Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE) PDF Download

Q1: The input x(t) and the output y(t) of a system are related as       (2024)
The system is       (2024)
(a) nonlinear
(b) linear and time-invariant
(c) linear but not time-invariant
(d) noncasual
Ans: (b)
Sol: System will follow law of superposition. Therefore, system will be linear.
Time-invariance check:
Since,
y(t) = y(t − t₀)
So, the system is time-invariant.

Q2: Which of the following statement(s) is/are true?      (2023)
(a) If an LTI system is causal, it is stable
(b) A discrete time LTI system is causal if and only if its response to a step input u[n] is 0 for n < 0
(c) If a discrete time LTI system has an impulse response h[n] of finite duration the system is stable  
(d) If the impulse response 0 < ∣h[n]∣ < 1 for all n, then the LTI system is stable.
Ans: (b)
Sol: For causal system, impuse response
h(n) = 0; n < 0
Therefore, for step input also
h(n) = 0; n < 0

Q3: Consider the system as shown below
where y(t) = x(e^t). The system is       (2022)
(a) linear and causal.
(b) linear and non-causal.
(c) non-linear and causal
(d) non-linear and non-causal
Ans: (b)
Sol: We know, a linear system follows the law of superposition.
It is a combination of two laws:
(i) Law of additivity:
Both results are same, hence, it follows law of additivity.
(ii) Law of Homogeneity:
Here also both results are same, hence it follows law of Homogeneity.
Therefore, System is linear.
We know, a causal system is independent of future values of input at each & every instant of time them system will be causal.
Given : y(t) = x(e^t)
Put t = 0
 y(0) = x(e⁰) = x(1)
Because its depends on future value.
Therefore, system is non-causal.

Q4: Let a causal LTI system be governed by the following differential equation   where x(t) and x(t) are the input and output respectively. Its impulse response is       (2022)
(a)
(b)
(c)
(d)
Ans: (d)
Sol: Given:
Taking Laplace transform,
Taking inverse Laplace, transform,

Q5: If the input x(t) and output y(t) of a system are related as y(t) = max(0, x (t)), then the system is       (2021)
(a) linear and time-variant
(b) linear and time-invariant
(c) non-linear and time-variant
(d) non-linear and time-invariant
Ans: (d)
Sol: Linearity check:
at input x₁(t) = −2, output y₁(t) = 0
at input x₂(t) = 1, output y₂(t) = 1
∴ system is non-linear because it violates law of additivity.
Check for time-invariance :
Delayed O/P:
Therefore, system is time-invariant.

Q6: Which of the following options is true for a linear time-invariant discrete time system that obeys the difference equation:
y[n] − ay[n − 1] = b₀x[n] − b₁x[n − 1]     (2020)
(a)  y[n] is unaffected by the values of x[n − k]; k > 2
(b) The system is necessarily causal.
(c) The system impulse response is non-zero at infinitely many instants.
(d) When x[n] = 0, n < 0, the function y[n]; n > 0 is solely determined by the function x[n].
Ans: (c)
Sol: By taking right-sided inverse ZT,
h(n) = b₀aⁿu(n)−b₁aⁿ⁻¹u(n − 1)
By taking left-sided inverse ZT,
 h(n) = −b₀aⁿu(−n − 1) + b₁aⁿ⁻¹u(−n)
Thus system is not necessarily causal.
The impulse response is non-zero at infinitely many instants.  

Q7: A continuous-time input signal x(t) is an eigenfunction of an LTI system, if the output is       (2018)
(a) k x (t) , where k is an eigenvalue
(b) ke ^jωt x(t), where k is an eigenvalue and e^jωt is a complex exponential signal
(c) x(t) e^jωt, where e^jωt is a complex exponential signal
(d) k H(ω) ,where k is an eigenvalue and H(ω) is a frequency response of the system
Ans: (a)
Sol: Eigen function is a type of input for which output is constant times of input.
i.e.
Where,
x(t) = System input = eigen function
H(s) = transfer function of system
y(t) =  system output
Here,
where,
k =  eigen-value = H(s)∣_{s = a}
x(t) = eigen-function input  

Q8: Let z(t) = x(t) * y(t) , where "*" denotes convolution. Let c be a positive real-valued constant. Choose the correct expression for z(ct).       (SET-1 (2017))
(a) c x(ct)*y(ct)
(b) x(ct)*y(ct)
(c) c x(t)*y(ct)
(d) c x(t)*y(ct)
Ans: (a)
Sol: Time scaling property of convolution.

