The impulse response of ideal filters is infinite duration (IIR). We cannot evaluate the corresponding frequency response and, especially, implement the filter by hardware and/or software. Thus we must truncate the impulse response at both ends with respect to the central. Even we truncate the impulse response when it is small enough, but such a sudden cutoff will cause some undesired effects. The window method will reduce them.

In the time domain, windowing means that the we multiply the desired (usually ideal) infinite duration impulse response hd(n)hd(n) size 12{h rSub { size 8{d} } \( n \) } {} by a finite duration window (or window function) w(n)w(n) size 12{w \( n \) } {} to get a soft truncation. The resulted impulse response h(n)h(n) size 12{h \( n \) } {} of the designed filter is the product

Here we assume that all desired impulse responses and windows are causal, from n=0n=0 size 12{n=0} {} to n=Mn=M size 12{n=M} {}, i.e. window length is M+1M+1 size 12{M+1} {} samples (time indices). Many authors start the design with hd(n)hd(n) size 12{h rSub { size 8{d} } \( n \) } {} and w(n)w(n) size 12{w \( n \) } {} bi-sided (noncausal), i.e. defined in the interval then shift to the right M/2M/2 size 12{M/2} {} indices to make them causal.

Multiplying in time domain corresponds to convolution in frequency domain. Thus the frequency response of the designed filter (corresponding to the windowed impulse response h(n)h(n) size 12{h \( n \) } {}) is

where W(ω)W(ω) size 12{W \( ω \) } {} is the Fourier transform (DTFT) of the window w(n)w(n) size 12{w \( n \) } {}.

In the window design method , we first evaluate the desired filter impulse response h d (n) h d (n) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaadIgadaWgaaWcbaGaamizaaqabaGccaGGOaGaamOBaiaacMcaaaa@3A3B@ from the given desired frequency response H d (ω) H d (ω) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeqabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIeadaWgaaWcbaGaamizaaqabaGccaGGOaGaeqyYdCNaaiykaaaa@3B16@ , and then apply an appropriate window . Thus the method should be called the Fourier – window method.

Retangular window

Actually, the sudden truncation mentioned earlier is the simplest window called rectangular window, defined as (Figure 3)

Figure 3: Rectangular window of order M or length M + 1

The Fourier transform of this window is

This is the same as Equation (3.54) when replacing MM size 12{M} {} by 2N2N size 12{2N} {}. Also, instead of the above expression we can use the sum of cosines (Equations (3.46), (3.53)). W(ω)W(ω) size 12{W \( ω \) } {} has includes a phase factor showing the time shift of a two-sided symmetric window to a causal window. The magnitude and phase responses are, respectively,

Figure 4: Normalized frequency response of rectangular window for M = 30 and 50

where ββ size 12{β} {} is 0 or ππ size 12{π} {} as discussed in previous section. Thus the window is linear phase. Fig.3.27 illustrates the variation of the amplitude of ∣W(ω)∣∣W(ω)∣ size 12{ lline W \( ω \) rline } {} (remember to replace 2N2N size 12{2N} {} by MM size 12{M} {}). It is maximum and equals to M+1M+1 size 12{M+1} {} at ω=0ω=0 size 12{ω=0} {}. The zero-crossing points are multiples of 2π/(M+1)2π/(M+1) size 12{2π/ \( M+1 \) } {}. The response consists of a main lobe and many sidelobes. As MM size 12{M} {} gets larger the main-lobe gets smaller, and there are more sidelobes, also narrower but the amplitude of the first sidelobe remains the same (Fig.5.11). As M→∞M→∞ size 12{M rightarrow infinity } {}, W(ω)W(ω) size 12{W \( ω \) } {} tends to be an unit sample δ(ω)δ(ω) size 12{δ \( ω \) } {}.

