1.Truncating the Impulse FIR Digital Filter Design Response Chapter 10 Let H")denote the desired frequency The transfer function is a polynomial in response function.H")is periodic ●●● Basic approaches in designing FIR filters function of with period 2 and can be Truncating the Fourier series representation of expressed as a Fourier series FIR Digital Filter Design the desired frequency response=>Window H.(e)=h(n)e method The Fourier coefficients (hAn))are the Computer-aided design based on optimization impulse response samples m=云上心-a-snsa 1.Truncating the Impulse 1.Truncating the Impulse 1.Truncating the Impulse Response Response Response Thus,given H)we can compute h (n) Minimizing the integral squared error The best finite-length approximation is and the corresponding A(z) -2-fa obtained by truncating the impulse response .Usually,He")is piecewise constant with where -三e .A causal impulse response /(n)can be ideal(or sharp)transitions between bands=> (hAn))sequence is of infinite length and obtained from h(n)by delaying it with M .Using the Parseval's relation samples noncausal 中=∑k(m)-h,f (n)=h,(n-M) The objective is to find a finite-duration impulse response (h,(n)of length 2M+I c)-wco 芝m+2m .h(n)has the same magnitude response as h(n) whose DTFTH,()approximates the M+l but its phase response has a linear phase shift desired DTFT H) 中is minimum when con时ant term of @Mradians PU间r一Mm≤就
Chapter 10 FIR Digital Filter Design 2 FIR Digital Filter Design The transfer function is a polynomial in The transfer function is a polynomial in z-1. Basic approaches in designing FIR filters Basic approaches in designing FIR filters Truncating the Fourier series representation of the desired frequency response => Window method Computer Computer-aided design based on optimization 3 1. Truncating the Impulse Response Let Hd(ej¹) denote the desired frequency response function. Hd(ej¹) is periodic function of ¹ with period 2±and can be expressed as a Fourier series The Fourier coefficients {hd(n)} are the impulse response samples ( ) () j jn d d n H e h ne 1 () ( ) , 2 j jn d d hn He e d n 4 1. Truncating the Impulse Response Thus, given Hd(ej¹) we can compute hd(n) and the corresponding Hd(z) Usually, Hd(ej¹) is piecewise constant with ideal (or sharp) transitions between bands => {hd(n)} sequence is of infinite length and infinite length and noncausal noncausal The objective is to find a finite-duration impulse response {ht(n)} of length 2M+1 whose DTFT Ht(ej¹) approximates the desired DTFT Hd(ej¹) 5 1. Truncating the Impulse Response Minimizing the integral squared error where Using the Parseval’s relation 1 2 () () 2 j j Ht d e He d ( ) () M j jn t t n M H e h ne 2 1 2 2 2 1 () () () () () () t d n M M td d d n M n nM hn h n hn h n h n h n is minimum when constant term ht(n)= hd(n) for ˉM İ n İ M, 6 1. Truncating the Impulse Response The best finite-length approximation is obtained by truncating the impulse response truncating the impulse response A causal impulse response h(n) can be obtained from ht(n) by delaying it with M samples h(n) has the same magnitude response as ht(n) but its phase response has a linear phase shift linear phase shift of ¹M radians () ( ) t hn h n M
1.Truncating the Impulse 2.Impulse Response of Ideal 2.Impulse Response of Ideal Response Lowpass Filters Lowpass Filters The group delay of h(n)is Msamples The ideal lowpass filter has a zero-phase ·Truncating to range-M≤n≤Mand delaying r@=-d（-oM=M frequency response (1. with Msamples yields the causal FIR lowpass Hzp(em）= filter do sin(@(n-M)) where the linear phase response is-@M 0,包.<回sπ The corresponding impulse response ir(n)= .0≤n≤2M (n-M) coefficients 0 otherwise h(n)= sin on -,-00≤n≤∞ The truncation of the impulse response is doubly infinite,not absolutely summable, coefficients of the ideal filters exhibit an and therefore unrealizable oscillatory behavior in the respective magnitude responses 3.Gibbs Phenomenon 3.Gibbs Phenomenon 3.Gibbs Phenomenon Gibbs phenomenon-Oscillatory behavior in the magnitude responses of causal FIR filters As can be seen,as the length of the lowpass .Truncation of hn)can be expressed by obtained by truncating the impulse response filter is increased,the number of ripples in windowing operation,i.e.,by multiplying the coefficients of ideal filters both passband and stopband increases,with a h(n)sequence with a finite-length sequence corresponding decrease in the ripple widths Impact of the langth of the window function w(n) (1)Narrower transition band Height of the largest ripples remain the same h(n)=h(n)w(n) 2)Mare ripp电es independent of length where w(n)is a window function (3)Smaller rpplo width Similar oscillatory behavior observed in the 周Same largest poak中pb magnitude responses of the truncated versions The perommance is beftor. of other types of ideal filters Haw to reduce the highest ripple?
