Checking the periodicity of a signal

Question

We're given this signal:

$X_a(t) = x(t)\cdot rect(\frac{t}{100})$, where $x(t) = cos(3t)$

And we have to check if it's periodic or not, and find the period if it is.

Now, for $cos(3t)$, the period $T_A$ is $\frac{2π}{ω_1} = \frac{2π}{3}$.

For $rect(\frac{t}{100})$, we can use Fourier transformation to get $100sinc(100πf)$, if I'm not mistaken.

But then I don't know how to continue. What do i do from here?

Answer 1

If your rect function is defined as $$\operatorname{rect}(t) = \begin{cases} 0, & \text{if } |t| > \frac{1}{2} \\ \frac{1}{2}, & \text{if } |t| = \frac{1}{2} \\ 1, & \text{if } |t| < \frac{1}{2}. \end{cases}$$ So $$\operatorname{rect}(\frac{t}{100}) = \begin{cases} 0, & \text{if } |t| > 50 \\ \frac{1}{2}, & \text{if } |t| = 50 \\ 1, & \text{if } |t| < 50. \end{cases}$$ Then your periodicity is ruined because of this rect function.