Digital Filter Design

Digital Filter Design

Program to illustrate the design of a Butterworth bandstop filter.

% Program

% Design of a Butterworth Bandstop Digital Filter

Ws = [0.4 0.6]; Wp = [0.3 0.7]; Rp = 0.4; Rs = 50;

% Estimate the Filter Order

[N1, Wn1] = buttord(Wp, Ws, Rp, Rs);

% Design the Filter

[num,den] = butter(N1,Wn1,’stop’);

% Display the transfer function

disp(’Numerator coefficients are ’);disp(num);

disp(’Denominator coefficients are ’);disp(den);

% Compute the gain response

[g,w] = gain(num,den);

% Plot the gain response

plot(w/pi,g);grid

axis([0 1 -60 5]);

xlabel(’\omega /\pi’); ylabel(’Gain, dB’);

title(’Gain Response of a Butterworth Bandstop Filter’);

The order N of the FIR filter to meet the given specifications can be estimated using either Kaiser’s formula or Herrmann’s formula .The MATLAB function kaiord given below implements Kaiser’s formula:

function N = kaiord(Fp, Fs, dp, ds, FT)

% Computation of the length of a linear-phase

% FIR multiband filter using Kaiser’s formula

% dp is the passband ripple

% ds is the stopband ripple

% Fp is the passband edge in Hz

% Fs is the stopband edge in Hz

% FT is the sampling frequency in Hz.

% If none specified default value is 2

% N is the estimated FIR filter order

if nargin == 4,

FT = 2;

end

if length(Fp) > 1,

TBW = min(abs(Fp(1) - Fs(1)), abs(Fp(2) - Fs(2)));

else

TBW = abs(Fp - Fs);

end

num = -20*log10(sqrt(dp*ds)) - 13;

den = 14.6*TBW/FT;

N = ceil(num/den);

The function kaiserord in the Signal Processing Toolbox can also be used for estimating

the filter order using Kaiser’s formula. It can be used in one of the following forms:

[N, Wn, beta, ftype] = kaiserord(fedge, aval, dev)

[N, Wn, beta, ftype] = kaiserord(fedge, aval, dev, FT)

c = kaiserord(fedge, aval, dev, FT, ’cell’)

where FT is the sampling frequency in Hz whose default value is 2 Hz if not specified;

fedge is a vector of bandedge frequencies in Hz, in increasing order between 0 and FT/2;

and aval is a vector specifying the desired values of the magnitude response at the specified bandedges given by fedge

The MATLAB function firpmord implements the formula. It can be used in

one of the following forms:

[N,fts,mval,wgts] = firpmord(fedge,aval,dev)

[N,fts,mval,wgts] = firpmord(fedge,aval,dev,FT)

where FT is the sampling frequency in Hz whose default value is 2 Hz if not specified,

fedge is a vector of bandedge frequencies in Hz, in increasing order between 0 and FT/2;

and aval is a vector specifying the desired values of the magnitude response at the specified bandedges given by fedge. The length of fedge is 2 less than twice the length of aval and therefore must be even. dev is a vector of maximum deviations or ripples in dB allowable for each band