# Signals and Systems (Test 5)

## Gate Exam : Ee Electrical Engineering

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Signals and Systems
| Signals and Systems |
Q.1
The 3-dB bandwidth of the low-pass signal e-t u(t), where u(t) is the unit step function, is given by
A. 1/2π Hz
B.
C.
D. 1 Hz
Explaination / Solution:

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Q.2
The result of the convolution  is
A. x ( t + t0)
B. x ( t - t0)
C. x ( -t + t0)
D. x ( -t - t0)
Explaination / Solution:

From the convolution property,

Now, we replace t by -t to obtain

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Q.3
Let the Laplace transform of a function F(t) which exists for t > 0 be F1(s) and the Laplace transform of its delayed version  be the complex conjugate of F1(s) with the Laplace variable set as s=σ + jw. If G(s) = , then the inverse Laplace transform of G(s) is
A.
B.
C. An ideal step function u(t)
D.
Explaination / Solution:

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Q.4
Consider a linear system whose state space representation is x(t) = Ax(t). If the initial state vector of the system is  then the system response is   If the itial state vector of the system changes to  the system response becomes
The eigenvalue and eigenvector pairs (λivi)for the system are
A.
B.
C.
D.
Explaination / Solution:

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Q.5
Consider a causal LTI system characterized by differential equation  The response of the system to the input  where u(t) denotes the unit step function, is
A.
B.
C.
D.
Explaination / Solution:

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Q.6
A discrete time signal x[n] = sin (π2n) n being an integer, is
A. periodic with period π
B.  periodic with period π2
C. periodic with period π/2
D. not periodic
Explaination / Solution:

In the given options (A), (B) and (C), we have the periods respectively as
N1 = π

N2 = π2

N3 = π/3

None of the above period is an integer. Since, a discrete time signal has its period an integer. So, all the three options are incorrect. Hence, we are left with the option (D). i.e. the discrete time signal x[n] = sin (π2n) is not periodic.

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Q.7
Consider a linear time invariant system x = Ax, with initial condition x(0) at t = 0. Suppose α and β are eigenvectors of (2 x 2) matrix A corresponding to distinct eigenvalues λ1 and λ2 respectively. Then the response x(t) of the system due to initial condition x(0) = α is
A.
B.
C.
D.
Explaination / Solution:

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Q.8
The state variable description of an LTI system is given by
where y is the output and u is the input. The system is controllable for

A.
B.
C.
D.
Explaination / Solution:

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Q.9
For a periodic signal  the fundamental frequency in rad/s
A. 100
B. 300
C. 500
D. 1500
Explaination / Solution:

Given, the signal

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Q.10
The period of the signal  is
A. 0.4πs
B. 0.8πs
C. 1.25s
D. 2.5s