Signals and Systems - B.Tech 4th Semester Examination, 2024

If \( X(f) \) is the Fourier transform of \( x(t) \), then what is the Fourier transform of \( x(2t) \)?

1. \( X(f/2) \)
2. \( 0.5X(f/2) \)
3. \( 2X(2f) \)
4. \( X(f) \)

Which of the following statements is true?

1. Discrete-time signals are always periodic
2. Continuous-time signals cannot be periodic
3. Discrete-time signals can be periodic or aperiodic
4. Continuous-time signals are always aperiodic

Find the fundamental period of \( x(t) = \sin(5\pi t) + \cos(7\pi t) \).

1. 0.2 sec
2. 0.4 sec
3. 0.8 sec
4. 1.0 sec

If a system's impulse response is \( h(t) = \delta(t) + \delta(t-1) \), the system is:

1. Causal
2. Anti-causal
3. Non-causal
4. Time-invariant

Which system property ensures that the output depends only on past and present inputs?

1. Linearity
2. Causality
3. Stability
4. Time-invariance

Which of the following is an example of a deterministic signal?

1. White noise
2. Sinusoidal signal
3. Random signal
4. Gaussian noise

The impulse response of an LTI system is given by \( h(t) = e^{-2t}u(t) \). Find the system response to an input \( x(t) = e^{-t}u(t) \).

1. \( \frac{e^{-t}-e^{-2t}}{3} \)
2. \( \frac{e^{-t}-e^{-2t}}{2} \)
3. \( e^{-t} - e^{-2t} \)
4. \( \frac{e^{-t}-e^{-2t}}{e^{-3t}} \)

The step response of an LTI system is obtained by:

1. Taking the derivative of the impulse response
2. Integrating the impulse response
3. Taking the Fourier transform of the impulse response
4. Adding two impulse responses

The Laplace transform of \( e^{-at}u(t) \) is:

1. \( \frac{1}{s+a} \)
2. \( \frac{1}{s-a} \)
3. \( \frac{s}{s+a} \)
4. \( \frac{s}{s-a} \)

If a signal contains frequency components up to 5 kHz, what should be the minimum sampling rate?

1. 2.5 kHz
2. 5 kHz
3. 10 kHz
4. 20 kHz

Find the inverse Laplace transform of: \( F(s) = \frac{s(s+2)}{(s+1)(s+3)(s+5)} \).

Given a system with a difference equation \( y[n] - 0.5y[n-1] = x[n] \), determine its stability.

Find the state space model of the system represented by the differential equation: \( \frac{d^3y}{dt^3} + 8\frac{d^2y}{dt^2} + 9\frac{dy}{dt} = 3x(t) \).

A system is described by the equation: \( y(t) = 3x(t) + 2 \). Check whether the system is linear or not.

Compute the poles and zeros of \( F(s) = \frac{s+1}{s^3 + 7s^2 + 10s + 18} \).

Consider a system defined by \( y(t) = tx(t) \). Determine whether it is linear and shift-invariant.

Compute the output \( y[n] \) of an LTI system with impulse response \( h[n] = \{1, 2, 1\} \) and input \( x[n] = \{2, 1, 3\} \) using convolution.

Compute the energy and power of the signal \( x(t) = e^{-t}u(t) \).

Find the Fourier transform of \( x(t) = e^{-2|t|} \).

Find the Laplace Transform of \( x(t) = e^{-t}u(t) \) and determine its region of convergence.

Given \( x(t) = \sin(10t) + \cos(15t) \), determine whether it is periodic and find the fundamental period.

Compute the z-transform of \( x[n] = -b^n u[n-1] \).

Find the step response of a system with impulse response \( h(t) = e^{-3t}u(t) \).

Determine the even and odd components of the signal \( x(t) = t^2 + 3t + 2 \).