Figure 5: Magnitude response of designed lowpass filter when using rectangular window with M = 44

The oscillatory ∣W(ω)∣∣W(ω)∣ size 12{ lline W \( ω \) rline } {} when convolved with the ideal lowpass response Hd(ω)Hd(ω) size 12{H rSub { size 8{d} } \( ω \) } {} will result in a response H(ω)H(ω) size 12{H \( ω \) } {} having a non-zero transition width and ripples is both the passband and stopband. Fig.5.12 shows the designed lowpass filter when using rectangular window having M=44M=44 size 12{M="44"} {} to abruptly truncate the impulse response of an ideal lowpass filter with cutoff frequency ωc=π/2ωc=π/2 size 12{w rSub { size 8{c} } =π/2} {}. Fig.5.13 shows the magnitude in linear scale for two case of filter order, M = 10 and M = 18.

In order that H(ω)H(ω) size 12{H \( ω \) } {} approaches Hd(ω)Hd(ω) size 12{H rSub { size 8{d} } \( ω \) } {} the rectangular window must be of infinite duration, i.e. we must take into account the whole impulse response without truncation. In frequency domain this means W(ω)W(ω) size 12{W \( ω \) } {} is an unit sample δ(ω)δ(ω) size 12{δ \( ω \) } {} as said. We can reason in the reverse direction: Since H(ω)=Hd(ω)∗δ(ω)=Hd(ω)H(ω)=Hd(ω)∗δ(ω)=Hd(ω) size 12{H \( ω \) =H rSub { size 8{d} } \( ω \) * δ \( ω \) =H rSub { size 8{d} } \( ω \) } {} then δ(ω)δ(ω) size 12{δ \( ω \) } {} must be a unit sample and the windows must be of infinite duration. When we simulate H(ω)H(ω) size 12{H \( ω \) } {}on a computer using Maltlab sofware, or else, with MM size 12{M} {} in the hundreds we will still see the non-zero transition width and ripples due to the Gibbs phenomenon (section 3.1.4). An infinitely long windows is not practical, so the idea is to look for finite duration windows which perform better than the rectangular.

Actually in examples 5.2.1 and 5.2.2 the truncation of the impulse responses of the ideal filters meant the rectangular window had been applied.

Figure 6: Magnitude response of designed filter for M = 10 and M = 18

Other windows

The Gibbs phenomenon can be reduced considerably by the use of a softer (less abrupt) truncation, i.e. by tapering the rectangular smootly to zero at both ends. Unfortunately, the reduction in amplitude of sidelobes of H(ω)H(ω) size 12{H \( ω \) } {}is accompanied by a widening of its mainlobe. For comparision, the window responses W(ω)W(ω) size 12{W \( ω \) } {}are usually plotted in dBdB size 12{ ital "dB"} {} scale.

Figure 7: Triangular window (Bartlett window)

Several smooth window functions have been proposed and used. It might come to our mind that the first applicant would be the triangular window, also called Bartlett window , depicted in Fig.5.14. From the figure we can write the window function ( table 5.1). Table 5.1 lists common fixed windows and Fig.5.15 plots their functions. Fig.5.16 plots the dB magnitude for and Fig.5.17 for M = 50.The first sidelobe is at -25dB compared to the -13dB of the corresponding (same length) rectangular window. However the mainlobe width is about twice as large.

Figure 8: Common window plotted as function of continuous time n (whereas actual windows are function of discrete index n)

The sidelobe of the triangular window is still high because the tapering is still rather coarse. For smoother tapering, cosinusoid is incorporated into the window function. This observation has led to the three well known windows: Hanning (or von Hann), Hamming, and Blackman, all defined in the interval 0≤n≤M0≤n≤M size 12{0 <= n <= M} {}, otherwise zero (table 5.1). We can check that the functions are normalized (peak value of 1 at n=M/2n=M/2 size 12{n=M/2} {}) and zero at both ends except the rectangular and the Hamming (0.08 instead of 0). Besides, there are many other less-used fixed windows.

