Question

Determine whether the following discrete-time signals are periodic or not?

For the periodic ones, find their fundamental period.

a. X1 [n] =cos(0.3mn +/4)

b. x2[n] = ejn/16

c. X3[n] = 8sin (in/3) + 4cos(2 /7) - 2cos(in/9)

Answer 1

Answer:

The three of them are periodic discrete-time signals. Their respective

fundamental periods are:

$T_1=\frac{2 \pi}{0.3m}\\T_2=32\pi\\T_3=\frac{6\pi}{i}$

Step-by-step explanation:

If a discrete-time signal X is a periodic one, it must exist a non-zero number T such that for any t time given X(t) is equal to X(t+T). The minimum number T that meets this equation is called a period. Let's test each signal.

Signal a:

Is there any T that X1[n]=X1[n+T]?

cos(0.3mn +1/4) = cos(0.3m[T+n] +1/4) =

cos(0.3mn +1/4) = cos(0.3mT +0.3mn] +1/4)

If 0.3mT=2π both sides of the equation will be balanced. Therefore T exists, this is a periodic discrete-time signal and its period is:

$T=\frac{2 \pi}{0.3m}$

Signal b:

$X_2[n]=e^{j\frac{n}{16}}=cos(\frac{n}{16})+jsin(\frac{n}{16})$

This is a sum of complex sinusoidal functions with the same frequency, therefore this is a periodic discrete-time signal and its period is:

$X_2[n]=X_2[n+T]$

$e^{j\frac{n}{16}}=e^{j\frac{n+T}{16}}$

$e^{j\frac{n}{16}}=e^{j(\frac{n}{16}+\frac{T}{16})}$ if $\frac{T}{16}=2\pi$

T=32π

Signal c:

This is a sum between sinusoidal functions with the different frequencies and a constant (4cos(2 /7) is a number). The constant don´t affect the fact whether this is a periodic signal or not, so we don´t need to consider it.

For the two sinusoidal functions, their frequencies are different but these are multiples of each other and both of them have a null phase, so the period of the function with the lowest frequency will be the period of the function X3[n]. Therefore this is a periodic discrete-time signal and its period is:  

$8sin(\frac{i}{3}n)=8sin(\frac{i}{3}(n+T))$ if $\frac{iT}{3}=2\pi$

$T=\frac{6\pi}{i}$