Answer to Question #129881 in Electrical Engineering for Md.Mostafijur Rahman

Question

Determine whether the following signals are periodic or non periodic if the signal is periodic find the fundamental period of
1. x (t) = 6cos 5/4 t + 2 cos 9/7 t
2. x (t) = sin√3t +cos √3t

Answer 1

Consider the function

$$x (t) = 6\text{cos} \bigg(\frac{5}{4}t\bigg) + 2 \text{cos }\bigg(\frac{9}{7} t\bigg).$$

This function is periodic with a period of 175.9

To avoid ambiguity, consider two functions as a second function:

$$2.1\space\space x(t)=\text{sin}(\sqrt3\cdot t)+\text{cos}(\sqrt3\cdot t),\\ 2.2\space\space x(t)=\text{sin}(\sqrt{3t})+\text{cos}(\sqrt{3t}).$$

The first one is a periodic function with a period of

$$T=\frac{2\pi}{\sqrt3}\approx3.63.$$

How did we find it? Simplify the sum of a sine and cosine of the same argument, you get

$$f(t)=\text{sin}(kt)+\text{cos}(kt)=\sqrt2\text{ sin}\bigg(kt+\frac{\pi}{4}\bigg).$$

This is a simple sine function that is certainly periodic. The period of a periodic function is

$$T=\frac{2\pi}{k}.$$

The function 2.2 is not a periodic function.