# How to find inverse of filter such that product of filter and its inverse results in a flat spectrum

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DSP Seeker on 26 Jul 2022
Answered: Star Strider on 26 Jul 2022
Hello All,
Suppose I have a Low pass filter which would generate a frequency spectrum of two peaks at +x and -x KHz.
Now, I have to create an opposite filter which generates dip at those frequencies so that when I pass the signal through these two filters it gives me a nearly flat spectrum.
I would want to know what is the easiest way to generate/implement this filter in MATLAB.
##### 3 CommentsShow 1 older commentHide 1 older comment
DSP Seeker on 26 Jul 2022
Let me put my question in other way:
If I have a sine wave with multiple frequency components and frequency spectrum generates multiple tones, lets say 1, 2 ,4 KHz.
If I have to design a filter which would generate dips at the above mentioned frequencies so that when the sine wave is passed through this filter, it gives nearly flat spectrum.
What is the easy way to do this in MATLAB?
Walter Roberson on 26 Jul 2022
If the frequencies are not at regular intervals, band notch filter. iirnotch
If the frequences are at regular intervals, comb filter. iircomb

Bruno Luong on 26 Jul 2022
Edited: Bruno Luong on 26 Jul 2022
If you are working if IIR filter faminy then the answer is trivial yes, since the inverse of the franfert function if a fraction of two polynomials
H(s) = Y(s)/X(s)
is just the
Hi(s) := 1/H(s) = X(s)/Ys(s)
When you apply both filters succesive you'll get the identity. So write in differnce equation recusive form you simply need to swap a and b, the coefficients of the original filter.
The problem is you might get unstable filter. But that is life you cannot do whatever you want.
DSP Seeker on 26 Jul 2022
Edited: DSP Seeker on 26 Jul 2022
Thanks for your answer Bruno. This answers my question partially. I should have mentioned my question in more depth. Let me edit it
DSP Seeker on 26 Jul 2022
Let me put my question in other way:
If I have a sine wave with multiple frequency components and frequency spectrum generates multiple tones, lets say 1, 2 ,4 KHz.
If I have to design a filter which would generate dips at the above mentioned frequencies so that when the sine wave is passed through this filter, it gives nearly flat spectrum.
What is the easy way to do this in MATLAB?

Star Strider on 26 Jul 2022
The easiest way to find out is to do the experiment —
Fs = 1E+5;
t = linspace(0, 1E+5, 1E+6)/Fs;
fc = [1 2 4]*1E+4;
s = sum(sin(fc(:)*2*pi*t));
Fn = Fs/2;
L = numel(t);
NFFT = 2^nextpow2(L);
FTs = fft(s,NFFT)/L;
Fv = linspace(0, 1, NFFT/2+1)*Fn;
Iv = 1:numel(Fv);
figure
plot(Fv, abs(FTs(Iv))*2)
grid
xlim([0 5E+3])
xlabel('Frequency')
ylabel('Amplitude')
title('Original Signal Spectrum')
fcomb = [[-9 -5 5 9]+1000, [-9 -5 5 9]+2000, [-9 -5 5 9]+4000]; % Design Multiband FIR Notch Filter
mags = [[1 0 1], [0 1], [0 1]];
dev = [[0.5 0.1 0.5], [0.1 0.5], [0.1 0.5]];
[n,Wn,beta,ftype] = kaiserord(fcomb,mags,dev,Fs);
hh = fir1(n,Wn,ftype,kaiser(n+1,beta),'noscale');
figure
freqz(hh, 1, 2^20, Fs)
set(subplot(2,1,1), 'XLim',[0 5000])
set(subplot(2,1,2), 'XLim',[0 5000])
sgtitle('Filter Bode Plot')
sf = filtfilt(hh,1,s);
figure
subplot(2,1,1)
plot(t,s)
grid
xlim([0 0.0005])
ylim([-2.5 2.5])
ylabel('Amplitude')
title('Original Signal')
subplot(2,1,2)
plot(t,sf)
grid
xlim([0 0.0005])
ylim([-2.5 2.5])
xlabel('Time')
ylabel('Amplitude')
title('Filtered Signal')
Attenuation_dB = 10*log10(sf.^2 / s.^2) % Measure Overall Result
Attenuation_dB = -56.5949
The filter meets the design requirements reasonably well, and produces an acceptable result. This filter uses the default values for a number of otherwise unspecified variables. A more specific design, for example using more options such as provided by designfilt could do even better. (I do not have the DSP System Toolbox, so I am not familiar with its functions. It has other functions and options as well.)
It is likely not possible to completely eliminate the frequencies-of-interest.
.

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