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Fir Filter Design: The Frequency Sampling Method

In the Fourier-Window design method the desired frequency response of the FIR filter is given in the continuous frequency, then the inverse DTFT is used to deduce the filter impulse response, followed by a windowing. Now, in the frequency-sampling method the desired frequency response of the filter is given at discrete frequencies which are the uniform samples of the continuous DTFT frequency response, then the inverse DFT (section 3.9) is used to obtain the corresponding discrete-time impulse response. From this impulse response we can implement the FIR filter, and, if recessary (for purpose of comparision or verification, say) the DTFT is used to find the continuous-frequency response.

The frequency sampling method allows us to design frequency selective filters having linear phase. Actually the method is not all that simple and not effective, it does not allow the control of the ripples. The real advantage is that it leads to a recursive implementation of FIR filters whose coefficients are integers, making the computation very fast even less precise.

We thus present the frequency sampling method rahter briefly.

Basic frequency sampling method

For the desired continuous-frequency magnitude response Hd(ω)

Hd(ω) size 12{H rSub { size 8{d} } \( ω \) } {}

of the filter (Fig.5.24a), we specify its magnitude at N

N size 12{N} {}

equally spaced frequency ωk=(2π/N)k

ωk=(2π/N)k size 12{ω rSub { size 8{k} } = \( 2π/N \) k} {}

, k=0,1,2,...,

k=0,1,2,..., size 12{k=0,1,2, "." "." "." ,} {}

in the frequency period 0≤ω≤2π

0≤ω≤2π size 12{0 <= ω <= 2π} {}

(Fig.5.24b), that is we specify

Figure 1: Basic frequency sampling method for ideal lowpass filter

∣H(k)∣=∣Hd(ωk)∣ωk=2πNk=∣Hd2πNk∣,k=0,1,2,...,N−1

∣H(k)∣=∣Hd(ωk)∣ωk=2πNk=∣Hd2πNk∣,k=0,1,2,...,N−1 size 12{ lline H \( k \) rline = lline H rSub { size 8{d} } \( ω rSub { size 8{k} } \) rline rSub { size 8{ωk= { {2π} over {N} } k} } = lline H rSub { size 8{d} } left ( { {2π} over {N} } k right ) rline matrix { , {} # {} # {} } k=0,1,2, "." "." "." ,N - 1} {}

(1)

Notice that for convenience we write H(k)

H(k) size 12{H \( k \) } {}

to mean H(ωk)

H(ωk) size 12{H \( ω rSub { size 8{k} } \) } {}

with ωk=(2π/N)k

ωk=(2π/N)k size 12{ω rSub { size 8{k} } = \( 2π/N \) k} {}

The impulse response h(n)

h(n) size 12{h \( n \) } {}

of the designed filter is the inverse DFT of the frequency response H(k)

H(k) size 12{H \( k \) } {}

h(n)=1N∑k=0N−1H(k)ej(2π/N)kn,0≤n≤N−1

h(n)=1N∑k=0N−1H(k)ej(2π/N)kn,0≤n≤N−1 size 12{h \( n \) = { {1} over {N} } Sum cSub { size 8{k=0} } cSup { size 8{N - 1} } {H \( k \) e rSup { size 8{j \( 2π/N \) ital "kn"} } } matrix { {} # , matrix { {} # 0 <= n <= N - 1{} } {} } } {}

(2)

This gives rise to a FIR filter of length N (or order N – 1). Depending on the type of linear-phase FIR filter (section 5.23) the summation is reduced accordingly. In order to check the design result, we the DTFT of the above impulse response to obtain the designed continuous frequency response H(ω)

H(ω) size 12{H \( ω \) } {}

(Fig 5.24c). If we do not make any mistake on the way, the designed H(ω)

H(ω) size 12{H \( ω \) } {}

and the desired Hd(ω)

Hd(ω) size 12{H rSub { size 8{d} } \( ω \) } {}

will match exactly at the specified discrete frequencies ωk

ωk size 12{ω rSub { size 8{k} } } {}

. This can also be shown theoretically. However, between the sampling points, the response may differ significantly. The obvious solution is to take more frequency samples, the transition width will be reduced accordingly.

As we know, there are four types of linear phase and generalized linear phase FIR filters, depending on the symmetry (symmetric or antisymmetric) and the order (even or odd) of the filter impulse response, the frequency sampling should be one of four types, resulting in four different expressions of the designed impulse response.

Optimization of the amplitude response

In the frequency sampling method there is a tradeoff between the amplitude response and the transition width. To reduce the amplitude ripples, at the expense of larger transition width (that is the filter is no longer ideal as far as the transition width is concerned) we take one sample or more within the transition band (Fig.5.25). In the case of lowpass filters, for each additional sample in the transition band the stopband attenuation increases about 20dB.

Figure 2: Addion of samples in the transition band

There is an optimization process to evaluate recursively the number of transition band frequency samples so that the stopband attenuation is maximized. Actually, Rabiner et al have developed rather fully the process for to use.

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Last edited by Nguyen Huu Phuong on Jun 27, 2008 12:52 am GMT-5.