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How to determine if the equation is a periodic signal?

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sky_tm

Junior Member level 1
(2What are periodic signals? How to determine if the equation is a periodic signal?

Find the fundamental period:
i) x[n] = sin(2n)
ii) x(t) = cos³(4t)
iii) $x(t) = 1 + \sin^2(2 \pi t)$

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Re: Periodic Signals

Periodic signals are signals that repeat of after a fixed time.

for example sin(x) is periodic with period 2*pi.

to prove a signal is periodic, you need to show

that f(x) = f(x+T) for some T for all x.

sin(x) = sin(x+2*pi) so let T = 2*pi .
this proves sin(x) is periodic.

x[n] is probably not periodic. why ?

I'll let you work out your own hw problems. good luck.

Re: Periodic Signals

Note that

For a discrete-time signal to be periodic it has to satisfy
$x[n+N]=x[n]$ where $N$ is the fundamental period and the condition on it is that it should be an integer.

For a continuous-time signal to be periodic it has to satify
$x(t+T)=x(t)$ where $T$ is the fundamental period and there is no restriction on this as in the case of DT signal.

Coming to the given problems

(i) The given signal is $x[n]=\sin(2n)$. For it to be periodic it has to satisfy the following equation $\sin(2n)=\sin(2n+2N)$ for integer values of $N$. Note that no value of $N$ will satisfy the equation. Thus it is aperiodic.

(ii) The given signal is $x(t)=\cos^3(4t)$. First let us simplify it in to the form $x(t)=\frac{1}{4}(\cos(3t)+3\cos(t))$. So for this to be periodic it has to satisfy the following equation.$x(t+T)=x(t)$

Thus $\cos(3t)$ is periodic with period $T_1 = 2\frac{\pi}{3}$ and $\cos(t)$ is periodic with period $T_2 = 2\pi$. Thus total period of the signal is (T = LCM($2\frac{\pi}{3},2\pi)$. This turns out to be
$T = 2\pi$

(iii) The given signal is $x(t)=1+\sin^2 (2\pi t)$. This can be written as $x(t)=\frac{3}{2}-\frac{1}{1}\cos(4\pi t)$. The period of $\cos(4\pi t)$ is $T_1 = \frac{1}{2}$. Thus total period is $T = \frac{1}{2}$.

thnx

purna!

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yousufsaleem

yousufsaleem

Points: 2
Re: Periodic Signals

Conclusion:

i) period is $\frac{1}{\pi }$ , therefore it is aperiodic

ii) period is $\frac{\pi }{2}$

iii) period is $\frac{1}{2}$

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