Question

Let x(t) be a continuos time periodic signal with fundamental period = 1 seconds. Let $a_k$ be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t) :

I. The complex Fourier series coefficients of x(3t) are $a_k$ where k is integer valued.

II. The complex Fourier series coefficients of X(3t) are 3$a_k$ where k is integer valued.

III. The fundamental angular frequency of x(3t) is $6\pi$ rad/s.

For the three statements above, which one of the following correct ?

• A. Only III is true
• B. Only I is true
• C. Only II and III are true
• D. Only I and III are true

Answer 1

The correct option is D Only I and III are true

Initially T = 1 sec, so ${\omega }_o=2\pi$ rad/sec. When x(t) is compressed by 3, frequency will expand by same factor but there is no change in value of $a_k$.
So, both statement I and III are correct.