Why is the number of periods of a cosine wave in the plot changing with the number of samples ?

Hello,

I am trying to plot a cosine wave of frequency 7.9 GHz on Matlab, from time interval - 8 to 8 so I use "linspace" to define my time vector.

However I noticed, If I change the number of points (or samples) in my "linspace" vector, the number of periods of the cosine I see in the plot changes. I don't understand why this number of periods would change when it's the same time interval, any idea on why this happens ?

Please see below my code for both cases (samples number = 400 and 500) and the plots that I get!

Thank you in advance for your help,

f=7.9*10^9; % frequency of my cosine 7.9 GHz 
w=2*pi*f;
t=linspace(-8,8,400); % time interval with 400 is the number of samples that I change 
y=cos(w*t);
plot(t,y)
grid on

t2=linspace(-8,8,500); % I change from 400 to 500 samples
y2=cos(w*t2);
plot(t2,y2)
grid on

Answer 1

[Les Beckham]

This is called aliasing. I recommend reading the linked Wikipedia article. The sample rate (in this case the number of time samples in your t vector, per second), must be at least twice the frequency of the signal in order to even approximately represent the signal. A better rule of thumb is at least 10 samples per period of the signal.

So, for a signal at approximately 8 GHz you need to sample it at around 80 GHz to get a relatively clean representation of the signal. This means that, for an 16 second time span you would need this many samples:

num_samples = 16*80e9

num_samples = 1.2800e+12

Of course that is going to take about a terabyte of memory and, if you had that much memory, your plot would look like a blue rectangle and you couldn't see the wave shape at all.

Each period of an 8 GHz sine wave is only

period = 1 / 8e9

period = 1.2500e-10

In other words, 0.125 nanoseconds.

Shorten the time span of your plot. For example:

f=7.9*10^9; % frequency of my cosine 7.9 GHz 
w=2*pi*f;
t=linspace(-8e-10,8e-10,400); % time interval with 400 is the number of samples that I change 
y=cos(w*t);
plot(t,y)
grid on

[myoui]

Thank you so much for the explanation. If I understood well, the reason why a number of 400 samples has worked correctly in t=linspace(-8e-10,8e-10,400) is because the required number of samples would be (16e-10)*80e9 which gives us 128 and 400 is higher than 128.

If I would like to change my time span to t=linspace(-8e-9,8e-9,Nsamples), then I would need (16e-9)*(80e9) = 1280 samples to not see aliasing / see my signal correctly. Is this correct? Thank you for your help!

[Les Beckham]

@myoui

That is correct, though I would recommend that as an absolute minimum if you want smooth looking plots.

If this answer solved your issue, please Accept it. Thanks.

[myoui]

I understand better, thanks a lot for your help!

[Les Beckham]

You are quite welcome.