Notice that all windows mentioned, from rectangular to Blackman, are simple functions easily evaluated, and their frequency responses concentrate around ω=0ω=0 size 12{ω=0} {}(the mainlobe) as expected. Also notice that all the windows mentioned are symmetric about the mid-point n=M/2n=M/2 size 12{n=M/2} {}, this when combined with the symmetry or antisymmetry of the filter’s impulse response will make the corresponding designed filter linear phase or generalized linear phase.

Figure 9: dB magnitude frequency response of fixed windows for M = 20 (a) Bartlett, (b) Hanning, (c) Hamming, (d) Blackman

Figure 10: dB magnitude frequency response of various windows for M = 50 (a) Bartlett, (b) Hanning, (c) Hamming, (d) Blackman

Table 5.2 lists various features of common windows for comparison. The passband ripple and the stopband ripple δsδs size 12{δ rSub { size 8{s} } } {} and the stopband ripple δ s δ s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBaaaleaacaWGZbaabeaaaaa@38AE@ , are taken equal (the smaller of the two):

Fig.5.16, Fig.5.17 and Table 5.2 show that no window is the best in all aspects but there is a tradeoff of features, and the choice of an appropriate window depending on our requirement. For smallest mainlobe width it’s the rectangular window, for best sidelobe attenuation it’s the Blackman. In between, the Hamming is a good choice. The Bartlett is the transition from the rectangular to the other windows. See section 5.3.3 for design examples.

The windows discussed so far are fixed windows, only the length (M+1)(M+1) size 12{ \( M+1 \) } {} is adjustable. The Kaiser window has an additional ripple parameter ββ size 12{β} {} , enabling the designer to tradeoff the transition and ripple. It is defined in the interval 0≤n≤M0≤n≤M size 12{0 <= n <= M} {}, otherwise zero:

where I0(x)I0(x) size 12{I rSub { size 8{0} } \( x \) } {} is the modified zeroth-order Bessel function, computed by a power series expansion:

Following is the values of I0(x)I0(x) size 12{I rSub { size 8{0} } \( x \) } {} for small xx size 12{x} {}:

X     I 0 (n)    X     I 0 (n) 0.0  1.000    1.2   1.3938 0.1  1.002    2.0   2.2796 0.2  1.010    3.0   4.3306 0.4  1.040    4.0   11.302 0.6  1.092    5.0   27.240 0.8  1.166    6.0   67.235 1.0  1.266    7.0   168.59 X     I 0 (n)    X     I 0 (n) 0.0  1.000    1.2   1.3938 0.1  1.002    2.0   2.2796 0.2  1.010    3.0   4.3306 0.4  1.040    4.0   11.302 0.6  1.092    5.0   27.240 0.8  1.166    6.0   67.235 1.0  1.266    7.0   168.59 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@0ADE@

The upper limit of the summation does not need to be too large, say about 20 at the most.

Figure 11: (a) Kaiser window for β=0, 3 β=0, 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabek7aIjabg2da9iaaicdacaGGSaGaaGjbVlaaiodaaaa@3C3D@ and 6 and M = 20, (b) Magnitude response corresponding to the windows in (a), (c) magnitude response for β=6 β=6 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabek7aIjabg2da9iaaiAdaaaa@3949@ and M = 10, 20,

For β=0β=0 size 12{β=0} {} both the numerator and denominator are 1 and the Kaiser window becomes the rectangular window, and for β=5.44β=5.44 size 12{β=5 "." "44"} {} it is near to the Hamming window. The parameter ββ size 12{β} {}is so chosen that the magnitude response of the filter lies in the allowed region (see Fig.5.1). Fig.5.18 shows the Kaiser for various values of ββ size 12{β} {} and MM size 12{M} {}. Notice that the Kaiser is also symmetric. It has minimum stopband attenuation from 50dB (for β=4.54β=4.54 size 12{β=4 "." "54"} {}) to 90dB (for β=8.96β=8.96 size 12{β=8 "." "96"} {}), so we can choose a Kaiser window to have maximum stopband atternuation equal to the Blackman window (74, Table 5.2) or higher.

