Consider a single input single output discrete-time system with x[n] as input y[n] as output, where the two are related as:$$y[n]= begin{cases} n ,|x[n]|, & ext{for } 0 leq n leq 10 [6pt] x[n]-x[n-1], & ext{otherwise} end{cases}$$Which one of the following statements is true about the systems?(GATE 2017 || EC || PYQ || MCQ || 1 MARKS)

It is neither causal nor stable

The input x(t) and output y(t) of a system are related as $$y(t)= int_{- infty}^{t} x( au) cos(3 au) ,d au$$The system is(GATE 2012 || EC || PYQ || MCQ || 2 MARKS)

not time-invariant and not stable

A system with input x[n] and output y[n] is given as $$y(n)= sin left( frac{5 pi}{6}n ight)x(n)$$The system is(GATE 2006 || EC || PYQ || MCQ || 2 MARKS)

Linear, stable and non-invertible

Non-linear, stable and non-invertible

Linear, unstable and invertible

GATE EC||SIGNALS & SYSTEMS ||BASIC OF SIGNALS & SYSTEMS|| PYQS(2000-2025)

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Question 1

(GATE 2019 || EC || PYQ || NAT || 1 MARKS)

Question 2

Let the input be u and the output be y of a system, and the other parameters are real constants. Identify which among the following systems is not a linear system:

(GATE 2018 || EC || [PYQ || MCQ || 1 MARKS)

[Tex]\frac{d^3y}{dt^3} +a_1\frac{d^2y}{dt^2} +a_2\frac{dy}{dt} +a_3y = b_3u +b_2\frac{du}{dt} +b_1\frac{d^2u}{dt^2}[/Tex]
with initial rest conditions
[Tex]y(t)=\int_{0}^{t} e^{\alpha (t-\tau)}\,b\,u(\tau)\,d\tau[/Tex]
y = au + b, b ≠ 0
y = au

Question 3

Consider a single input single output discrete-time system with x[n] as input y[n] as output, where the two are related as:

[Tex]y[n]= \begin{cases} n\,|x[n]|, & \text{for } 0\leq n\leq 10\\[6pt] x[n]-x[n-1], & \text{otherwise} \end{cases}[/Tex]

Which one of the following statements is true about the systems?

(GATE 2017 || EC || PYQ || MCQ || 1 MARKS)

It is causal and stable
It is causal but not stable
It is not causal but stable
It is neither causal nor stable

Question 4

The input x(t) and output y(t) of a system are related as

[Tex]y(t)=\int_{-\infty}^{t} x(\tau)\cos(3\tau)\,d\tau[/Tex]

The system is

(GATE 2012 || EC || PYQ || MCQ || 2 MARKS)

time-invariant and stable
stable and not time-invariant
time-invariant and not stable
not time-invariant and not stable

Question 5

Let x(t) be the input and y(t) be the output of a continuous time system. Match the system properties P₁, P₂ and P₃ with system relations R₁, R₂, R₃, R₄.

Properties

P₁: Linear but NOT time-invariant

P₂: Time-invariant but NOT linear

P₃: Linear and time-invariant

Relations

R₁: y(t) = t²x(t)

R₂: y(t) = t|x(t)|

R₃: y(t) = |x(t)|

R₄: y(t) = x(t – 5)

(GATE 2008 || EC || PYQ || MCQ || 2 MARKS)

(P₁, R₁), (P₂, R₃), (P₃, R₄)
(P₁, R₂), (P₂, R₃), (P₃, R₄)
(P₁, R₃), (P₂, R₁), (P₃, R₂)
(P₁, R₁), (P₂, R₂), (P₃, R₃)

Question 6

The input and output of a continuous time system are respectively denoted by x(t) and y(t). Which of the following descriptions corresponds to a causal system?

(GATE 2008 || EC || PYQ || MCQ || 1 MARKS)

y(t) = x(t – 2) + x(t + 4)
y(t) = (t – 4) x(t + 1)
y(t) = (t + 4) x(t – 1)
y(t) = (t + 5) x(t + 5)

Question 7

A system with input x[n] and output y[n] is given as

[Tex]y(n)=\sin\left(\frac{5\pi}{6}n\right)x(n)[/Tex]

The system is

(GATE 2006 || EC || PYQ || MCQ || 2 MARKS)

Linear, stable and invertible
Non-linear, stable and non-invertible
Linear, stable and non-invertible
Linear, unstable and invertible

Question 8

Let P be linearity, Q be time-invariance, R be causality and S be stability. A discrete-time system has the input-output relationship

[Tex]y(n)= \begin{cases} x(n), & n\geq 1\\ 0, & n=0\\ x(n+1), & n\leq -1 \end{cases}[/Tex]

where x(n) is the input and y(n) is the output. The above system has the properties

(GATE 2003 || EC || PYQ || MCQ || 2 MARKS)

P, S but not Q, R
P, Q, S but not R
P, Q, R, S
Q, R, S but not P

Question 9

A system with an input x(t) and output y(t) is described by the relation: y(t) = t x(t). This system is

(GATE 2000 || EC || PYQ || MCQ || 1 MARKS)

linear and time-invariant
linear and time varying
non-linear and time-invariant
non-linear and time-varying

Question 10

(GATE 2001 || EC || PYQ || MCQ || 1 MARKS)

1
-1
0
0

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