Apply frequency sampling filter on audio file

Question

I am trying to apply a double-bandstop filter using frequency sampling from a given data file. The method I am using is as follows

1. Inverse fft ( ifft ) of the given frequency sampling file

2. Circular shifting of the real values given from step 1

3. Perform an fft of the result from step 2

4. Apply the filter on the audio file using convolution. (frequency domain)

The problem is that the bandstop frequencies (925Hz & 2090Hz) still exist. Is there any problem with my code or do I miss something?

[wave,fs]=audioread('audio.WAV'); 
data=importdata ('freqSampling.txt')

y=(ifft(data,401))
x=real (y)

r=circshift (x,200)

f=fft (r,4096);

new_sound=conv (wave, f)
sound(new_sound,fs,16);

Can anyone help me with that?

Answer 1

The first two steps

1. Inverse fft ( ifft ) of the given frequency sampling file
2. Circular shifting of the real values given from step 1

should correctly give you the time-domain coefficients of a filter constructed from your specifications, provided the freqSampling.txt file correctly specifies the desired full double-sided spectrum (see "Validating specifications" below). The number of ifft points may also need to be adjusted/increased if the frequency specification contains steep transients. Performing the convolution in the frequency domain as you indicated in step 4 however does not correspond to a typical filtering operation, but would rather be equivalent to a time-domain multiplication of the two signals.

Time-domain filtering

From the result of step 2 you could filter your wave data directly in the time-domain using either conv :

new_sound = conv(r, wave);

or filter :

new_sound = filter(r, 1, wave);

conv would be giving you the full length(wave)+length(r)-1 convolution, whereas filter is a more signal processing oriented function returning the first length(wave) samples of the convolution (and can also handle recursive filters which conv does not support directly).

Frequency-domain filtering

Alternatively to perform the filtering in the frequency-domain you would

3. Perform an FFT of the result from step 2, using a size that is at least length(r)+length(wave)-1
4. Perform an FFT of the wave data, using the same size as in step 3
5. Apply the filter on the audio file using multiplication (frequency domain).
6. Compute the inverse FFT ( ifft ) of the result from step 5
7. Take the real part of the result from step 6 (technically not required, but often needed due to small numerical round-off errors)

This can be implemented using the following:

N = length(wave)+length(r)-1;
wave_fd = fft(wave, N);              % step 3
filter_fd = fft(r, N);               % step 4

filtered_fd = wave_fd .* filter_fd;  % step 5
new_sound = real(ifft(filtered_fd)); % step 6 & 7

Note that you could also perform this frequency-domain filtering operation in smaller chunks using the overlap-add method .

Validating specifications

Based on your comments, the data imported from the freqSampling.txt file could be reconstructed with:

N = 401;
data = ones(N,1);
data(19:23) = [2 1 0 1 2]/3;
data(51:56) = [2 1 0 1 2]/3;
data(N-[2:(N+1)/2]+2) = data([2:(N+1)/2]);

To validate that this would filter the desired frequencies, we can plot this specification as a function of frequency. To do this we would need the sampling rate used ( fs ), which seem to be 22050 according to your graph. You should then be able to plot these with:

hold off; plot([0:N-1]*fs/N, data);
hold on;  plot([925 925;2090 2090]', [0 1.2;0 1.2]', 'k:');
axis([0 3000 0 1.2]);
xlabel('Frequency (Hz)');
ylabel('Amplitude');
legend('Specs', 'Tones');

Which should give a plot that looks like:

Based on this, it seems that the specifications do not provide any attenuation at the tone frequencies. A better fit could be constructed with:

N = 401;
data = ones(N,1);
data(round(925*N/fs)+1+[-2:2]) = [2 1 0 1 2]/3;  % data([16:20])
data(round(2090*N/fs)+1+[-2:2]) = [2 1 0 1 2]/3; % data([37:41])
data(N-[2:(N+1)/2]+2) = data([2:(N+1)/2]);

Yielding a validation plot that looks like:

PS: based on your signal's frequency-domain graph, it would seem that the second tone is closer to 2600Hz rather than the indicated 2090Hz.