For ideal filters (Fig.5.1 and Fig.5.2) the Kaiser window has equal ripple δδ size 12{δ} {} in both the passband and stopband. For real filters the ripples are slightly different . In design we take the common ripple as δ=min(δp,δs)δ=min(δp,δs) size 12{δ="min" \( δ rSub { size 8{p} } ,δ rSub { size 8{s} } \) } {} of the desired values δp,δsδp,δs size 12{δ rSub { size 8{p} } ,δ rSub { size 8{s} } } {}. As said earlier, actually δδ size 12{δ} {} is usually specified in term of dBs: A=−20log10δA=−20log10δ size 12{A= - "20""log" rSub { size 8{"10"} } δ} {}, for example δ=0.01δ=0.01 size 12{δ=0 "." "01"} {} then A=40dBA=40dB size 12{A="40""dB"} {}, δ=0.001δ=0.001 size 12{δ=0 "." "001"} {} then A=60dBA=60dB size 12{A="60""dB"} {}. Since δδ size 12{δ} {} is a fraction of 1 then A is always ponsitive.

Kaiser has developed empirical formulae and guide for the application of the window. Following is design steps for lowpass filters (similarly for other frequency selective filters).

1/ Specifications of edge frequencies ωp,ωsωp,ωs size 12{ω rSub { size 8{p} } ,ω rSub { size 8{s} } } {}; ripples δp,δsδp,δs size 12{δ rSub { size 8{p} } ,δ rSub { size 8{s} } } {}

2/Evaluate the minimum ripple A=−20log10[min(δp,δs)]A=−20log10[min(δp,δs)] size 12{A= - "20""log" rSub { size 8{"10"} } \[ "min" \( δ rSub { size 8{p} } ,δ rSub { size 8{s} } \) \] } {}, the cutoff frequency ω c =( ω p + ω s )/2 ω c =( ω p + ω s )/2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWGJbaabeaakiabg2da9iaacIcacqaHjpWDdaWgaaWcbaGaamiCaaqabaGccqGHRaWkcqaHjpWDdaWgaaWcbaGaam4CaaqabaGccaGGPaGaai4laiaaikdaaaa@4371@ , the transition width △ω= ω s − ω p △ω= ω s − ω p MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabloBjwjabeM8a3jabg2da9iabeM8a3naaBaaaleaacaWGZbaabeaakiabgkHiTiabeM8a3naaBaaaleaacaWGWbaabeaaaaa@414B@

3/ Determine the filter filter order M

4/ Determine the parameter ββ size 12{β} {}, depending on A:

5/ Computer the window coefficients w(n)w(n) size 12{w \( n \) } {}from the formula (5.40)

6/ Compute the impulse response hd(n)hd(n) size 12{h rSub { size 8{d} } \( n \) } {}of the desired filter which is linear phase

7/ Compute the designed impulse response h(n)=hd(n)w(n)h(n)=hd(n)w(n) size 12{h \( n \) =h rSub { size 8{d} } \( n \) w \( n \) } {}.

8/ Find the frequency response H(ω)=IDTFT[h(n)] H(ω)=IDTFT[h(n)] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIeacaGGOaGaeqyYdCNaaiykaiabg2da9iaadMeacaWGebGaamivaiaadAeacaWGubGaai4waiaadIgacaGGOaGaamOBaiaacMcacaGGDbaaaa@43E8@ to check with specifications. For bandpass filter the two transition bandwidths are usually equal, if not we take the smaller one.