7 1. Truncating the Impulse Response The group delay of h(n) is M samples where the linear phase response is ˉ¹M () ( ) d M M d 8 2. Impulse Response of Ideal Lowpass Filters The ideal lowpass filter has a zero-phase frequency response The corresponding impulse response coefficients is doubly infinite doubly infinite, not absolutely not absolutely summable summable, and therefore unrealizable unrealizable 1, ( ) 0, j c LP c c H e sin () , c LP n hn n n 9 2. Impulse Response of Ideal Lowpass Filters Truncating to range ˉMİnİM and delaying with M samples yields the causal FIR lowpass filter The truncation of the impulse response coefficients of the ideal filters exhibit an oscillatory behavior oscillatory behavior in the respective magnitude responses sin ( ) ,0 2 ˆ ( ) ( ) 0, otherwise c LP n M n M h n n M 10 0 0.2 0.4 0.6 0.8 1 0 0.5 1 Normalized Frequency Magnitude N=20 N=60 3. Gibbs Phenomenon Gibbs phenomenon - Oscillatory behavior in the magnitude responses of causal FIR filters obtained by truncating the impulse response coefficients of ideal filters Impact of the length of the window function (1) Narrower transition band (2) More ripples (3) Smaller ripple width (4) Same largest peak ripple The performance is better. How to reduce the highest ripple? 11 3. Gibbs Phenomenon As can be seen, as the length of the lowpass filter is increased, the number of ripples in both passband and stopband increases, with a corresponding decrease in the ripple widths Height of the largest ripples remain the same independent of length Similar oscillatory behavior observed in the magnitude responses of the truncated versions of other types of ideal filters 12 3. Gibbs Phenomenon Truncation of hd(n) can be expressed by windowing operation, i.e., by multiplying the hd(n) sequence with a finite-length sequence w(n) where w(n) is a window function () () () t d h n h n wn
3.Gibbs Phenomenon 3.Gibbs Phenomenon 3.Gibbs Phenomenon Multiplication in the time domain corresponds For a rectangular window to convolution in the frequency domain -M≤n≤M w(n)= H(e)= Aee一dp 0.otherwise where H(e)=F(h(n)Y(e)=Fw(m) .The Gibbs phenomenon can be explained in the frequency domain by the convolution 。H(e“)is obtained by a periodic continuous theorem convolution of the frequency response H") with the Fourier transform ()of the window 4 3.Gibbs Phenomenon 3.Gibbs Phenomenon 4.Fixed Window Functions The frequency response (e") Rectangular window has an abrupt transition has a narrow mainlobe Symmetric window functions are used in FIR centered at w=0 to zero outside the range -Msn M, filter design in order to guarantee the linear All the other ripples in the which results in Gibbs phenomenon in H(") phase response frequency response are called Gibbs phenomenon can be reduced either: 。Smoother behavior sidelobes (a)Using a window that tapers smoothly to cutoff frequency is The main lobe is characterized by its width zero at each end,or obtained by using 4/(2M+1)defined by the first zero crossings on different cosine-type both sides of w=0 (b)Providing a smooth transition from functions instead of >As Mincreases the width of the main lobe decreases passband to stopband in the magnitude specifications the rectangular The area under each lobe remains constant,while the window width of each lobe decreases with increasing M
