Z Transform Practice Questions

Let H(z) be the z-transform of a real-valued discrete-time signal h[n]. If $P(z)=H(z)H\left (\frac{1}{z} \right )$ has a zero at $z=\frac{1}{2}+\frac{1}{2}j$ , and P(z) has a total of four zeros, which one of the following plots represents all the zeros correctly?

A discrete-time all-pass system has two of its poles at 0.25 $\angle 0^{\circ}$ and 2 $\angle 30^{\circ}$ . Which one of the following statements about the system is TRUE?

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

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

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

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

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[n] 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 $z = \pm 2j$ . Which one of the following statements is TRUE for the signal x[n]?

Two causal discrete-time signals x[n]and y[n] are related as $y[n]=\sum_{m=0}^{n}x[m]$ . If the z-transform of y[n] is $\frac{2}{z(z-1)^{2}}$ , the value of x[2] is _______.

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 canconclude

Consider a four-point moving average filter defined by the equation $y[n]=\sum_{i=0}^{3}\alpha _{i} x[n-i]$ , The condition on the filter coefficients that results in a null at zero frequency is

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

Let x[n] = x[-n]. Let X(z) be the z -transform of x[n]. If 0.5+j0.25 is a zero of X(z), which one of the following must also be a zero of X(z).

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

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

The input-output relationship of a causal stable LTI system is given as $y[n] = \alpha$ y[n -1]+ $\beta x[n]$ If the impulse response h[n] of this system satisfies the condition $\sum_{n=0}^{\infty }h[n]=2$ , the relationship between $\alpha$ and $\beta$ is

Let $H_{1}(z)=(1-pz^{-1})^{-1}$ , $H_{2}(z)=(1-qz^{-1})^{-1}$ , $H(z)=H_{1}(z)+rH_{2}(z)$ . The quantities $p, q, r$ are real numbers. Consider $p=\frac{1}{2}, q=-\frac{1}{4},|r|\lt 1$ . If the zero of $H(z)$ lies on the unit circle, then $r$ = ____.

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

Two systems $H_{1}(z)$ and $H_{2}(z)$ are connected in cascade as shown below. The overall output $y(n)$ is the same as the input $x(n)$ with a one unit delay. The transfer function of the second system $H_{2}(z)$ is