Solution

The relation between the analog linear frequency FF size 12{F} {} samples/sec (or Hz) and the digital angular frequency ωω size 12{ω} {} radians/sample is (Equation 1.40) is

ω = 2π F f s ω = 2π F f s size 12{ω=2π { {F} over {f rSub { size 8{s} } } } } {}

where fsfs size 12{f rSub { size 8{s} } } {} is the sampling frequency (samples/sec, or Hz). Thus

Table 5.2 shows that the Hamming satisfy (so can the Blackman) the maximum stopband attenuation ( −20log10δs)−20log10δs) size 12{ - "20""log" rSub { size 8{"10"} } δ rSub { size 8{s} } \) } {} requirement. Table 5.2 includes the relation between transition width ΔωΔω size 12{Δω} {} and window order MM size 12{M} {}:

Δω = 6 . 6π M ⇒ M = 6 . 6π 0 . 33 π = 20 Δω = 6 . 6π M ⇒ M = 6 . 6π 0 . 33 π = 20 size 12{Δω= { {6 "." 6π} over {M} } matrix { {} # drarrow {} # {} } M= { {6 "." 6π} over {0 "." "33"π} } ="20"} {}

Now we compute the causal impulse response hLP(n)hLP(n) size 12{h rSub { size 8{ ital "LP"} } \( n \) } {}of ideal lowpass filter (Equation (5.32)) for 0≤n≤200≤n≤20 size 12{0 <= n <= "20"} {}. Notice that for MM size 12{M} {} even and h(n)h(n) size 12{h \( n \) } {}symmetric we have linear phase FIR-1 (Fig.5.4a). The result is plotted in Fig.5.19a.

Next we evaluate the Hamming window

w(n)=0.54−0.46cos⁡ 2πn M , 0≤n≤M=20 w(n)=0.54−0.46cos⁡ 2πn M , 0≤n≤M=20 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadEhacaGGOaGaamOBaiaacMcacqGH9aqpcaaIWaGaaiOlaiaaiwdacaaI0aGaeyOeI0IaaGimaiaac6cacaaI0aGaaGOnaiGacogacaGGVbGaai4CamaalaaabaGaaGOmaiabec8aWjaad6gaaeaacaWGnbaaauaabeqabeaaaeaaaaGaaiilauaabeqabiaaaeaaaeaaaaGaaGimaiabgsMiJkaad6gacqGHKjYOcaWGnbGaeyypa0JaaGOmaiaaicdaaaa@515E@

The result is plotted in Fig.5.19b.

The impulse response of the designed filter is the product

h(n)= h LP (n)w(n) , 0≤n≤M=20 h(n)= h LP (n)w(n) , 0≤n≤M=20 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaadIgacaGGOaGaamOBaiaacMcacqGH9aqpcaWGObWaaSbaaSqaaiaadYeacaWGqbaabeaakiaacIcacaWGUbGaaiykaiaadEhacaGGOaGaamOBaiaacMcafaqabeqabaaabaaaaiaacYcafaqabeqacaaabaaabaaaaiaaicdacqGHKjYOcaWGUbGaeyizImQaamytaiabg2da9iaaikdacaaIWaaaaa@4BCB@

We can evaluale w(n) and h LP (n) h LP (n) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIgadaWgaaWcbaGaamitaiaadcfaaeqaaOGaaiikaiaad6gacaGGPaaaaa@3AF8@ separately as above then take the product, or evaluate the product directly . The result is plotted in Fig.5.19c. We then take the inverse transform of h(n)h(n) size 12{h \( n \) } {} to get the frequency response H(ω)H(ω) size 12{H \( ω \) } {}:

H ( ω ) = ∑ n = − ∞ ∞ h ( n ) e − jωn = ∑ n = 0 M h ( n ) e − jωn , M = 20 H ( ω ) = ∑ n = − ∞ ∞ h ( n ) e − jωn = ∑ n = 0 M h ( n ) e − jωn , M = 20 size 12{H \( ω \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {h \( n \) } e rSup { size 8{ - jωn} } = Sum cSub { size 8{n=0} } cSup { size 8{M} } {h \( n \) e rSup { size 8{ - jωn} } } matrix { } , matrix { {} # {} } M="20"} {}

Fig.5.19d shows |H(ω)| |H(ω)| MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacYhacaWGibGaaiikaiabeM8a3jaacMcacaGG8baaaa@3BD6@ , and Fig.5.19c shows |H(ω) | dB |H(ω) | dB MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacYhacaWGibGaaiikaiabeM8a3jaacMcacaGG8bWaaSbaaSqaaiaadsgacaWGcbaabeaaaaa@3DB2@ .

