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Optimal Equiripple Design Technique 367
differentiator, the m vector does not specify the desired slope in each
band but the desired magnitude.
• [h] = firpm(N,f,m,weights,ftype) is similar to the above case except
that the array weights specifies the weighting function in each
band.
To estimate the filter order N, the SP toolbox provides the function
firpmord, which also estimates other parameters that can be used in the
firpm function. The basic syntax is
[N,f0,m0,weights] = firpmord(f,m,delta);
The function computes the window order N, the normalized frequency
band edges in f0, amplitude response in a0, and the band weights in
weights. The vector f is a vector of normalized band edges and m is a
vector specifying the desired amplitude on the bands defined by f. The
length of f is two less than twice the length of m; i.e., f does not contain 0
or 1. The vector delta specifies tolerances in each band (not in decibels).
The estimated parameters can now be used in the firpm function.
As explained during the description of the Parks-McClellan algorithm,
we have to first guess the order of the filter using (7.48) to use the function
firpm. After we obtain the filter coefficients in array h, wehave to check
the minimum stopband attenuation and compare it with the given A s
and then increase (or decrease) the filter order. We have to repeat this
procedure until we obtain the desired A s .Weillustrate this procedure
in the following several MATLAB examples. These examples also use the
ripple conversion function db2delta, which is developed in Problem P7.1.
□ EXAMPLE 7.23 Let us design the lowpass filter described in Example 7.8 using the Parks-
McClellan algorithm. The design parameters are
ω p =0.2π ,
ω s =0.3π ,
R p =0.25 dB
A s =50dB
We provide a MATLAB script to design this filter.
>> wp = 0.2*pi; ws = 0.3*pi; Rp = 0.25; As = 50;
>> [delta1,delta2] = db2delta(Rp,As);
>> [N,f,m,weights] = firpmord([wp,ws]/pi,[1,0],[delta1,delta2]);
>> h = firpm(N,f,m,weights);
>> [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> delta_w = 2*pi/1000; wsi=ws/delta_w+1; wpi = wp/delta_w;
>> Asd = -max(db(wsi:1:501))
Asd = 47.8404
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