GATE ECE

Signals and Systems

Discrete Time Linear Time Invariant Systems

Previous Years Questions

## Marks 1

Consider a single input single output discrete-time system with $$h\left[ n \right]\,$$ as input and $$y\left[ n \right]\,$$ as output, where the two...

An LTI system with unit sample response
$$h\left( n \right) = 5\delta \left[ n \right] - 7\delta \left[ {n - 1} \right] + 7\delta \left[ {n - 3} \rig...

A system is defined by its impulse response $$h\left( n \right) = {2^n}\,u\left( {n - 2} \right).$$ The system is

Two discrete time systems with impulse responses $${h_1}\left[ n \right]\, = \delta \left[ {n - 1} \right]$$ and $${h_2}\left[ n \right]\, = \delta \l...

The impulse response $$h\left[ n \right]$$ of a linear time-invariant system is given by $$h\left[ n \right]$$ $$ = u\left[ {n + 3} \right] + u\left[ ...

A sequence $$x\left( n \right)$$ with the $$z$$-transform $$X\left( z \right)$$ $$ = {z^4} + {z^2} - 2z + 2 - 3{z^{ - 4}}$$ is applied as an input to ...

## Marks 2

Two sequence $${x_1}\left[ n \right]$$ and $${x_2}\left[ n \right]$$ have the same energy.
Suppose $${x_1}\left[ n \right]$$ $$ = \alpha \,{0.5^n}\,u...

Consider a discrete-time signal
$$x\left[ n \right] = \left\{ {\matrix{
{n\,\,for\,\,0 \le n \le 10} \cr
{0\,\,otherwise} \cr
} } \right...

Let $$y\left[ n \right]$$ denote the convolution of $$h\left[ n \right]$$ and $$g\left[ n \right]$$, where $$h\left[ n \right]$$ $$ = \,{\left( {1/2} ...

Two system $${H_1}\left( z \right)$$ and $${H_2}\left( z \right)$$ are connected in cascade as shown below. The overall output $$y\left( n \right)$$ ...

The transfer function of a discrete time LTI system is given by
$$H\left( z \right) = {{2 - {3 \over 4}{z^{ - 1}}} \over {1 - {3 \over 4}{z^{ - 1}} +...

A discrete time linear shift - invariant system has an impulse response $$h\left[ n \right]$$ with $$h\left[ 0 \right]$$ $$ = 1,\,\,h\left[ 1 \right]\...

A system with input $$x\left( n \right)$$ and output $$y\left( n \right)$$ is given as $$y\left( n \right)$$ $$ = \left( {\sin {5 \over 6}\,\pi \,n} \...

A causal LTI system is described by the difference equation $$2y\left[ n \right] = ay\left[ {n - 2} \right] - 2x\left[ n \right] + \beta x\left[ {n - ...

The impulse response $$h\left[ n \right]$$ of a linear time invariant system is given as
$$h\left[ n \right] = \left\{ {\matrix{
{ - 2\sqrt 2 ,} &...

Let P be linearity, Q be time-invariance, R be causality and S be stability.
A discrete time system has the input-output relationship,
$$y\left( n \r...

If the impulse response of a discrete-time system is $$h\left[ n \right]\, = \, - {5^n}\,\,u\left[ { - n\, - 1} \right],$$ then the system function $$...

A linear discrete - time system has the characteristic equation, $${z^3} - 0.81\,\,z = 0.$$ The system

Consider the system shown in the Fig.1 below. The transfer function $$Y\left( z \right)/X\left( z \right)$$ of the system is
...

## Marks 4

The output of a system is given in difference equation form as $$y\left( k \right) = \,a\,\,y\left( {k - 1} \right) + x\left( k \right),$$ where $$x\...

## Marks 5

In Fig. 1, a linear time invariant discrete system is shown. Blocks labeled D represent unit delay elements. For $$n\, < 0,$$ you may assume that $...

A system having a unit impulse response $$h\left( n \right)$$ = $$u\left( n \right)$$ is excited by a signal $$x\left( n \right)$$ $$ = \,{\alpha ...

In the linear time-invariant system shown in Fig. 1, blocks labeled D represent unit delay elements. Find the expression for $$y\left( n \right),$$ an...