Signals and Systems - B.Tech. 3rd Semester Examination, 2023

The minimum sampling rate to avoid aliasing for \( x(t) = 5 \cos(400\pi t) \) is:

1. 100 Hz
2. 400 Hz
3. 800 Hz
4. 300 Hz

Which of the following system is memory less?

1. \( h(t) = 0 \) for \( t \neq 0 \)
2. \( h(t) = x(t-1) \)
3. \( h(t) = 0 \) for \( t = 0 \)
4. \( h(t) = kx(t+2) \)

The ROC of \( x(t) = e^{-2t}u(t) + e^{-3t}u(t) \) is:

1. \( \sigma > 2 \)
2. \( \sigma > 3 \)
3. \( \sigma > -3 \)
4. \( \sigma > -2 \)

Time period of: \( x(t) = 3 \cos(20t+5) + \sin(8t-3) \) is:

1. \( \pi/10 \) sec
2. \( \pi/20 \) sec
3. \( 2\pi/5 \) sec
4. \( 2\pi/15 \) sec

The power of the signal: \( x(t) = \cos(t) \) is:

1. 1/2
2. 1
3. 2
4. 4

The Fourier transform exponential signal \( f(t) = e^{-at}u(t), a > 0 \) is:

1. \( \frac{1}{-a+j\omega t} \)
2. \( \frac{1}{a-j\omega t} \)
3. \( \frac{1}{a+j\omega t} \)
4. \( \frac{1}{-a-j\omega t} \)

\( x(5t) \) is:

1. Compressed signal
2. Expanded signal
3. Time shifted signal
4. Amplitude scaled signal by factor 1/5

Which one of the following is an example of a bounded signal?

1. \( e^t \cos(\omega t) \)
2. \( e^{-t} \cos(\omega t) \)
3. \( e^t \cos(-\omega t) \)
4. \( e^t \sin(\omega t) \)

The simplified valve of \( X(n) = \sum_{n=-5}^{5} \sin(2n)\delta(n+7) \) is:

1. 0
2. -sin 10
3. sin 10
4. 1

If \( X(S) = \frac{4S+1}{S^2+6S+3} \), then initial value of \( x(0) \) will be:

1. 1/3
2. 1/4
3. 4
4. 3

Find the inverse Z-transform of \( X(Z) = \frac{1}{1+3Z^{-1}+2Z^{-2}} \), \( ROC: |Z| > 2 \).

For the system \( y(t) = 12x(t) + 7 \), check whether the system is (i) time variant/time-invariant (ii) causal/non-causal (iii) linear/non-linear.

Find the even and odd components of the sequence \( X(n) = 5\delta(n+4) + 4\delta(n+3) + 3\delta(n+2) + \delta(n+1) \).

Determine the power of the signal \( x(t) = e^{j\alpha t}\cos(\omega_o t) \).

Find the Fourier transform of \( x(t) = \frac{1}{a^2+t^2} \).

Find the time response of LTI system with impulse response \( h(t) = 2u(t) - 2u(t-3) \) & input is \( x(t) = 8u(t) - 8u(t-5) \).

Sketch the signal \( x(-4t-3) \) as shown in figure.

Find the convolution of the following sequence \( x(n) = 2\delta(n+1) - \delta(n) + \delta(n-1) + 3\delta(n-2) \) and \( h(n) = 3\delta(n-1) + 4\delta(n-2) + 2\delta(n-3) \).

Compute the DFT of \( x(n) = \{0, 1, 2, 4\} \).

Compute the output of the following signals whose impulse response and input are given by \( h(t) = e^{-at}u(t) \); \( x(t) = e^{at}u(-t), a>0 \) respectively.

Find the Laplace Transform of signal in the figure.

Calculate the fundamental period of \( x(t) = 1 + \sin(\frac{2\pi}{3}t)\cos(\frac{4\pi}{5}t) \).

Determine the Nyquist sampling rate and Nyquist sampling intervals for the signal \( x(t) = \text{sinc}(100\pi t)\text{sinc}(200\pi t) \).

Compute the state transition matrix \( \Phi(t) \) for the system represented by state equation: \( \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -2 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \).