How to apply a bandstop filter after performing FFT

How to apply a bandstop filter after performing FFT on the this data, to remove the spikes ? SC is the data matrix.

rows1 =length(SC(:,2));
incr1=1/(rows1+1)*12;
fmax1=12*(1-1/(rows1+1));
X1=[0:incr1:fmax1/2]';
Y1 = abs(fft(SC(1:rows1,2),rows1));

[Walter Roberson]

Do not do the abs() at this stage.

Using the sampling frequency and the number of points to figure out the resolution of each bin of the fft result. https://www.researchgate.net/post/How-can-I-define-the-frequency-resolution-in-FFT-And-what-is-the-difference-on-interpreting-the-results-between-high-and-low-frequency-resolution

Now, knowing that, and knowing that the first bin is "0 Hz", you can figure out which bins in the fft cover the frequencies you want to stop. You can now zero those locations or multiply them by an fraction to diminish them; you could use a curve to diminish more towards the center of the group if you wanted. After that, take the indices of those bins, subtract 1, and apply the same curve in reverse to the bins starting from the end of the fft result. For example, you might effectively multiply the fft result by

1 1 .7 .3 .1 .6 1 1 1 [...] 1 1 1 .6 .1 .3 .7 1

Now you can ifft back.

[Joydeb Saha]

rows1 =length(SC(:,2));
incr1=1/(rows1+1)*12;
fmax1=12*(1-1/(rows1+1));
X1=[0:incr1:fmax1/2]';
Y1 = abs(fft(SC(1:rows1,2),rows1));
Y1=Y1/rows1;

for i=1:rows1/2
    per1(i,1)=1/X1(i+1,1);
end
% plot(per,Y(2:(rows/2+1),1),'marker', '*');
dd1=max(Y1(2:(rows1/2+1),1));
ddd1=(Y1(2:(rows1/2+1),1))/dd1;
plot(per1,ddd1,'marker', '*');

This is the code I am using. We need to apply a bandstop filter after performing FFT on the data (attaching the main matrix as 'SC'), to remove the spikes with a narrow bandpass filter.

After applying the bandstop filter we need to apply an inverse FFT algorithm to recreate the time-series data (filtered) from the frequency domain to the time domain.

So how would be the code?

[Walter Roberson]

Why is it mandatory to take abs() of the fft() ?

When you use abs() of the fft() then in order to get real-valued components back after the ifft, then you have to force the second half of the spectra to be the complex conjugate mirror of the first half... except you don't even know what phase to use.

For example:

t = linspace(0,1,100);
y = sin(2*pi*5*t + pi/7) + sin(2*pi*3*t + pi/11);
plot(t,y)

Y1 = abs(fft(y));
Y2 = ifft(Y1);
whos Y2

  Name      Size             Bytes  Class     Attributes

  Y2        1x100              800  double              

plot(t, Y2)

Wrong magnitude, wrong phase.

[Joydeb Saha]

OK, I understood. But how to use the bandstop filter after the fft on this data.

[Walter Roberson]

Example:

data = randn(1, 50);
plot(data)
title('original data');

Y1 = fft(data);
bins_of_interest = 5:22;
damping = exp(-(1:length(bins_of_interest)));

Y1(bins_of_interest) = Y1(bins_of_interest) .* damping;
Y1(end+2-bins_of_interest) = conj(Y1(bins_of_interest));
back_data = ifft(Y1);
whos back_data

  Name           Size            Bytes  Class     Attributes

  back_data      1x50              400  double              

plot(back_data)
title('reconstructed')

Your bins_of_interest should be chosen according to the sampling frequency and the number of points, and which frequencies you want to filter out; one of the links I posted shows how "wide" each bin is. Your "damping" could be constant or could be any curve that is appropriate for your circumstances, in recognition that you might not be wanting to filter each frequency equally.