GATE ECE
Signals and Systems
Discrete Time Signal Z Transform
Previous Years Questions

## Marks 1

A discrete time all-pass system has two of its poles at 0.25$$\angle 0^\circ$$ and $$\angle 30^\circ$$. Which one of the following statements about ...
A discrete-time signal$$x\left[ n \right]\, = \delta \left[ {n - 3} \right]\, + 2\delta \left[ {n - 5} \right]$$ has z-transform x(z). If Y (z)=X (-z)...
Consider the sequence $$x\left[ n \right]$$= $${a^n}u\left[ n \right] + {b^{\partial n}}u\left[ n \right]$$ , where u[n] denotes the unit step seque...
Two casual discrete-time signals $$x\left[ n \right]$$ and $$y\left[ n \right]$$ =$$\sum\limits_{m = 0}^n x \left[ m \right]$$. If the z-transform of ...
The sequence x $$\left[ n \right]$$ = $${0.5^n}$$ u[n], where u$$\left[ n \right]$$ is the unit step sequence, is convolved with itself to obtain y ...
For an all-pass system H(z)= $${{({z^{ - 1}} - b)} \over {(1 - a{z^{ - 1}})}}$$ where $$\left| {H({e^{ - j\omega }})} \right| = \,1$$ , for all $$\o... Let x$$\left[ n \right]$$= x$$\left[- n \right]$$. Let X(z) be the z-transform of x$$\left[ n \right]$$. if 0.5 +j 0.25 is a zero of X(z), which one... An FIR system is described by the system function$$$H(z) = 1 + {7 \over 2}{z^{ - 1}} + {3 \over 2}{z^{ - 2}}$$If$$x\left[ n \right]$$=$${(1/3)^{\left| n \right|}} - {(1/2)^n}u\left[ n \right]$$, then the region of convergence (ROC) of its Z- transform in th... Consider the z-transform X(z)=5$${z^2} + 4{z^{ - 1}} + 3;0 < \left| z \right| < \infty $$. The inverse z - transform x$$\,\left[ n \right]$$i... The ROC of Z-transform of the discrete time sequence x(n)=$${\left( {{1 \over 3}} \right)^{n}}u(n) - {\left( {{1 \over 2}} \right)^{ n}}\,u( - n - 1... If the region of convergence of $${x_1}\left[ n \right]$$ + $${x_2}\left[ n \right]$$ is 1/3< $$\left| {z\,} \right|$$<2/3, then the region of ... The region of convergence of z-transform of the sequence $${\left( {{5 \over 6}} \right)^n}u(n) - {\left( {{6 \over 5}} \right)^n}u( - n - 1)$$ must ... The z transform of a system is H(z) = $${z \over {z - 0.2}}$$ . If the ROC is $$\left| {z\,} \right|$$ < 0.2, then the impulse response of the s... The region of convergence of the z- transform of a unit step function is The z-transform F(z) of the function f(nT) = $${a^{nT}}$$ is The z - transform of the time function $$\sum\limits_{k = 0}^\infty {\delta \left( {n - k} \right)}$$ is ## Marks 2 The ROC (region of convergence) of the z-transform of a discrete-time signal is represented by the shaded region in the z-plane. If the signal $$x\lef... A sequence x$$\left[ n \right]$$is specified as$$\left[ {\matrix{ {x\left[ n \right]} \cr {x\left[ {n - 1} \right]} \cr } } \right] = {\... A realization of a stable discrete time system is shown in the figure. If the system is excited by a unit step sequence input x[n ] , the response y[... Suppose x $$\left[ n \right]$$ is an absolutely summable discrete-time signal. Its z-transform is a rational function with two poles and two zeroes. T... The pole-zero diagram of a causal and stable discrete-time system is shown in the figure. The zero at the origin has multiplicity 4. The impulse resp... For the discrete-time system shown in the figure, the poles of the system transfer function are located at ... The z-transform of the sequence x$$\left[ n \right]$$ is given by x(z)= $${1 \over {{{(1 - 2{z^{ - 1}})}^2}}}$$ , with the region of convergence $$\le... Let$${H_1}(z) = {(1 - p{z^{ - 1}})^{ - 1}},{H_2}(z) = {(1 - q{z^{^{ - 1}}})^{ - 1}}$$, H(z) =$${H_1}(z)$$+r$${H_2}$$. The quantities p, q, r are r... The input-output relationship of a causal stable LTI system is given as 𝑦[𝑛] = 𝛼 𝑦[𝑛 − 1] +$$\beta $$x[n]. If the impulse response h[n] of t... Let x$$\left[ n\right]$$=$${\left( { - {1 \over 9}} \right)^n}\,u(n) - {\left( { - {1 \over 3}} \right)^n}u( - n - 1).$$The region of Convergence (... In the following network (Fig .1), the switch is closed at t = 0- and the sampling starts from t = 0. The sampling frequency is 10 Hz. The expressio... In the following network (Fig.1), the switch is closed at t = 0 and the sampling starts from t=0. The sampling frequency is 10 Hz. The samples x (n)... The z-transform X (z) f a sequence x$$\left[ n \right]$$is given by =$${{0.5} \over {1 - 2{z^{ - 1}}}}$$. It is given that the region of convergen... The z-transform of a signal is given by c(z)=$${1 \over 4}{{{z^{ - 1}}(1 - {z^{ - 4}})} \over {{{(1 - {z^{ - 1}})}^2}}}$$. Its final value is The Z-transform of the following real exponential sequence: x(nT) =$${a^n}$$, nT$$ \ge$\$ 0 =0, nT<0, a> 0 gives us by
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