmetry we can choose it to be c 0. 5 Compute A 20 log10 60 s p 0. 2 And Kaiser window parameters 5. 653 M 37 Then the impulse response is given as 2 n 18. 5 I 0 5. 653 1 18 . 5 hn sin 0. 5n 18. 5

n 18. 5 I 0 5. 653 0 Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 0 n M else 14 Example Contd Approximation Error Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 15 General Frequency Selective Filters A general multiband impulse response can be written as Nmb hmb n Gk Gk 1 k 1

sin k n M / 2 n M / 2 Window methods can be applied to multiband filters Example multiband frequency response Special cases of Bandpass Highpass Bandstop Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 16

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