How to insert noise in a signal using matlab?

Hi. I want to create a signal with noise. I found this coding but I don't know how it works. Can somebody explain this to me?

x= cos(2*pi*12*[0:0.001:1.23]);
x(end) = [];
[b a] = butter(2,[0.6 0.7],'bandpass');
filtered_noise = filter(b,a,randn(1, length(x)*2));
x = (x + 0.5*filtered_noise(500:500+length(x)-1))/length(x)*2;

This is the result when I plot the x.

Answer 1

[Star Strider]

It creates a signal in ‘x’, then creates band-limited noise by bandpass-filtering a Gaussian random vector (created by the randn call), the same length as ‘x’, and stores it in the ‘filtered_noise’ variable. It then adds that to the original ‘x’ and scales it to create the final value of ‘x’. (There is a bit more involved in the code than I described, but that is the essence of it.)

[Nur Fauzira Saidin]

Thank you for your answer. If I'm not mistaken, the general equation for sinusoidal signal is 2*pi*f*t, so the frequency of the signal is 12 Hz. Am I right? And from the code, can I get the information about the noise? What is the frequency of the noise?

[Star Strider]

My pleasure.

If the time is in seconds, that would be correct (as I read it).

The sampling interval is Ts=0.001 (again I assume seconds), so the sampling frequency, Fs=1000 Hz, and the Nyquist frequency, Fn=500 Hz. Since the passband frequencies for the filter are normalised to Fn, the frequency bandwidth of the noise would be [0.6 0.7]*Fn, or [300 350] Hz.

[Nur Fauzira Saidin]

Again, thank you so much for your answer. I can understand it very well. How about the final amplitude? From the result, it is almost 2x10-3. I also do not understand the x-axis parameter, which is samples. How did it turn out that way?

[Star Strider]

My pleasure.

That is correct, since the amplitude is normalised by the half the length of ‘x’.

The x-axis is defined in the command that plotted the figure, so you would have to see the code. The samping interval is 1 ms, so if you want the value of the x-axis in seconds (again assuming those are the correct units), multiply the values by 0.001.

[Nur Fauzira Saidin]

The length of 'x' is 1230, so the final amplitude is 1/(1230/2) = 1.63x10-3. Okay I understand it now. Thanks! Now if you don't mind, can you help me with the next step? I want to plot the frequency spectrum of signal 'x'.

%plot magnitude spectrum of the signal
X_mags = abs(fft(x));
figure,plot(X_mags);
xlabel('DFT Bins');
ylabel('Magnitude');

Then, this code is to plot the first half of DFT. I don't quite understand this part. Can you explain this to me?

%plot first half of DFT (normalised frequency)
num_bins = length(X_mags);
figure,plot([0:1/(num_bins/2 -1):1], X_mags(1:num_bins/2));
xlabel('Normalised frequency (\pi rads/sample)');
ylabel('Magnitude');

[Star Strider]

That appears to be correct.

If you want the frequency in Hz (again assuming the time unit is seconds), multiply the normalised frequency by Fn, or 500 Hz.