Solution

The passband of the signal is from 300 – 200/2 to 300 + 200/2, i.e. from 200Hz to 400Hz. We are to design a bandpass FIR filter to meet the specifications

Figure 12: Example 1(linear phase lowpass filter with ω c =0.5π, Δω=0.33π ω c =0.5π, Δω=0.33π MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWGJbaabeaakiabg2da9iaaicdacaGGUaGaaGynaiabec8aWjaacYcacaaMe8UaeuiLdqKaeqyYdCNaeyypa0JaaGimaiaac6cacaaIZaGaaG4maiabec8aWbaa@48D5@ ) using the Hamming window

Figure 13: Example 2(linear phase bandpass filter using Blackman window

From Table 5.2 see that only the Blackman window can satisfy the stopband ripple (so can the Kaiser window, see next subsection). Since the passband and stopband attenuations are different, we take the smaller one in the design, i.e. AA size 12{A} {}= 60dB. The transition width and the filter order are, respectively,

Δω = 2π 100 1200 = π 6 Δω = 2π 100 1200 = π 6 size 12{Δω=2π { {"100"} over {"1200"} } = { {π} over {6} } } {} M= 11.1π π/6 =66.6≈67 M= 11.1π π/6 =66.6≈67 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaad2eacqGH9aqpdaWcaaqaaiaaigdacaaIXaGaaiOlaiaaigdacqaHapaCaeaacqaHapaCcaGGVaGaaGOnaaaacqGH9aqpcaaI2aGaaGOnaiaac6cacaaI2aGaeyisISRaaGOnaiaaiEdaaaa@46C4@

Thus the Blackman window is

W(n)=0.42−0.5cos⁡ 2πn M +0.08cos⁡ 4πn M , 0≤n≤M=67 W(n)=0.42−0.5cos⁡ 2πn M +0.08cos⁡ 4πn M , 0≤n≤M=67 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeqabiGaciaacaqaaeaadaqaaqaaaOqaaiaadEfacaGGOaGaamOBaiaacMcacqGH9aqpcaaIWaGaaiOlaiaaisdacaaIYaGaeyOeI0IaaGimaiaac6cacaaI1aGaci4yaiaac+gacaGGZbWaaSaaaeaacaaIYaGaeqiWdaNaamOBaaqaaiaad2eaaaGaey4kaSIaaGimaiaac6cacaaIWaGaaGioaiGacogacaGGVbGaai4CamaalaaabaGaaGinaiabec8aWjaad6gaaeaacaWGnbaaauaabeqabeaaaeaaaaGaaiilauaabeqabiaaaeaaaeaaaaGaaGimaiabgsMiJkaad6gacqGHKjYOcaWGnbGaeyypa0JaaGOnaiaaiEdaaaa@5B75@

Ideal bandpass filter impulse response (Equation 5.10b)

h BP (n)= sin⁡ ω u n πn − sin⁡ ω l n πn h BP (n)= sin⁡ ω u n πn − sin⁡ ω l n πn MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIgadaWgaaWcbaGaamOqaiaadcfaaeqaaOGaaiikaiaad6gacaGGPaGaeyypa0ZaaSaaaeaaciGGZbGaaiyAaiaac6gacqaHjpWDdaWgaaWcbaGaamyDaaqabaGccaWGUbaabaGaeqiWdaNaamOBaaaacqGHsisldaWcaaqaaiGacohacaGGPbGaaiOBaiabeM8a3naaBaaaleaacaWGSbaabeaakiaad6gaaeaacqaHapaCcaWGUbaaaaaa@4FE7@