13 3. Gibbs Phenomenon For a rectangular window The Gibbs phenomenon can be explained in the frequency domain by the convolution theorem 1, ( ) 0, otherwise R MnM w n 14 3. Gibbs Phenomenon Multiplication in the time domain corresponds to convolution in the frequency domain where Ht(ej¹) is obtained by a periodic continuous a periodic continuous convolution of the frequency response Hd(ej¹) with the Fourier transform (ej¹) of the window 1 ( ) ( ) ( )( ) 2 j jj He H e e d t d ( ) () j H e Fh n d d ( ) () j e F wn 15 3. Gibbs Phenomenon ( ) j H e d ( ) ( ) j e ( ) ( ) j e c ( ) ( ) j e c ( ) ( ) j e 0 c ( ) j H e t 16 -3 -2 -1 0 1 2 3 -10 -5 0 5 10 15 20 25 30 Normalized Frequency Amplitude Rectangular Window M=10 M=4 3. Gibbs Phenomenon The frequency response (ej¹) has a narrow mainlobe centered at ¹=0 All the other ripples in the frequency response are called sidelobes sidelobes The main lobe is characterized by its width 4±/(2M+1) defined by the first zero crossings on both sides of ¹=0 As M increases the width of the main lobe decreases The area under each lobe remains constant, while the width of each lobe decreases with increasing M 17 3. Gibbs Phenomenon Rectangular window has an abrupt transition to zero outside the range ˉMİnİ M , which results in Gibbs phenomenon in Ht(ej¹) Gibbs phenomenon can be reduced either: (a) Using a window that tapers smoothly to zero at each end, or (b) Providing a smooth transition from passband to stopband in the magnitude specifications 18 4. Fixed Window Functions Symmetric window functions are used in FIR filter design in order to guarantee the linear phase response Smoother behavior cutoff frequency is obtained by using different cosine-type functions instead of the rectangular window 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 Time Amplitude Different Window Functions: N=25 Hamming Hann Blackman
4.Fixed Window Functions 4.Fixed Window Functions 4.Fixed Window Functions Various window functions:(rised cosine) Plots of magnitudes of the DTFTs of these windows for M=25 are shown below: Magnitude spectrum of each window Hann: characterized by a main lobe centered at =0 (n)= 1+c08 MsnsM followed by a series of sidelobes with 2 2M+1 decreasing amplitudes Hamming: 4n)=0.54+0.46c0s 2πn -AM≤n≤M Parameters predicting the performance of a 2M+1 Blackman: window in filter design are: 2元 1)Main lobe width n=0.42+0.5cos +0.08c0s 2M+1 2M+1 2)Relative sidelobe level -MsnsM 4.Fixed Window Functions 4.Fixed Window Functions 4.Fixed Window Functions 。Main lobe width-△given by the distance Lowpass Filter Design by Windowing ·Observe H,(ea+a+H,(ea-aa三l between zero crossings on both sides of main ·Thus H,(em)≥0.5 lobe 1+6 Passband and stopband ripples are the same Relative sidelobe level-A given by the difference in dB between amplitudes of largest Distance between the locations of the sidelobe and main lobe maximum passband deviation and minimum stopband value≈△w 。Width of transition band△w=w,一op<△n 4
19 4. Fixed Window Functions Various window functions: (rised cosine) Hann: Hamming: Hamming: Blackman: Blackman: 1 2 ( ) 1 cos , 2 21 n wn M n M M ! $ % " # 2 ( ) 0.54 0.46cos , 2 1 n wn M n M M " # 2 4 ( ) 0.42 0.5cos 0.08cos 21 21 n n w n M M M nM "# "# 20 4. Fixed Window Functions Plots of magnitudes of the DTFTs of these windows for M = 25 are shown below: 0 0.5 1 1.5 2 2.5 3 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Rectangular Window 0 0.5 1 1.5 2 2.5 3 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Hann Window 0 0.5 1 1.5 2 2.5 3 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Blackman Window 0 0.5 1 1.5 2 2.5 3 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Hamming Window 21 4. Fixed Window Functions Magnitude spectrum of each window characterized by a main lobe centered at¹=0 followed by a series of sidelobes with decreasing amplitudes Parameters predicting the performance of a window in filter design are: 1) Main lobe width 2) Relative 2) Relative sidelobe sidelobe level 22 4. Fixed Window Functions Main lobe width Main lobe width - ML given by the distance between zero crossings on both sides of main lobe Relative Relative sidelobe sidelobe level - Asl given by the difference in dB between amplitudes of largest sidelobe and main lobe 23 4. Fixed Window Functions Lowpass Lowpass Filter Design by Windowing ( ) j H e t ( ) j H e d s p 1 & 1& & c & ' 'ML c ( ) ( ) c j e 24 4. Fixed Window Functions Observe Thus Passband and stopband ripples are the same Distance between the locations of the maximum passband deviation and minimum stopband value ĬML Width of transition band ¹= ¹sˉ¹p<ML ( ) ( )1 c c j j He He t t ' ' ( ( ) 0.5 c j H e t (
