It's very easy to filter a signal by performing an FFT on it, zeroing out some of the bins, and then performing an IFFT. For instance:
t = linspace(0, 1, 256, endpoint=False)
x = sin(2 * pi * 3 * t) + cos(2 * pi * 100 * t)
X = fft(x)
X[64:192] = 0
y = ifft(X)
The high frequency component is completely removed by this "brickwall" FFT filter.
But I've heard this is not a good method to use.
Zeroing bins in the frequency domain is the same as multiplying by a rectangular window in the frequency domain. Multiplying by a window in the frequency domain is the same as circular convolution by the transform of that window in the time domain. The transform of a rectangular window is the Sinc function ($\sin(\omega t)/\omega t$). Note that the Sinc function has lots of large ripples and ripples that extend the full width of time domain aperture. If a time-domain filter that can output all those ripples (ringing) is a "bad idea", then so is zeroing bins.
These ripples will be largest for any spectral content that is "between bins" or non-integer-periodic in the FFT aperture width. So if your original FFT input data is a window on any data that is somewhat non-periodic in that window (e.g. most non-synchronously sampled "real world" signals), then those particular artifacts will be produced by zero-ing bins.
Another way to look at it is that each FFT result bin represents a certain frequency of sine wave in the time domain. Thus zeroing a bin will produce the same result as subtracting that sine wave, or, equivalently, adding a sine wave of an exact FFT bin center frequency but with the opposite phase. Note that if the frequency of some content in the time domain is not purely integer periodic in the FFT width, then trying to cancel a non-integer periodic signal by adding the inverse of an exactly integer periodic sine wave will produce, not silence, but something that looks more like a "beat" note (AM modulated sine wave of a different frequency). Again, probably not what is wanted.
Conversely, if your original time domain signal is just a few pure unmodulated sinusoids that are all exactly integer periodic in the FFT aperture width, then zero-ing FFT bins will remove the designated ones without artifacts.
This question has also confused me for a long time. @hotpaw2's explanation is good. You may be interested in the simple experiment using matlab.
https://poweidsplearningpath.blogspot.com/2019/04/dftidft.html
updated information.
To verify this fact is simple, we just need to cautiously observe the spectrum of impulse response of an ideal(?) band pass filter which just zeros out FFT bins. Why do I need to add the adverb "cautiously"? If we just use the same size of the FFT to observe the response of the impulse, we will be deceived as shown in Fig 1. Nonetheless, if we add the order of DFT when observing the output of the filter, that is, zero padding the impulse response, we can find the so called Gibbs phenomenon, ripples in frequency domain, as depicted in Fig 2.
The results in fact comes from the windowing effect. If you want to entirely understand the problem, please refer to chapter 7.6 and chapter 10.1-10.2 of the bible of DSP (1). To sum up, three key points are noted here.
So, with the help of denser spectrum in Fig.2, we can see through the mask of ideal(fake) Band pass filter.
Deceitfully Freq. Response.
Gibbs phenomenon in Freq. Response.
(1) Alan V. Oppenheim and Ronald W. Schafer. 2009. Discrete-Time Signal Processing (3rd ed.). Prentice Hall Press, Upper Saddle River, NJ, USA.
fps = 15;
LPF = 1;
HPF = 2;
n = -511:512;
n0 = 0;
imp = (n==n0);
NyquistF = 1/2*fps;
%% Ideal BPF
tmp_N = 512;
tmp_n = 0:1:tmp_N-1;
freq = ( n .* fps) ./ tmp_N;
F = fft(imp, tmp_N);
F_bpf = IdealBandpassFilter(F, fps, LPF, HPF);
imp_rep =[real(ifft(F_bpf))'];
% Zero padding.
imp_rep2 =[zeros(1,2048) real(ifft(F_bpf))' zeros(1,2048)];
N = 2^nextpow2(length(imp_rep));
F = fft(imp_rep,N);
freq_step = fps/N;
freq = -fps/2:freq_step:fps/2-freq_step;
freq = freq(N/2+1:end)';
figure;
plot(freq,abs(F(1:N/2)));
xlabel('freq(Hz)');
ylabel('mag');
title('Mis leading Freq Response');
N = 2^nextpow2(length(imp_rep2));
F = fft(imp_rep2,N);
freq_step = fps/N;
freq = -fps/2:freq_step:fps/2-freq_step;
freq = freq(N/2+1:end)';
figure;
plot(freq,abs(F(1:N/2)));
xlabel('freq(Hz)');
ylabel('mag');
title('Zero Padding (DFT) with more points');
%% Function
function filered_signal = IdealBandpassFilter(input_signal, fs, w1, w2)
N = length(input_signal);
n = 0:1:N-1;
freq = ( n .* fs) ./ N;
filered_signal = zeros(N, 1);
for i = 1:N
if freq(i) > w1 & freq(i) < w2
filered_signal(i) = input_signal(i);
end
end
end
FFT gives poor time resolution i.e it doesn't give information at what time that particular frequency exist. It gives information on existing frequency components for given signal duration.
By zeroing bins in FFT gives poor resolution after IFFT in time domain.
Zeroing is a good idea if
1 is actually optimal and is what's proven by the sampling theorem. It's not a "no-op" in that it could be a step in decimation - then doing nothing corresponds to subsampling. 2 works because the DFT is stable in sense of norms, roughly meaning small change in one domain $\Leftrightarrow$ small change in other. These are important to note because any kind of filtering is, in almost all cases, a deformation, and there's many ways to filter wrong.
The next-best alternative is what's flat in frequency domain over frequencies of interest, so freqs aren't scaled relative to each other - but the filter's time-domain behavior also matters, so Layman's safest is to go with known filters, e.g. Hann-windowed sinc. Since it's convolution, choice of padding also becomes important.
Why it's bad in general: refer to hotpaw's answer.