where

ω u = 2π 400 1200 = 2π 3 ω u = 2π 400 1200 = 2π 3 size 12{ω rSub { size 8{u} } =2π { {"400"} over {"1200"} } = { {2π} over {3} } } {} ω l =2π 200 1200 = π 3 ω l =2π 200 1200 = π 3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWGSbaabeaakiabg2da9iaaikdacqaHapaCdaWcaaqaaiaaikdacaaIWaGaaGimaaqaaiaaigdacaaIYaGaaGimaiaaicdaaaGaeyypa0ZaaSaaaeaacqaHapaCaeaacaaIZaaaaaaa@4512@

Since M is odd we have linear phase type 2 (FIR-2) . The designed impulse response is

h ( n ) = h BP ( n ) w ( n ) h ( n ) = h BP ( n ) w ( n ) size 12{h \( n \) =h rSub { size 8{ ital "BP"} } \( n \) w \( n \) } {}

We then take the inverse transform of h(n) to get or the frequency response of the designed filter (Fig.5.20).

Solution

First let’s derive the expression of the impule response of ideal highpass filter (Fig.5.1b) with generalized linear phase (Fig.5.4) (Fig.5.1b) with generalized linear phase (Fig.5.4). The frequency response of the causal highpass filter in the interval ω=[0,π] ω=[0,π] MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3jabg2da9iaacUfacaaIWaGaaiilaiabec8aWjaac2faaaa@3D9D@ is

or related to the frequency response of the lowpass filter HLP(ω)HLP(ω) size 12{H rSub { size 8{ ital "LP"} } \( ω \) } {}(Equation (5.32)) as

H HP ( ω ) = e − j M 2 ω − H LP ( ω ) H HP ( ω ) = e − j M 2 ω − H LP ( ω ) size 12{H rSub { size 8{ ital "HP"} } \( ω \) =e rSup { size 8{ - j { {M} over {2} } ω} } - H rSub { size 8{ ital "LP"} } \( ω \) } {}

Taking the inverse DTFT of either above expression we will get the impulse response of the ideal highpass filter

Notice that we cannot just shift the result of the two-sided symmetric impulse response Equation (5.10a) to get the above causal result (but must replace δ(n)δ(n) size 12{δ \( n \) } {} by sinπn/πnsinπn/πn size 12{"sin"πn/πn} {} before shifting).

The given specifications are interpreted as

Thus the attenuation is

A=−20 log⁡ 10 [min⁡( δ p , δ s )]=−20 log⁡ 10 δ p =60 dB A=−20 log⁡ 10 [min⁡( δ p , δ s )]=−20 log⁡ 10 δ p =60 dB MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaadgeacqGH9aqpcqGHsislcaaIYaGaaGimaiGacYgacaGGVbGaai4zamaaBaaaleaacaaIXaGaaGimaaqabaGccaGGBbGaciyBaiaacMgacaGGUbGaaiikaiabes7aKnaaBaaaleaacaWGWbaabeaakiaacYcacqaH0oazdaWgaaWcbaGaam4CaaqabaGccaGGPaGaaiyxaiabg2da9iabgkHiTiaaikdacaaIWaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaaigdacaaIWaaabeaakiabes7aKnaaBaaaleaacaWGWbaabeaakiabg2da9iaaiAdacaaIWaGaaGjbVlaadsgacaWGcbaaaa@5B5B@

Knowing ΔωΔω size 12{Δω} {} and AA size 12{A} {} we can determine the window order

