Let $f(t)$ be a periodic signal with fundamental period $T_0>0$. Consider the signal $y(t)=f(\alpha t)$, where $\alpha>1$.

The Fourier series expansions of $f(t)$ and $y(t)$ are given by

$$ f(t)=\sum\limits_{k = - \infty }^\infty c_k e^{j \frac{2 \pi}{T_0} k T} \text { and } y(t)=\sum\limits_{k = - \infty }^\infty d_k e^{j \frac{2 \pi}{T_0} \alpha k T} . $$

Which of the following statements is/are TRUE?

Let $$\mathrm{x_1(t)=u(t+1.5)-u(t-1.5)}$$ and $$\mathrm{x_2(t)}$$ is shown in the figure below. For $$\mathrm{y(t)=x_1(t)~*~x_2(t)}$$, the $$\int_{ - \infty }^\infty {y(t)dt} $$ is ____________ (rounded off to the nearest integer).

The exponential Fourier series representation of a continu-ous-time periodic signal $X(t)$ is defined as

$$ x(t)=\sum\limits_{k=-\infty}^{\infty} a_k e^{j k w_0 t} $$

Where $\omega_0$ is the fundamental angular frequency of $x(t)$ and the coefficients of the series are $a_k$. The following information is given about $x(t)$ and $a_k$.

I. $x(t)$ is real and even, having a fundamental period of 6

II. The average value of $x(t)$ is 2

III. $a_k=\left\{\begin{array}{c}k, 1 \leq k \leq 3 \\ 0, k>3\end{array}\right.$

The average power of the signal $x(t)$ (rounded off one decimal place) is $\_\_\_\_$

Let x(t) be a continuous time periodic signal with fundamental period T = 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 {3a_k} where k is integer valued

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

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