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...

## 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...
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