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FIR FILTER DESIGN: THE WINDOW DESIGN METHOD

The design of a FIR filter starts with its specifications in either discrete-time domain or DTFT frequency domain, or both. In the time domain, the design objective is the impulse response. In the frequency domain, the requirement is on various parameters of the magnitude response, shown in Fig.5.8 for a lowpass filter. Important parameters are band edge frequencies ωp

ωp size 12{ω rSub { size 8{p} } } {}

and ωs

ωs size 12{ω rSub { size 8{s} } } {}

, passband

Figure 1: Typical normalized magnitude response specfications of a lowpass filter

[0, ω p ]

[0, ω p ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacUfacaaIWaGaaiilaiabeM8a3naaBaaaleaacaWGWbaabeaakiaac2faaaa@3C03@

, transition band Δω=ωs−ωp

Δω=ωs−ωp size 12{Δω=ω rSub { size 8{s} } - ω rSub { size 8{p} } } {}

, stopband , cutoff frequency ωc=(ωp+ωc)/2

ωc=(ωp+ωc)/2 size 12{ω rSub { size 8{c} } = \( ω rSub { size 8{p} } +ω rSub { size 8{c} } \) /2} {}

, passband ripple (or deviation) δp

δp size 12{δ rSub { size 8{p} } } {}

and stopband ripple δs

δs size 12{δ rSub { size 8{s} } } {}

. The two ripples are usually assumed equal . Notice that for ideal (desired) filter, the passband frequency magnitude is normalized to 1 and the stopband to 0 , and the frequency response of the designed filter oscillates between the high amplitude of 1 or the low amplitude of 0.

Figure 2 illustrates the specifications of bandpass filter. Here there are two sets of edge frequencies, lower and upper. The rising bandwith Δ ω l

Δ ω l MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiabfs5aejabeM8a3naaBaaaleaacaWGSbaabeaaaaa@3A52@

and the falling bandwidth Δ ω u

Δ ω u MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiabfs5aejabeM8a3naaBaaaleaacaWG1baabeaaaaa@3A5B@

are usually assumed equal.

Even the initial requirement is imposed on magnitude response but we must design the filter having linear phase or generalized linear phase.

Overall filter design

From the filter specifications, the first step is to choose between FIR and IIR filters based on their advantages and disadvantages. This chapter only concerns FIR filters. The next step is to select the proper linear phase FIR filters (section 5.2.3). A rather complete procedure for the design of FIR filters are as follows.

Specifications of the filter
Choosing an appropriate linear phase filter type (section 5.2.3)
Choosing the mothod of design such as window, optimal, frequency sampling…
Calculation the filter coefficients (impulse response)
Finding suitable structure
Analysis of the finite wordlength effects (chapter 7)
Implementation of the filter in hardware and/or software (chapter 7).Fig.5.9: Bandpass filter specifications

Figure 2: Bandpass filter specifications

Fixed windows

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

h(n)= h d (n)w(n) , 0≤n≤M

h(n)= h d (n)w(n) , 0≤n≤M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeqabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIgacaGGOaGaamOBaiaacMcacqGH9aqpcaWGObWaaSbaaSqaaiaadsgaaeqaaOGaaiikaiaad6gacaGGPaGaam4DaiaacIcacaWGUbGaaiykaGqabiaa=bcacaGGSaGaa8hiaiaa=bcacaWFGaGaa8hiaiaa=bcacaWFGaGaaGimaiabgsMiJkaad6gacqGHKjYOcaWGnbaaaa@4CCC@

(1)

Here we assume that all desired impulse responses and windows are causal, from n=0

n=0 size 12{n=0} {}

to n=M

n=M size 12{n=M} {}

, i.e. window length is M+1

M+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/2

M/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

H(ω)=Hd(ω)∗W(ω)=12π∫−ππHd(ω')W(ω−ω')dω'

H(ω)=Hd(ω)∗W(ω)=12π∫−ππHd(ω')W(ω−ω')dω' size 12{H \( ω \) =H rSub { size 8{d} } \( ω \) * W \( ω \) = { {1} over {2π} } Int rSub { size 8{ - π} } rSup { size 8{π} } {H rSub { size 8{d} } \( ω' \) W \( ω - ω' \) dω'} } {}

(2)

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

w(n)=1,0≤n≤M0,otherwise

w(n)=1,0≤n≤M0,otherwisealignl { stack { size 12{w \( n \) = matrix { {} # 1{} } matrix { {} } , matrix { {} # {} } 0 <= n <= M} {} # size 12{ matrix { {} # {} # matrix { {} # 0{} } {} } matrix { {} } , matrix { {} # {} } "otherwise"} {} } } {}

(3)

The Fourier transform of this window is

W(ω)= ∑ n=−∞ ∞ w(n) e −jωn = ∑ n=0 M e −jωn = 1− e −jω(M+1) 1− e −jω = e −j M 2 ω sin⁡ω(M+1)/2 sin⁡ω/2 , ω≠0 M+1 , ω=0

W(ω)= ∑ n=−∞ ∞ w(n) e −jωn = ∑ n=0 M e −jωn = 1− e −jω(M+1) 1− e −jω = e −j M 2 ω sin⁡ω(M+1)/2 sin⁡ω/2 , ω≠0 M+1 , ω=0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@9C45@

(4)