Q9: 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         (SET-2  (2016))
(a)
(b)
(c)
(d)
Ans: (d)
Sol: The differential equation

Q10: The output of a continuous-time, linear time-invariant system is denoted by T{x(t)} where x(t) is the input signal. A signal z(t) is called eigen-signal of the system T, when T{z(t)} = γz(t), where γ is a complex number, in general, and is called an eigenvalue of T. Suppose the impulse response of the system T is real and even. Which of the following statements is TRUE?        (SET-1 (2016))
(a) cos(t) is an eigen-signal but sin(t) is not
(b) cos(t) and sin(t) are both eigen-signals but with different eigenvalues
(c) sin(t) is an eigen-signal but cos(t) is not
(d) cos(t) and sin(t) are both eigen-signals with identical eigenvalues
Ans: (d)
Sol: Given that impulse response is real and even, Thus H(jω) will also be real and even.
Since, H(jω) is real and even thus,
So, sin(t) and cos(t) are eigen signal with same eigen values.  

Q11: Consider the following state-space representation of a linear time-invariant system.
The value of y(t) for t = log_e2 is______.       (SET-1 (2016))
(a) 4
(b) 5
(c) 6
(d) 7
Ans: (c)

Q12: Consider a continuous-time system with input x(t) and output y(t) given by
y(t) = x(t) cos(t)
This system is      (SET-1  (2016))
(a) linear and time-invariant
(b) non-linear and time-invariant
(c) linear and time-varying
(d) non-linear and time-varying
Ans: (c)
Sol: So, the system is linear, to check time invariance.
The delayed output,
System is time varying.

Q13: The following discrete-time equations result from the numerical integration of the differential equations of an un-damped simple harmonic oscillator with state variables x and y. The integration time step is h.
For this discrete-time system, which one of the following statements is TRUE?      (SET-2 (2015))
(a) The system is not stable for h > 0
(b) The system is stable for h > (1/π)
(c) The system is stable for 0 < h < (1/2π)
(d) The system is stable for (1/2π) < h < (1/π) 
Ans: (a)

Q14: For linear time invariant systems, that are Bounded Input Bounded Output stable, which one of the following statements is TRUE?      (SET-2 (2015))
(a) The impulse response will be integrable, but may not be absolutely integrable.
(b) The unit impulse response will have finite support.
(c) The unit step response will be absolutely integrable
(d) The unit step response will be bounded.
Ans: (d)

Q15: The impulse response g(t) of a system, G , is as shown in Figure (a). What is the maximum value attained by the impulse response of two cascaded blocks of G as shown in Figure (b)?      (SET-1 (2015))


(a) 2/3
(b) 3/4
(c) 4/5
(d) 1
Ans: (d)
Sol: Maximum value = 1

Q16: A moving average function is given by  If the input u is a sinusoidal signal of frequency (1/2τ) Hz, then in steady state, the output y will lag u (in degree) by ______ .      (SET-1 (2015))
(a) 30
(b) 60
(c) 90
(d) 120
Ans: (c)
Sol: System input: sin ω₀t
Therefore,

Q17: Consider an LTI system with impulse response h(t) = e^−5tu(t). If the output of the system is y(t) = e^−3tu(t) − e^−5tu(t) then the input, x(t), is given by       (SET-2 (2014))
(a) e^−3t u(t)
(b) 2e^−3t u(t)
(c) e^−5t u(t)
(d) $2𝑒-5𝑡𝑢\left(𝑡\right)$2e^−5t u(t)
Ans: (b)
Sol: Impulse response of an LTI system = transfer function
Q18: x(t) is nonzero only for T_x < t < T_x′, and similarly, y(t) is nonzero only for  T_y < t < T_y′. Let z(t) be convolution of x(t) and y(t). Which one of the following statements is TRUE ?         (SET-1 (2014))
(a) z(t) can be nonzero over an unbounded interval.
(b) z(t) is nonzero for t < T_x + T_y
(c) z(t) is zero outside of
(d) z(t) is nonzero for
Ans: (c)
Sol: x(t) is non-zero for and  y(t) is non-zero for then the limits of the resultant signal is

Q19: The impulse response of a continuous time system is given by h(t) = δ(t − 1)δ(t − 3). The value of the step response at t = 2 is        (2013)
(a) 0
(b) 1
(c) 2
(d) 3
Ans: (b)
Sol: For step response, impulse response can be integrated.

Q20: Two systems with impulse responses h₁(t) and h₂(t) are connected in cascade. Then the overall impulse response of the cascaded system is given by     (2013)
(a) product of h₁(t) and h₂(t)
(b) sum of h₁(t) and h₂(t)
(c) convolution of h₁(t) and h₂(t)
(d) subtraction of h₁(t) and h₂(t)
Ans: (c)
Sol: In cascade connection,