band limited white noise

I need to make random excitation and I usually do that by "randn" command. However, I have to make random excitation with broadband frequency components, namely to excite certain frequencies. Let's say my sampling frequency (fs) is 50Hz and I need to have random white noise in range of 0 to 8 Hz to make sure this range is adequately excited.

I appreciate if someone gives me a simple way to do that which could be applied to general cases.

Cheers

Answer 1

[Star Strider]

Create a lowpass filter with the appropriate passband and then use the filtfilt function to filter your noise signal. The output should be band-limited noise that matches your specifications. If you need help in designing and implementing your filter, my filter design procedure is here: How to design a lowpass filter for ocean wave data in Matlab?

See if this works for you:

Fs = 5E+4;                                  % Sampling Frequency (Hz)
Fn = Fs/2;                                  % Nyquist Frequency (Hz)
Fco = 8;                                    % Cutoff Frequency (Hz)
Wp = Fco/Fn;                                % Normalised Cutoff Frequency (rad)
Ws = 1.6*Wp;                                % Stopband Frequency (rad)
Rp =  3;                                    % Passband Ripple (dB)
Rs = 20;                                    % Stopband Ripple (dB)
[n,Wn]  = buttord(Wp,Ws,Rp,Rs);             % Calculate Filter Order
[b,a]   = butter(n,Wn);                     % Calculate Filter Coefficients
[sos,g] = tf2sos(b,a);                      % Convert To Second Order Section Representation

figure(1)
freqz(sos, 1024, Fs)                        % Plot Filter Response (Bode Plot)

L = 1E+6;                                   % Signal Length
noise = randn(1, L);                        % White Noise
filtered_noise = filtfilt(sos,g,noise);     % Band-Limited Noise

FTfn = fft(filtered_noise)/L;               % Fourier Transform Of Filtered Signal
Fv = linspace(0, 1, fix(L/2)+1)*Fn;         % Frequency Vector
Iv = 1:length(Fv);                          % Index Vector

figure(2)
plot(Fv, abs(FTfn(Iv)))
axis([0  50    ylim])
grid
title('Band-Limited Noise')
xlabel('Frequency (Hz)')
ylabel('Amplitude')

[Shz713]

Thanks for your input, frankly I did not understand your method (my background is not electrical at all). Is the code below right for this purpose?

   % total time steps size
dt = 0.005; % Time Step
T = 30; % Duration of Impact (sec)
t = 0:dt:T; % Time scale
n = length(t);
%
fs = 50;
d = fdesign.lowpass('Fp,Fst,Ap,Ast',1,10,0.5,40,fs);
B = design(d);
x = randn(n,1);
y = filter(B,x);
%
%saving file to txt to be imported to software for analysis
filename = 'dt0.005_30sec';
fid(1)=fopen([filename,'.txt'],'w');
    for j = 1:n
        fprintf(fid(1), '%f \t',t(j));
        fprintf(fid(1), '%f \n',y(j));
    end
fclose('all');

When I input Fp as zero, it gives error. Also, I am not sure about Ap and Ast values what should be for my case (here I used Help example values).

Thanks again

[Shz713]

One more thing, what if I simply multiply the randn output by let's say 0.05? Then all values reduced by that amount. Does this make sense?

[Star Strider]

My pleasure.

I generally don’t use the fdesign functions, preferring to design my own filters. (I misread your question, so change Fs=50; and it produces the desired result.)

The filter you created works as well as my design. If you use the Fs option, you can enter the frequencies in Hz. Instead of zero, use the eps function and your filter has a low-frequency cutoff that is essentially zero:

d = fdesign.lowpass('Fp,Fst,Ap,Ast',eps,10,0.5,40,fs);

Multiplying the randn output by 0.05 does not reduce its amplitude. It creates a normally-distributed random number sequence with a standard deviation of 0.05.

[Shz713]

Thanks again for your insight.

Lastly, May I know about 'Ap' and 'Ast' appropriate values for general cases? Are the values I employed adequate? I merely need this type of excitation to use it for ambient excitation of structures such as bridge. That's the reason I don't want to make it too detailed in electrical part.

Cheers

[Star Strider]

As always, my pleasure.

'Ap' (passband ripple) is only important in some filter designs, such as Chebyshev. It does not apply to maximally-flat designs such as Butterworth. It depends on what you’re using the filter for. To create band-limited noise, a higher value is not likely to affect the result significantly. That is the reason I usually use a Chebyshev Type II filter (flat passband with ripple in the stopband) for filtering signals, instead of a Type I that has the opposite characteristic. I usually use a 1 dB passband ripple for Type I Chebyshev designs for that reason.

'Ast' (stopband attenuation) of 40 dB is appropriate in your application, since you don’t want noise outside the passband. It could even be greater. As a general rule, like the ripple specification, it depends on what you’re doing. If you’re filtering two narrowly-separated signals (you’re not here), a higher attenuation is preferable because you also want a steeper ‘roll-off’ (passband-to-stopband) characteristic.

If you are exciting a structure such as a bridge, I would record everything, then take an fft or a spectrogram (if the frequencies differ over time) of the signal to see where the most prominent frequencies are, then design discrete filters to pass those frequencies and reject the noise before doing the analysis on the filtered signals.

[Shz713]

Heaps of thanks and Merry Xmas

Peace

[Star Strider]

As always, my pleasure. You, too.

Definitely Peace!