# Generation of 1/f noise using Matlab.

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Massilon Toniolo da Silva on 14 Jun 2017
Commented: Star Strider on 3 Oct 2023
Dear Colleagues, I have been trying to generate the 1/f noise, where f means frequency. I would appreciate any help and guidance. Kind regards,
Massilon

Star Strider on 14 Jun 2017
Probably the easiest way is to create a FIR filter that has a ‘1/f’ passband, then filter random noise through it:
fv = linspace(0, 1, 20); % Normalised Frequencies
a = 1./(1 + fv*2); % Amplitudes Of ‘1/f’
b = firls(42, fv, a); % Filter Numerator Coefficients
figure(1)
freqz(b, 1, 2^17) % Filter Bode Plot
N = 1E+6;
ns = rand(1, N);
invfn = filtfilt(b, 1, ns); % Create ‘1/f’ Noise
figure(2)
plot([0:N-1], invfn) % Plot Noise In Time Domain
grid
FTn = fft(invfn-mean(invfn))/N; % Fourier Transform
figure(3)
plot([0:N/2], abs(FTn(1:N/2+1))*2) % Plot Fourier Transform Of Noise
grid
It uses the firls function to design a FIR filter that closely matches the ‘1/f’ passband. See the documentation on the various functions to get the result you want.
Note: The filter is normalised on the open interval (0,1), corresponding to (0,Fn) where ‘Fn’ is the Nyquist frequency, or half your sampling frequency. It should work for any sampling frequency that you want to use with it.
This should get you started. Experiment to get the result you want.
Antonio D'Amico on 26 Aug 2020
Hello, thank you for your answer. If I understand the script correctly, it applies a 1/f (approximation) roll-off factor to the noise, whether it is uniformilly distributed (rand) or gaussian (randn). However what I would like to achieve is something like (From Wikimedia Commons)
I hope I was clearer
Antonio D'Amico on 26 Aug 2020
Edited: Antonio D'Amico on 26 Aug 2020
Ok, I think I got it, something like this could work
fv = linspace(0, 1, 20); % Normalised Frequencies
a = zeros(1,20);
a(1:10) = 1./(1 + fv(1:10)*2); % Amplitudes Of 1/fv until 0.5
a(11:20) = a(10); % after 0.5 it gets flat
b = firls(42, fv, a); % Filter Numerator Coefficients
figure(1)
freqz(b, 1, 2^17) % Filter Bode Plot
N = 1E+6;
ns = randn(1, N);
invfn = filtfilt(b, 1, ns); % Create ‘1/f’ Noise
figure(2)
plot([0:N-1], invfn) % Plot Noise In Time Domain
grid
FTn = fft(invfn-mean(invfn))/N; % Fourier Transform
figure(3)
plot([0:N/2], abs(FTn(1:N/2+1))*2) % Plot Fourier Transform Of Noise
grid

Ali Mostafa on 11 Jun 2018
f=0:1/fs:1-1/fs;S=1./sqrt(f); S(end/2+2:end)=fliplr(S(2:end/2)); S=S.*exp(j*2*pi*rand(size(f))); plot(abs(S)) S(1)=0;figure;plot(real(ifft(S)))
Massimo Ciacci on 10 Aug 2019
Quite ingenious to put the randomness in the phase, and this way the amplitude profile is exact, without the need to average out a lot of noise realizations. Thumbs up!
XIAOHUA HUA on 11 Mar 2020
Great, thank you very much for sharing this.

James on 3 Oct 2023
Hi were does 1./(1 + fv*2) come from?
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James on 3 Oct 2023
is there any paper or book I could look at to undestand that a bit more, or is this based on your own experience/skill?
Thank you very much for your response!
Star Strider on 3 Oct 2023
It’s entirely my own experience. I remember learning about noise in graduate school, in the context of biomedical instrumentation. I’m certain there must be more recent discussions of it, however I have no specific references.