Signals and Systems (Quiz-3)

1. The convolution process associated with the Laplace Transform, in time domain results into

Simple multiplication in complex frequency domain

Simple division in complex frequency domain

Simple multiplication in complex time domain

Simple division in complex time domain

The Laplace Transform is a mathematical tool used to convert functions from the time domain to the complex frequency domain. During this transformation, the convolution process, which involves the integration of the product of two functions, results in a simple multiplication in the complex frequency domain. This means that when applying the Laplace Transform, the convolution of two functions in the time domain can be simplified to a multiplication operation in the complex frequency domain.

Explanation

The Laplace Transform is a mathematical tool used to convert functions from the time domain to the complex frequency domain. During this transformation, the convolution process, which involves the integration of the product of two functions, results in a simple multiplication in the complex frequency domain. This means that when applying the Laplace Transform, the convolution of two functions in the time domain can be simplified to a multiplication operation in the complex frequency domain.

2. An LTI system with rational system function having poles at -19, -6 and -1 and ROC is on the right side of the rightmost pole. The system is

Causal-Unstable

Non-causal-stable

Causal-stable

Non-causal-unstable

The given system has poles at -19, -6, and -1. For a system to be causal, all the poles must be in the left half of the complex plane. Since the poles in this system are all negative, the system is causal. Additionally, the system is stable because the region of convergence (ROC) is on the right side of the rightmost pole. Therefore, the correct answer is causal-stable.

Explanation

The given system has poles at -19, -6, and -1. For a system to be causal, all the poles must be in the left half of the complex plane. Since the poles in this system are all negative, the system is causal. Additionally, the system is stable because the region of convergence (ROC) is on the right side of the rightmost pole. Therefore, the correct answer is causal-stable.

3. ROC in Z plane excludes unity circle, it signifies….

Z transform converges but Fourier transform does not converge

Fourier transform converges but Z transform does not converge

Both Fourier and Z transform converge

Both Fourier and Z transform do no converge

The statement "ROC in Z plane excludes unity circle" indicates that the region of convergence for the Z-transform does not include the unit circle in the complex plane. This implies that the Z-transform converges for certain values of the complex variable z, but not for points on the unit circle. On the other hand, the statement does not provide any information about the convergence of the Fourier transform. Therefore, the correct answer is that the Z-transform converges but the Fourier transform does not converge.

Explanation

The statement "ROC in Z plane excludes unity circle" indicates that the region of convergence for the Z-transform does not include the unit circle in the complex plane. This implies that the Z-transform converges for certain values of the complex variable z, but not for points on the unit circle. On the other hand, the statement does not provide any information about the convergence of the Fourier transform. Therefore, the correct answer is that the Z-transform converges but the Fourier transform does not converge.

4. An LTI system's transfer function is represented by (1/s). If unit step signal is applied at the input of this system, corresponding output will be

Sinc function

Cosine function

Unit impulse

Unit ramp function

The transfer function of an LTI system is given by (1/s), which represents a first-order integrator. When a unit step signal is applied at the input of this system, the output will be a unit ramp function. This is because the integral of a unit step function is a ramp function. Therefore, the correct answer is the unit ramp function.

Explanation

The transfer function of an LTI system is given by (1/s), which represents a first-order integrator. When a unit step signal is applied at the input of this system, the output will be a unit ramp function. This is because the integral of a unit step function is a ramp function. Therefore, the correct answer is the unit ramp function.

5. A signal x(t) is delayed by T time, corresponding ROC in the S – plane will shift by

Exponential (-sT)

Exponential (sT)

T

0

When a signal x(t) is delayed by T time, it means that the signal is shifted to the right by T units on the time axis. However, this delay does not affect the region of convergence (ROC) in the S-plane. The ROC is determined by the properties of the original signal and not by any time shift. Therefore, the correct answer is 0, indicating that the ROC does not shift when the signal is delayed.

Explanation

When a signal x(t) is delayed by T time, it means that the signal is shifted to the right by T units on the time axis. However, this delay does not affect the region of convergence (ROC) in the S-plane. The ROC is determined by the properties of the original signal and not by any time shift. Therefore, the correct answer is 0, indicating that the ROC does not shift when the signal is delayed.