Add complex valued bandpass filter

[zsoerenm]

This pull request allows to create ComplexBandpass filters for one sided pass bands (different pass bands for positive or negative frequencies)
I haven't implemented scalefactor(coefs::Vector, ftype::ComplexBandpass) yet, as I'm not really sure how to implement it for this case. So this currently only works for scale = false.
Also I haven't implemented tests, yet. Where do all the filter coefficients in the tests for the different filters come from? Maybe someone can help me out with this.
Here is an example:

using DSP, FFTW, Plots
signal = randn(ComplexF64, 10000)
filtered_signal = filt(digitalfilter(ComplexBandpass(-5e6, -3e6; fs = 10e6), FIRWindow(hanning(512), scale = false)), signal)
p = periodogram(filtered_signal, fs = 10e6)
Plots.plot(fftshift(freq(p)), 10 * log10.(fftshift(power(p))))

[martinholters]

Should we also put a lower bound on f? It seems that 0 <= f < 2 and -1 <= f < 1 would be reasonable limits, but depending on context, one or the other may be more useful. So require -1 <= f < 2? Looks rather arbitrary. Or maybe we put no restriction on f at all? Really, I don't know.

[martinholters]

Regarding tests, I think most of the reference data has been produced using comparable Matlab functions. Personally, I'm not too huge a fan of this. For the case at hand, I could imagine determining the frequency responses of a couple of designed filters and verifying that they fulfill the specifications (under some assumptions on what passband ripple and stopband gain can be reasonably expected).

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

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[martinholters]

I think constructor should return values of the type they're asked to construct, which appears not to be the case here.

[wheeheee]

I think this PR just needs a reference signal for the test, probably written with scipy (replacing the placeholder one I left). Not sure how the complex bandpass is supposed to work, so I'm unable to do that...

[martinholters]

LGTM modulo the minor docstring fix. Good to go then or are there any objections?

[wheeheee]

scalefactor isn't implemented for complex bandpass. Is that meaningful or is it ok as is now?

[martinholters]

Ah, good point. I think this should work:

function scalefactor(coefs::Vector{T}, ftype::ComplexBandpass, fs::Real) where T<:Number
    n = length(coefs)
    freq = normalize_complex_freq(middle(ftype.w1, ftype.w2), fs)
    c = zero(T)
    for k = 1:n
        c = muladd(coefs[k], cispi(-freq * (k - (n + 1) / 2)), c)
    end
    return abs(c)
end

[martinholters]

Trying to re-create the example from the OP, I got different results which lead me to what I suspect to be a bug in how fs is used.

[wheeheee]

Is this good to go? Or maybe an additional commit changing the bounds for f?

[martinholters]

LGTM.