any rules of thumb how to smooth FFT spectrum to prevent artifacts when hand-tweaking?

Question

I've got a FFT magnitude spectrum and I want to create a filter from it that selectively passes periodic noise sources (e.g. sinewave spurs) and zero's out the frequency bins associated with the random background noise. I understand sharp transitions in the freq domain will create ringing artifacts once this filter is IFFT back to the time domain... and so I'm wondering if there are any rules of thumb how to smooth the transitions in such a filter to avoid such ringing.

For example, if the FFT has 1M frequency bins, and there are five spurs poking out of the background noise floor, I'd like to zero all bins except the peak bin associated with each of the five spurs. The question is how to handle the neighboring spur bins to prevent artifacts in the time domain. For example, should the the bin on each side of a spur bin be set to 50% amplitude? Should two bins on either side of a spur bin be used (the closest one at 50%, and the next closest at 25%, etc.)? Any thoughts greatly appreciated. Thanks!

Answer 1

I like the following method:

• Create the ideal magnitude spectrum (remembering to make it symmetrical about DC)
• Inverse transform to the time domain
• Rotate the block by half the blocksize
• Apply a Hann window

I find it creates reasonably smooth frequency domain results, although I've never tried it on something as sharp as you're suggesting. You can probably make a sharper filter by using a Kaiser-Bessel window, but you have to pick the parameters appropriately. By sharper, I'm guessing maybe you can reduce the sidelobes by 6 dB or so.

Here's some sample Matlab/Octave code. To test the results, I used freqz(h, 1, length(h)*10);.

function [ht, htrot, htwin] = ArbBandPass(N, freqs)
%# N = desired filter length
%# freqs = array of frequencies, normalized by pi, to turn into passbands
%# returns raw, rotated, and rotated+windowed coeffs in time domain

if any(freqs >= 1) || any(freqs <= 0)
    error('0 < passband frequency < 1.0 required to fit within (DC,pi)')
end

hf = zeros(N,1); %# magnitude spectrum from DC to 2*pi is intialized to 0
%# In Matlabs FFT, idx 1 -> DC, idx 2 -> bin 1, idx N/2 -> Fs/2 - 1, idx N/2 + 1 -> Fs/2, idx N -> bin -1
idxs = round(freqs * N/2)+1; %# indeces of passband freqs between DC and pi
hf(idxs) = 1; %# set desired positive frequencies to 1
hf(N - (idxs-2)) = 1; %# make sure 2-sided spectrum is symmetric, guarantees real filter coeffs in time domain
ht = ifft(hf); %# this will have a small imaginary part due to numerical error
if any(abs(imag(ht)) > 2*eps(max(abs(real(ht)))))
    warning('Imaginary part of time domain signal surprisingly large - is the spectrum symmetric?')
end
ht = real(ht); %# discard tiny imag part from numerical error
htrot = [ht((N/2 + 1):end) ; ht(1:(N/2))]; %# circularly rotate time domain block by N/2 points
win = hann(N, 'periodic'); %# might want to use a window with a flatter mainlobe
htwin = htrot .* win;
htwin = htwin .* (N/sum(win)); %# normalize peak amplitude by compensating for width of window lineshape