This is the same as Equation (3.54) when replacing M

M size 12{M} {}

by 2N

2N 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

| W(ω) |=| sin⁡ω(M+1)/2 sin⁡ω/2 | , ω≠0 M+1 , ω=0

| W(ω) |=| sin⁡ω(M+1)/2 sin⁡ω/2 | , ω≠0 M+1 , ω=0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaWaaqWaaeaacaWGxbGaaiikaiabeM8a3jaacMcaaiaawEa7caGLiWoacqGH9aqpdaabdaqaamaalaaabaGaci4CaiaacMgacaGGUbGaeqyYdCNaaiikaiaad2eacqGHRaWkcaaIXaGaaiykaiaac+cacaaIYaaabaGaci4CaiaacMgacaGGUbGaeqyYdCNaai4laiaaikdaaaaacaGLhWUaayjcSdGaaGjbVlaacYcacaaMf8UaaGzbVlabeM8a3jabgcMi5kaaicdaaeaacaaMf8UaaGzbVlaaywW7caaMf8EbaeqabeGaaaqaaiaad2eacqGHRaWkcaaIXaaabaGaaiilaiaaywW7caaMf8UaaGzbVdaacaaMf8UaaGzbVlabeM8a3jabg2da9iaaicdaaaaa@6F96@

(5)

Φ(ω)=− M 2 ω+β

Φ(ω)=− M 2 ω+β MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agjaacIcacqaHjpWDcaGGPaGaeyypa0JaeyOeI0YaaSaaaeaacaWGnbaabaGaaGOmaaaacqaHjpWDcqGHRaWkcqaHYoGyaaa@4265@

(6)

where β

β size 12{β} {}

is 0 or π

π 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 2N

2N size 12{2N} {}

by M

M size 12{M} {}

). It is maximum and equals to M+1

M+1 size 12{M+1} {}

at ω=0

ω=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 M

M 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=44

M=44 size 12{M="44"} {}

to abruptly truncate the impulse response of an ideal lowpass filter with cutoff frequency ωc=π/2

ωc=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 M

M 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 dB

dB 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≤M

0≤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/2

n=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=0

ω=0 size 12{ω=0} {}

(the mainlobe) as expected. Also notice that all the windows mentioned are symmetric about the mid-point n=M/2

n=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 δs

δs 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):

A=−20 log⁡ 10 [min⁡( δ p , δ s )]

A=−20 log⁡ 10 [min⁡( δ p , δ s )] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadgeacqGH9aqpcqGHsislcaaIYaGaaGimaiGacYgacaGGVbGaai4zamaaBaaaleaacaaIXaGaaGimaaqabaGccaGGBbGaciyBaiaacMgacaGGUbGaaiikaiabes7aKnaaBaaaleaacaWGWbaabeaakiaacYcacqaH0oazdaWgaaWcbaGaam4CaaqabaGccaGGPaGaaiyxaaaa@4ACA@

(7)

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 Kaiser window

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 β

β size 12{β} {}

, enabling the designer to tradeoff the transition and ripple. It is defined in the interval 0≤n≤M

0≤n≤M size 12{0 <= n <= M} {}

, otherwise zero:

w(n)= I 0 { 1−( n−M/2 M/2 ) β 2 } I 0 (β) , 0≤n≤M

w(n)= I 0 { 1−( n−M/2 M/2 ) β 2 } I 0 (β) , 0≤n≤M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadEhacaGGOaGaamOBaiaacMcacqGH9aqpdaWcaaqaaiaadMeadaWgaaWcbaGaaGimaaqabaGcdaGadaqaamaakeaabaGaaGymaiabgkHiTmaabmaabaWaaSaaaeaacaWGUbGaeyOeI0Iaamytaiaac+cacaaIYaaabaGaamytaiaac+cacaaIYaaaaaGaayjkaiaawMcaaaWcbaqcLbyacqaHYoGycaaMe8oaaOqbaeqabeqaaaqaamaaxacabaaaleqabaGaaGOmaaaaaaaakiaawUhacaGL9baaaeaacaWGjbWaaSbaaSqaaiaaicdaaeqaaOGaaiikaiabek7aIjaacMcaaaqbaeqabeqaaaqaaaaacaGGSaqbaeqabeGaaaqaaaqaaaaacaaIWaGaeyizImQaamOBaiabgsMiJkaad2eaaaa@5897@

(8)

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:

I0(x)=1+∑k=1∞x(2)kk!2=1+x24+122⋅x24⋅x24+1z2⋅x24⋅x416+⋅

I0(x)=1+∑k=1∞x(2)kk!2=1+x24+122⋅x24⋅x24+1z2⋅x24⋅x416+⋅ size 12{I rSub { size 8{0} } \( x \) =1+ Sum cSub { size 8{k=1} } cSup { size 8{ infinity } } { left [ { {x \( 2 \) rSup { size 8{k} } } over {k!} } right ]} rSup { size 8{2} } =1+ { {x rSup { size 8{2} } } over {4} } + { {1} over {2 rSup { size 8{2} } } } cdot { {x rSup { size 8{2} } } over {4} } cdot { {x rSup { size 8{2} } } over {4} } + { {1} over {z rSup { size 8{2} } } } cdot { {x rSup { size 8{2} } } over {4} } cdot { {x rSup { size 8{4} } } over {"16"} } + cdot cdot cdot } {}

(9)

Following is the values of I0(x)

I0(x) size 12{I rSub { size 8{0} } \( x \) } {}

for small x

x 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=0

β=0 size 12{β=0} {}

both the numerator and denominator are 1 and the Kaiser window becomes the rectangular window, and for β=5.44

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