Design single pass band FIR filter

Question

I'm new in DSP. I want to design single pass band FIR filter using LS method. I have this information :

• one pass band $[-0.1\pi , 0.3\pi]$

• two stop bands $[-\pi , -0.18\pi]$, $[0.38\pi, \pi]$

• weighting constant in :

  pass band : $a = 1$ and stop band $b_1=b_2=2$.

In this case, can I use Matlab's firls function? Because we have negative frequency and f is a vector of pairs of frequency points, specified in the range between $0$ and $1$.

Answer 1

As mentioned in a comment, you can't use firls directly because this function only designs real-valued filters with (conjugate) symmetrical frequency responses. Your specification can only be met by a complex-valued filter because the frequency domain specification is not symmetrical.

In principle, you could use firls to design the real part and the imaginary part of the impulse response separately (by using the even and odd parts, respectively, of the frequency domain specification as the desired responses), but in that case you would get discontinuous desired responses, which would result in large approximation errors.

A better and simpler approach is to set up an overdetermined system of complex linear equations and use Matlab/Octave to solve them in a least squares sense. This is very straightforward if you use matrix commands:

N = 51;     % desired filter length
% frequency grid, desired frequency response, weighting function
f = [linspace(-1,-.18,164),linspace(-.1,.3,80),linspace(.38,1,124)];
d = [zeros(1,164),ones(1,80),zeros(1,124)].*exp(-1i*pi*f*(N-1)/2);
w = [2*ones(1,164),ones(1,80),2*ones(1,124)];
f = f(:); d = d(:); w = w(:);

% set up and solve overdetermined linear system
A = w(:,ones(1,N)) .* exp(-1i*pi*f*(0:N-1) );
h = A \ (w.*d);

The resulting filter is complex-valued and has a linear phase response, i.e., the real part of the impulse response is even, and the imaginary part is odd (see figure):

You can further decrease the approximation error by chosing a larger filter length.

I also have a corresponding Matlab/Octave function on GitHub: cfirls.m.