Signals and Systems

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

1. X(f/2)
2. 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(5πt) +cos(7πt).

1. 2 sec
2. 4 sec
3. 8 sec
4. 0 sec

If a system's impulse response is h(t) =δ(t) +δ(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^−2tu(t). Find the system response to an input x(t) = e^−tu(t).

1. [e^−t−e^−2t] / [3]
2. [e^−t−e^−2t] / [2]
3. e^−t− e^−2t
4. [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^−atu(t) is:

1. [1] / [s+a]
2. [1] / [s−a]
3. [s] / [s+a]
4. [s] / [s−a]

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

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

Find the inverse Laplace transform of: F(s) =[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: d³y/dt³+ 8d²y/dt²+ 9dy/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) =[s+1] / [s³+ 7s²+ 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^−tu(t).

Find the Fourier transform of x(t) = e^−2|t|.

Find the Laplace Transform of x(t) = e^−tu(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] = −bⁿu[n−1].

Find the step response of a system with impulse response h(t) = e^−3tu(t).

Determine the even and odd components of the signal x(t) = t²+ 3t + 2.