4.Fixed Window Functions 4.Fixed Window Functions 4.Fixed Window Functions To ensure a fast transition from passband to Table 10.2:Properties of fixed window functions In the case of rectangular,Hann,Hamming, stopband,window should have a very small and Blackman windows,the value of ripple ·Rectangular window-△a=4π/(2M+l) main-lobe width does not depend on filter length or cutoff 4t=13.3dB.a,=20.9dB.△o=0.92x/M To reduce the passband and stopband ripple 6, frequency ,and is essentially constant ·Hann window·△i位=8x/2M+) the area under the sidelobes should be very 。In addition,△w≈clM 4=31.5dB,a,=43.9dB.△0=311π/M small where c is a constant for most practical ·Hamming window-△z=8r/(2M+l) Unfortunately,these two requirements are purposes contradictory 4r=42.7dB.a,=54.5dB.△m=3.32x/M ·Blackman window-△=l2r/《2M+) 4=58.1dB.a,=753dB,△0=5.56π/M 27 4.Fixed Window Functions 4.Fixed Window Functions 5.Adjustable Window Functions Filter Design Steps- ·Lowpass filter of length51ando=π/2 Dolph-Chebyshev Window- ①Set0=(0。+0,)/2 减三 2M+1y台 2 Choose window based on specified a, amplitude of sidelobe -MSnsM main lobe amplitude ③Estimate Musing A@≈clM where B=cosh2 cosh-'》 1 An increase in the main lobe width is associated with an increase in the width of the transition band cos(/cos"r),for s1 A decrease in the sidelobe amplitude results in an increase in and T(r)= the stopband attenuation cosh(/cos-x),for>1
25 4. Fixed Window Functions To ensure a fast transition from passband to stopband, window should have a very small main-lobe width To reduce the passband and stopband ripple¥, the area under the the area under the sidelobes sidelobes should be very small Unfortunately, these two requirements are contradictory 26 4. Fixed Window Functions In the case of rectangular, Hann, Hamming, and Blackman windows, the value of ripple value of ripple does not depend on filter length or cutoff frequency¹c, and is essentially constant is essentially constant In addition, ¹Ĭc/M where c is a constant for most practical purposes 27 4. Fixed Window Functions Table 10.2: Properties of fixed window functions 28 4. Fixed Window Functions Filter Design Steps Filter Design Steps - ķ Set ĸ Choose window based on specified Ĺ Estimate M using ¹Ĭc/M ( )/2 c ps ) s 29 4. Fixed Window Functions Lowpass filter of length 51 and / 2 c An increase in the main lobe width is associated with an An increase in the main lobe width is associated with an increase in the width of the transition band A decrease in the A decrease in the sidelobe amplitude results in an increase in amplitude results in an increase in the stopband attenuation 0 0.2 0.4 0.6 0.8 1 -120 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Hann Window 0 0.2 0.4 0.6 0.8 1 -120 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Hamming Window 0 0.2 0.4 0.6 0.8 1 -120 -100 -80 -60 -40 -20 0 Normalized Frequency Gain in dB Blackman Window 30 5. Adjustable Window Functions Dolph-Chebyshev Window – where and 1 11 2 ( ) 2 cos cos 21 21 21 M k k k nk wn T M MM M nM * + ! $ % " #" # amplitude of sidelobe main lobe amplitude + 1 11 cosh cosh 2M * + " # 1 1 cos( cos ), for 1 ( ) cosh( cos ), for 1 l lx x T x lx x ,
