Periodicity of Discrete and Continuous-Time Signals
Published:
Part 2
For the signals given below, determine whether they are periodic. For the periodic signals, find their periods.
$$x_1[n] = \cos\left(\frac{8}{15}\pi n\right)$$
$$x_2(t) = \cos(2t) + \sin(3t)$$
$$x_3[n] = \sum_{k=-\infty}^{\infty} \{\delta[n - 3k] + \delta[n - k^2]\}$$
$$x_4(t) = \cos(t)u(t)$$
$$x_5(t) = v(t) + v(-t), \quad \text{where } v(t) = \cos(t)u(t)$$
Animated Video Solution
The first half plays free, the full solution is in the app.
Step by Step Written Solution
In this problem, we need to determine if each signal is periodic and find the fundamental period for those that are.
Periodicity of Signals
Let's begin with the first discrete-time signal, x sub one of n.
For a discrete-time sinusoid to be periodic, the frequency omega divided by two pi must be a rational number.
Here, omega is eight pi over fifteen. Dividing by two pi gives us four fifteenths.
Since four fifteenths is rational, the signal is periodic. The fundamental period N is the smallest integer that makes m an integer, which is fifteen.
Now let's look at the second signal, which is continuous-time. It's a sum of two cosines.
The frequencies are omega one equals two and omega two equals three. Their periods are pi and two pi over three, respectively.
A sum of periodic signals is periodic if the ratio of their periods is a rational number.
Three halves is rational, so the signal is periodic. The fundamental period is the least common multiple of the individual periods.
So, the fundamental period for x sub two is two pi.
Next is x sub three of n. This is a sum of impulses.
The rest of this solution is on Solvi
10 more steps are locked. Watch the full animated, narrated solution for free.
Snap a photo, solve any question like this.
Watch the Rest for Free
Free to download · First solutions are on us
