Scaling of Filter Coefficients in "fir1" function

Hello Everyone,

I had a quick question.

I was looking through the "fir1" function's implementation to see how the coeffs are being scaled. I came across this function:

function b = scalefilter(b,First_Band,ff,L)
%SCALEFILTER   Scale filter to have passband approx. equal to one.

if First_Band
    b = b / sum(b);  % unity gain at DC
else
    if ff(4)==1
        % unity gain at Fs/2
        f0 = 1;
    else
        % unity gain at center of first passband
        f0 = mean(ff(3:4));
    end
    b = b / abs( exp(-1i*2*pi*(0:L-1)*(f0/2))*(b.') );
end

end

Since there is no documentation for this particular line, I was curious to know where the following equation comes from:

scalingFactor = 1 / abs( exp(-1i*2*pi*(0:L-1)*(f0/2))*(b.') );

Does anyone know?

Answer 1

[Paul]

For a FIR filter

H(z) = b0 + b1*z^-1 + b2*z^-2 .... + b_n*z^(n-1)

its frequency response is

H(w) = b0 + b1(exp(-1j*w) + b2*exp(-1j*w)^2 + .... b_n*exp(-1j*w)^(n-1)

where w is has units of rad/sample.

The equation of interest is computing abs(H(w0)) where w0 (rad/sample) is the middle of pass band. Recall that on input to fir1 the frequencies are normalized such that Wn = 1 -> w = pi. So if f0, which is derived from Wn, is the middle of the pass band, it's related to w0 by: w0 = pi*f0. I'm not sure why it's coded with a mutliply by 2 followed by a divide by 2. I guess it can be viewed as f0/2 converts the input frequency to cycles/sample and then the multiply by 2*pi converts to rad/sample.

[Farshad Bolouri]

Thank You @Paul !

Makes sense now.