M= A−7.95 2.285Δω = 60−7.95 2.285(0.1π) =72.5≈72 M= A−7.95 2.285Δω = 60−7.95 2.285(0.1π) =72.5≈72 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaaiaad2eacqGH9aqpdaWcaaqaaiaadgeacqGHsislcaaI3aGaaiOlaiaaiMdacaaI1aaabaGaaGOmaiaac6cacaaIYaGaaGioaiaaiwdacqqHuoarcqaHjpWDaaGaeyypa0ZaaSaaaeaacaaI2aGaaGimaiabgkHiTiaaiEdacaGGUaGaaGyoaiaaiwdaaeaacaaIYaGaaiOlaiaaikdacaaI4aGaaGynaiaacIcacaaIWaGaaiOlaiaaigdacqaHapaCcaGGPaaaaiabg2da9iaaiEdacaaIYaGaaiOlaiaaiwdacqGHijYUcaaI3aGaaGOmaaaa@59CE@

Since A>50A>50 size 12{A>"50"} {}dB, the ripple parameter ββ size 12{β} {} is given by

β = 0 . 1102 ( A − 8 . 7 ) = 5 . 65 β = 0 . 1102 ( A − 8 . 7 ) = 5 . 65 size 12{β=0 "." "1102" \( A - 8 "." 7 \) =5 "." "65"} {}

With M/2 =33 M/2 =33 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaamaalyaabaGaamytaaqaaiaaikdaaaGaeyypa0JaaG4maiaaiodaaaa@3A07@ and β=5.65 β=5.65 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabek7aIjabg2da9iaaiwdacaGGUaGaaGOnaiaaiwdaaaa@3B7A@ , the Kaiser window (Equation (5.40)) is

W(n)= I 0 { 1− ( n−36 36 ) 2 5.65  } I 0 (5.65) W(n)= I 0 { 1− ( n−36 36 ) 2 5.65  } I 0 (5.65) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadEfacaGGOaGaamOBaiaacMcacqGH9aqpdaWcaaqaaiaadMeadaWgaaWcbaGaaGimaaqabaGccaGG7bWaaOqaaeaacaaIXaGaeyOeI0YaaeWaaeaadaWcaaqaaiaad6gacqGHsislcaaIZaGaaGOnaaqaaiaaiodacaaI2aaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaeaajugGbiaaiwdacaGGUaGaaGOnaiaaiwdacaaMe8oaaOGaaiyFaaqaaiaadMeadaWgaaWcbaGaaGimaaqabaGccaGGOaGaaGynaiaac6cacaaI2aGaaGynaiaacMcaaaaaaa@526F@

The impulse response of the FIR-1 highpass filter h HP (n) h HP (n) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIgadaWgaaWcbaGaamisaiaadcfaaeqaaOGaaiikaiaad6gacaGGPaaaaa@3AF4@ is computed for 0≤n≤66 0≤n≤66 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaaicdacqGHKjYOcaWGUbGaeyizImQaaGOnaiaaiAdaaaa@3C7A@ , and the designed impulse response is

h ( n ) = h HP ( n ) w ( n ) h ( n ) = h HP ( n ) w ( n ) size 12{h \( n \) =h rSub { size 8{ ital "HP"} } \( n \) w \( n \) } {}

As usual, the frequency response is the inverse DTFT of above impulse response.. Using Matlab we can simulate all the responses. ■

Let’s briefly consider the error in the Fourier design. Just like the Fourier series expansion of a signal (section 3.1), the Fourier design method give the least mean square (LMS) error as presented here. This also applies to the case of abupt truncation of the desired impulse , that is using the rectangular window. Let ε(ω) ε(ω) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabew7aLjaacIcacqaHjpWDcaGGPaaaaa@3AB0@ be the error between the desired ideal frequency response H d (ω) H d (ω) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIeadaWgaaWcbaGaamizaaqabaGccaGGOaGaeqyYdCNaaiykaaaa@3AF5@ and the designed frequency response H(ω) H(ω) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIeacaGGOaGaeqyYdCNaaiykaaaa@39D6@ :

This error varies with frequency and is a complex quantity , the same as the frequency response. We are interested in the error amplitude, so we take the square