# Questions & Answers of Z-transform

## Weightage of Z-transform

Total 26 Questions have been asked from Z-transform topic of Signals and Systems subject in previous GATE papers. Average marks 1.54.

Consider the sequence $x\left[n\right]={a}^{n}u\left[n\right]+{b}^{n}u\left[n\right]$, where $u\left[n\right]$ denotes the unit-step sequence and $0<\left|a\right|<\left|b\right|<1$. The region of convergence (ROC) of the z-transform of $x\left[n\right]$ is

A discreate-time signal  $x\left[n\right]=\delta \left[n-3\right]+2\delta \left[n-5\right]$ has z-transform $X(z).$ If $Y(z)=X(-z)$ is the z-transform of another signal $y[n]$,then

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\left[n\right]={\left(2.0\right)}^{\left|n\right|},-\infty , then the ROC of its z-transform is represented by

For the discrete time system shown in the figure, the poles of the system transfer function are Located at

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 response of the system is h[n]. If h[0] = 1, we can conclude.

Two casual discrete-time signals $x\lbrack n\rbrack$ and $y\lbrack n\rbrack$ are related as $y\left[n\right]=\sum _{m=0}^{n}x\left[m\right]$ . If the z-transform of , the value of $x\lbrack2\rbrack$ is _______.

The value of $\sum\limits_{n=0}^\infty n\left(\frac12\right)^n$ is _____.

Consider a four-point moving average filter defined by the equation . The condition on the filter coefficients that results in a null at zero frequency is

Suppose x[n] is an absolutely summable discrete-time signal. Its z-transform is a rational function with two poles and two zeroes. The poles are at $z=±2j$ . Which one of the following statements is TRUE for the signal x[n]?

C is a closed path in the z-plane given by |z|=3. The value of the integral $\oint_C\left(\frac{z^2-z+4j}{z+2j}\right)dz$

Let  $x\left[n\right]={\left(-\frac{1}{9}\right)}^{n}u\left(n\right)-{\left(-\frac{1}{3}\right)}^{n}u\left(-n-1\right)$ The Region of Convergence (ROC) of the z-transform of x[n]

Let x[n]=x[-n]. Let X(z) be the z-transform of x[n]. If 0.5+j 0.25 is a zero of ,X(z) which one of the following must also be a zero of X(z)

The input-output relationship of a causal stable LTI system is given as

If the impulse response h[n] of this system satisfies the condition ${\sum }_{n=0}^{\infty }h\left[n\right]=2$, the relationship between α and β is

For an all-pass system $H\left(z\right)=\frac{\left({z}^{-1}-b\right)}{\left(1-a{z}^{-1}\right)}$, where $\left|H\left({e}^{-j\omega }\right)\right|=1$ for all $\omega$
if $Re\left(a\right)\ne 0,Im\left(a\right)\ne 0,$then b equals

Let ${H}_{1}\left(z\right)={\left(1-p{z}^{-1}\right)}^{-1},{H}_{2}\left(z\right)={\left(1-q{z}^{-1}\right)}^{-1},H\left(z\right)={H}_{1}\left(z\right)+r{H}_{2}\left(z\right)$. The quantities $p,q,r$ are real numbers. Consider $p=\frac{1}{2}$,$q=-\frac{1}{4}$,$\left|r\right|<1$. If the zero of $H\left(z\right)$ lies on the unit circle, then r = ________

The z-transform of the sequence x[n] is given by $X\left(z\right)=\frac{1}{{\left(1-2{z}^{-1}\right)}^{2}}$  , with the region of convergence $\left|z\right|>2$. Then, $x\left[2\right]$ is ________.

If $x\left[n\right]={\left(1/3\right)}^{\left|n\right|}-{\left(1/2\right)}^{n}u\left[n\right]$, then the region of convergence (ROC) of its Z-transform in the Z-plane will be

Two systems H1 (z) and H2 (z) are connected in cascade as shown below. The over all output y(n) is the same as the input x(n) with a one unit delay. The transfer function of the second system H2 (z) is

Consider the z-transform X(z) = 5z2 + 4z-1 + 3; 0<|z| < ∞ . The inverse z transform x[n] is

Two discrete time systems with impulse responses h1[n] = δ [n -1] and h2[n] = δ [n – 2] are connected in cascade. The overall impulse response of the cascaded system is

The transfer function of a discrete time LTI system is given by

$H\left(z\right)=\frac{2-\frac{3}{4}{z}^{-1}}{1-\frac{3}{4}{z}^{-1}+\frac{1}{8}{z}^{-2}}$

Consider the following statements:

S1: The system is stable and causal for ROC:$\left|z\right|>\frac{1}{2}$

S2: The system is stable but not causal for ROC:$\left|z\right|<\frac{1}{4}$

S3: The system is neither stable nor causal for ROC:$\frac{1}{4}<\left|z\right|<\frac{1}{2}$

Which one of the following statements is valid?

The ROC of Z-transform of the discrete time sequence $x\left(n\right)={\left(\frac{1}{3}\right)}^{n}u\left(n\right)-{\left(\frac{1}{2}\right)}^{n}u\left(-n-1\right)$ is

A system with transfer function H(z) has impulse response h(·) defined as h(2) = 1, h(3) = -1 and h(k) = 0 otherwise. Consider the following statements
S1 : H(z) is a low pass filter
S2 : H(z) is a FIR filter
Which of the following is correct?

In the following network, the switch is closed at t = 0- and the sampling starts from t = 0. The sampling frequency is 10 Hz.

The samples x(n) (n = 0, 1, 2, ...) are given by

In the following network, the switch is closed at t = 0- and the sampling starts from t = 0. The sampling frequency is 10 Hz.

The expression and the region of convergence of the z-transform of the sampled signal are

The z-transform X[z] of a sequence x[n] is given by $X\left[z\right]=\frac{0.5}{1-2{z}^{-1}}$. It is given that the region of convergence of X[z] includes the unit circle. The value of x[0] is