help me to find the last required in this question please

I have this question ... I solved the first three required

but I dont know what to do with the last one ( last required )

please help me to solve the last one

x1=sin(10.*pi.*t)

x2=(1/3)*sin(30.*pi.*t)

x3=(1/5)*sin(50.*pi.*t)

A. Generate and plot x1(t) for one period.

B. Generate and plot xb(t)=x1(t)+x2(t) for one period.

C. Generate and plot xc(t)=x1(t)+x2(t) +x3(t) for one period. Show all the results on one figure using subplot

D. Determine, using Matlab plots, if the generated signals are periodic or not.

syms x1 x2 x3 m n t;
x1=sin(10.*pi.*t);
x2=(1/3)*sin(30.*pi.*t);
x3=(1/5)*sin(50.*pi.*t);
xb=x1+x2;
xc=x1+x2+x3;

%The period of signals, x1, xb and xc, is 1/5
% (5 is the frequency for x1)

subplot(2,2,1);
fplot(x1,[0 0.2]);
xlabel('t');
ylabel('x1(t)');
title ('x1=sin(10.*pi.*t)');

subplot(2,2,2); 
fplot(xb,[0 0.2]);
xlabel('t');
ylabel('xb(t)');
title ('xb= x1+x2');

subplot(2,2,[3 4]); 
fplot(xc,[0 0.2]);
xlabel('t');
ylabel('xc(t)');
title ('xc=x1+x2+x3');

Answer 1

[Sulaymon Eshkabilov]

Here is a complete answer to your exercise:

https://www.mathworks.com/matlabcentral/answers/618588-determine-whether-or-not-the-following-signals-are-periodic

Answer 2

[Walter Roberson]

A signal is periodic if at some time t, it starts to repeat the same values, with time t*2 being another repeat and so on.

A sine wave with time 2*pi*F1 repeats itself F1 times per second. A sine wave with time 2*pi*F2 repeats itself F2 times per second. At time F1*F2 with F1 and F2 integer, you will have repeated the first signal due to F1, and the second signal due to F2. Therefore sin(2*pi*F1*t) + sin(2*pi*F2*t) will repeat itself no later than t = F1*F2 and must be periodic.

You know have a periodic signal based upon the sum of the first two signals, and you have a third signal that is another sine wave. By the same reasoning as before, there exists some time for which the composite signal and the third signal both repeat. Therefore the sum of the three signals is periodic.

This logic can be made more rigourous. The actual time required is much less than hinted at above. The logic does not actually require that F1, F2, F3 all be integers: it does require, though, that F1, F2, F3 all be rational (unless... well, I have to leave some of the reasoning to you.)

The question only asks whether the signal is periodic. You are not required to figure out the period.

Answer 3

[Chunru]

Changing the plotting time from 0.2 to a larger value (for example, 1), Matlab plots will give you the visual ouptput to show if the signal is periodic or not (signal repeating itself or not).

[Walter Roberson]

Not really. Consider a system something like sin(2*pi*100*(t+exp(-1000*t.^2)) . Unless you look at the plot pretty close to t=0 then the exp() part would contribute less than floating point round-off and visually you would get the impression that it was periodic, but it is not.

You cannot count on graphs to determine periodicity: you have to prove it mathematically.

[Chunru]

It's science you prove the periodicity mathematically. It's engineering that you oberve the result visually (anyway, it's the requirement of the original question).

[Walter Roberson]

If you were to rewrite from

x1=sin(10.*pi.*t);

to

x1=sinpi(10.*t);

and likewise for x2, x3, then you could show peridocity graphically by determining the period, P, and graphing x(t) - x(t+N*P) for integer N (preferably larger N). The fact that the difference was 0 to within round-off would be good evidence, especially if you tried several different random N (if you only tried one fixed N, then you could be accused of having accidentally or deliberately referred to a section of signal that just looked similar to the original signal even though the signal was (hypothetically) not really periodic.)