5.Adjustable Window Functions 5.Adjustable Window Functions 5.Adjustable Window Functions Dolph-Chebyshev window can be designed Gain response of a Dolph-Chebyshev window Properties of Dolph-Chebyshev window: with any specified relative sidelobe level of length 51 and relative sidelobe level of 50 .All sidelobes are of equal height while the main lobe width adjusted by dB is shown below choosing length appropriately Stopband approximation error of filters 2.056a-16.4 Filter order is estimated using N designed have essentially equiripple behavior where Aois the normalized transition 2.285(△0) For a given window length,it has the smallest bandwidth,e.g.for a lowpass filter A=,- main lobe width compared to other windows resulting in filters with the smallest transition band 5.Adjustable Window Functions 5.Adjustable Window Functions 5.Adjustable Window Functions Kaiser Window- Filter order is estimated using w)i ·In practice -M SnSM o W=g-8 1() A controls the minimum stopband 2.285(△四) where is an adjustable parameter and lo(u)is attenuation of the windowed filter response where A is the normalized transition the modified zeroth-order Bessel function of ·B is estimated using bandwidth the first kind: 6)=1+u2yTr 0.1102(a.-8.7八. for a,>50 」 B={0.5824a,-21)+0.07886(a,-21).for21sa,≤50 .Note lou)>0 for u being real 0. for a.<21
31 5. Adjustable Window Functions Dolph-Chebyshev window can be designed with any specified relative sidelobe level while the main lobe width adjusted by choosing length appropriately Filter order is estimated using where is the normalized transition bandwidth, e.g, for a lowpass filter 2.056 16.4 2.285( ) s N ) ' ' ' s p 32 5. Adjustable Window Functions Gain response of a Dolph-Chebyshev window of length 51 and relative sidelobe level of 50 dB is shown below 0 0.2 0.4 0.6 0.8 1 -80 -70 -60 -50 -40 -30 -20 -10 0 Normalized Frequency Gain in dB Dolph-Chebyshev Window 33 5. Adjustable Window Functions Properties of Properties of Dolph-Chebyshev window: window: All sidelobes sidelobes are of equal height Stopband approximation error of filters designed have essentially equiripple equiripple behavior For a given window length, it has the smallest smallest main lobe main lobe width compared to other windows resulting in filters with the smallest transition smallest transition band 34 5. Adjustable Window Functions Kaiser Window Kaiser Window - where is an adjustable parameter and I0(u) is the modified zeroth-order Bessel function of the first kind: Note I0(u)>0 for u being real 2 0 0 1(/ ) () , ( ) I nM wn M n M I * * * 2 0 1 ( / 2) () 1 ! r r u I u r ! $ % 35 5. Adjustable Window Functions In practice £ controls the minimum stopband attenuation of the windowed filter response £ is estimated using 2 20 0 1 ( / 2) () 1 ! r r u I u r ! $ % 0.4 0.1102( 8.7), for 50 0.5824( 21) 0.07886( 21), for 21 50 0, for 21 s s s ss s ) ) *) ) ) ) , 36 5. Adjustable Window Functions Filter order is estimated using where is the normalized transition bandwidth 8 2.285( ) s N ) ' '
6.Impulse Responses of FIR Filters 6.Impulse Responses of FIR Filters 6.Impulse Responses of FIR Filters with a Smooth Transition with a Smooth Transition with a Smooth Transition First-order spline passband-to-stopband Example-=0.35x @,=0.45 transition H .Pth-order spline passband-to-stopband transition 三阁 a=(%+a,)/2 / 月=0 h(n)= 2sin(△an/2P) sin(o.n ,>0 △期 江程 包, 月=0 h(n)= 2sin(Aom/2)sin(
37 6. Impulse Responses of FIR Filters with a Smooth Transition First-order spline passband-to-stopband transition ( )/2 c sp ' s p /, 0 ( ) 2sin( / 2) sin( ) . ,0 c LP c n h n n n n n n ' , ' ( ) j H e L ' s p s p p s 0 38 6. Impulse Responses of FIR Filters with a Smooth Transition Pth-order spline passband-to-stopband transition /, 0 ( ) 2sin( / 2 ) sin( ) . ,0 c P LP c n h n n P n n n n ' , " # ' 39 6. Impulse Responses of FIR Filters with a Smooth Transition Example Example- 0.45 p 0.35 s
