FIR Filter Design - input parameters conversion

Question

I am new to DSP and filter design. I have developed a code in C++ to calculate FIR coefficients using Parks-McClellan algorithm. The inputs to calculations are:

1. Filter type (Low-Pass, High-Pass)
2. Passband Freq [Hz]
3. Stopband Freq [Hz]
4. Passband Ripple [dB]
5. Stopband Attenuation [dB].

I got a new requirement for inputs, namely to use: 1. Cutoff freq ([email protected]) 2. Slope [24dB/octave].

Is there a way to convert these inputs into the inputs I used in my C++ code, as listed above?

Or, more general question, how to design FIR filter with custom slope?

Thank you.

Answer 1

As I've mentioned in a comment, the Parks McClellan algorithm is usually used to design frequency selective filters with a fixed maximum stopband error, which results in an equiripple behaviour in the stopband. Note that the algorithm can in principle approximate any desired frequency response shape. However, many implementations just allow for piecewise constant desired responses (which includes all standard frequency selective filters, such as low pass, high pass, etc.).

If you want a frequency response that decays monotonically from the cut-off frequency, you probably want something like a Butterworth filter. This is an IIR filter which is usually designed via the bilinear transform from an analog prototype filter. If you want an FIR filter, then one option would be to truncate (or window) the infinite impulse response of the Butterworth filter.

In Matlab/Octave this can be easily done as follows. First design a digital Butterworth filter with the given specifications, compute its impulse response, and truncate it. Where to truncate depends on the error that you allow. For your example a third order Butterworth filter is sufficient. In the following example I assume a sampling frequency of $8\,\text{kHz}$:

[b,a] = butter(3,1.6/4);    % Butterworth IIR filter
h = impz(b,a,30);           % impulse response (30 coeffs)
f = logspace(0,3.6,400);    % log. frequency grid [1,4000]Hz
H = freqz(b,a,f*pi/4000);   % Butterworth frequency response
Hf = freqz(h,1,f*pi/4000);  % FIR frequency response
semilogx(f,20*log10(abs(H)),f,20*log10(abs(Hf)))
axis([1,4000,-80,5]), grid on

You see just a single curve because both filters (IIR and FIR) have virtually the same magnitude response.

Answer 2

To do any algorithmic FIR filter design you must specify or assume (or just take some default for) a minimum stop-band attenuation. Once you provide a realistic stopband attenuation, control the transition slope from your generator algorithm by adjusting the FIR filter length and the delta between the pass band and stop band frequencies until you reach the desired result. Perhaps iteratively.

There exist some rule-of-thumb approximations for the minimum FIR filter length that will allow a given transition slope. Perhaps that's what you are looking for?