Consider a real-valued base-band signal $x(t)$. band limited to 10 kHz . The Nyquist rate for the signal $y(t)=x(t) \times \left(1+\frac{t}{2}\right)$ is

Consider the signals $x[n]=2^{n-1} u[-n+2]$ and $y[n]=2^{-n+2} u[n+1]$, where $u[n]$ is the unit step sequence. Let $X\left(e^{j \omega}\right)$ and $Y\left(e^{j \omega}\right)$ be the discrete-time Fourier transform of $x[n]$ and $y[n]$, respectively. The value of the integral

$$ \frac{1}{2 \pi} \int_o^{2 \pi} X\left(e^{j \omega}\right) Y\left(e^{-j \omega}\right) d \omega $$

(round off to one decimal place) is $\_\_\_\_$

For a unit step input $u[n]$, a discreate-time $L T I$ system produces an output signal $(2 \delta[n+1]+\delta[n]+\delta[n-1])$. Let $y[n]$ be the output of the system for an input $\left(\left(\frac{1}{2}\right)^n u[n]\right)$.

The value of $y[0]$ is The value of $y[0]$ is $\_\_\_\_$

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 $\_\_\_\_$