Question

hey hey! mechanical engineering student who is VERY confused and has absolutely no idea what any of this means but really wants to know. could you please help me! this is the question. given M=4,5,7,10 and N=12.

Question 1: Explain how fundamental period of discrete-time sinusoidal signal can be calculated and calculate fundamental period for four cases in exercise

**100% RATING WILL BE GIVEN IF CLEAR, CORRECT ANSWERS FOR EACH FOUR CASES OF M ARE GIVEN

this is my matlab code for the M =5 value.

N=12;
n=0:1:N-1; %n is defined with step 1
M=5;
x=sin(2*pi*M.*n./N);%defines dependant variable x[n]
[~,T0]= rat(M/N) %using rational approximation the fundamental period is calculated.
%nosemicolon means matlab will display result.
stem(n,x) %using stem the discrete signal is drawn
xlabel ('n');
ylabel('y[n]');
title ('discrete sinusoidal signal example');

Answer 1

Consider the function

In continuous time signals, for a sinusoidal signal of the form

the period is simply given by

However, this is not the same for discrete time signals as the period must be an integer for discrete time signals.

Here we have,

So.

The function is periodic if there exists an integer such that,

Therefore,

This is the same if

is a multiple of

or , where is some integer

If can be represented in such form, then the function is indeed periodic and with period

1. for

. Clearly, this is true for

Therefore, the fundamental period is 3

2. for

or . Since, we know that is an integer, we must have

Therefore, the fundamental period is 12

3.

Similar to above observation, the fundamental period is 12

4.

Here, . For to be an integer, the minimum value of .

Therefore, the fundamental period is 6.