ECE04: Z-Transforms and DT-LTI Systems

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Questions and Answers

What is the primary role of the z-transform in the context of DT-LTI systems?

To simplify the design of analog filters.
To analyze and characterize discrete-time linear time-invariant systems. (correct)
To analyze the frequency response of continuous-time systems.
To convert continuous-time signals into discrete-time signals.

What does the notation $X(z) \equiv Z{x[n]}$ represent?

The region of convergence of x[n].
The z-transform of the discrete-time signal x[n]. (correct)
The derivative of x[n].
The inverse z-transform of a signal x[n].

In the equation $X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$, what does 'z' represent?

A real-valued constant.
A time-domain variable.
An integer representing the time index.
A complex variable. (correct)

Why is the Region of Convergence (ROC) important for the z-transform?

It determines the set of 'z' values for which the infinite power series converges. (A) Signup and view all the answers

What is the mathematical definition of the z-transform of a discrete-time function x[n]?

$X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$ (B) Signup and view all the answers

What is the specific term used to describe the collection of 'z' values for which the z-transform of a sequence x[n] converges to a finite value?

Region of Convergence (ROC) (D) Signup and view all the answers

If the z-transform is an infinite power series, what dictates if it exists?

If the series converges for values of z (C) Signup and view all the answers

What is indicated by the relationship $x[n] \overset{z}{\longrightarrow} X(z)$?

The z-transform of the signal x[n] is represented by X(z). (D) Signup and view all the answers

What condition ensures that the Z-transform, $X(z)$, is finite?

When $x[n]r^{-n}$ is absolutely summable (C) Signup and view all the answers

What must be true about the value of $r$ in the expression $x[n]r^{-n}$ for the first term of the Z-transform to be finite?

$r$ must be small enough (B) Signup and view all the answers

In the context of Z-transforms, what is the primary concern regarding the values of $r$?

Finding values of $r$ that make the $x[n]r^{-n}$ absolutely summable (A) Signup and view all the answers

Why must $r$ not be too small when calculating the second term of the Z-transform?

To ensure that sum of the second term does not vanish (B) Signup and view all the answers

If $r = 0.5$, and $x[n] = 2^n$ for $n>0$, what would happen to the second term of the Z Transform when summing from 0 to $\infty$?

It will diverge to infinity (C) Signup and view all the answers

What expression is used to represent the condition for a finite Z-transform?

$|X(z)|$ is finite when $x[n]r^{-n}$ is absolutely summable (A) Signup and view all the answers

What is the relationship between $r$ and the convergence of the Z transform’s individual terms?

A smaller $r$ is required for convergence in the first term and a larger $r$ is required for the second term. (A) Signup and view all the answers

If a signal $x[n]$ results in $x[n]r^{-n}$ not being absolutely summable, what does this imply for its Z-transform $X(z)$?

$X(z)$ does not converge (A) Signup and view all the answers

What is a key consideration when specifying a z-transform?

Its Region of Convergence (ROC). (D) Signup and view all the answers

For a finite duration signal, what is a typical characteristic of its Region of Convergence (ROC)?

It is the entire z-plane, excluding possibly z = 0 and/or z = ∞. (A) Signup and view all the answers

Why are the points z = 0 and z = ∞ sometimes excluded from the Region of Convergence (ROC) of a z-transform?

Powers of $z^k$ (for k > 0) become unbounded at $z = ∞$ and powers of $z^{-k}$ (for $k > 0$) become unbounded at $z = 0$. (B) Signup and view all the answers

If the z-transform can be expressed as a closed expression, what does this imply?

The summation of the z-transform series can be simplified to a formula. (D) Signup and view all the answers

Given z is a complex variable expressed in polar form as $z = re^{j\theta}$, what does 'r' represent?

The radius of z. (C) Signup and view all the answers

How is $X(z)$ expressed with a complex variable in polar form $z = re^{j\theta}$?

$X(z) = \sum_{n=-\infty}^{\infty} x[n] (re^{j\theta})^{-n}$. (A) Signup and view all the answers

What characterizes the Region of Convergence (ROC) regarding the value of $X(z)$?

It includes values where $X(z)$ is a finite constant. (D) Signup and view all the answers

What is the z-transform of $x[n] = \delta[n]$?

$1$ (C) Signup and view all the answers

For a finite-duration, unilateral causal signal $x[n]$ where $0 \le n \le N$, what is the correct Region of Convergence (ROC)?

$z \ne 0$ (D) Signup and view all the answers

Given a signal $x[n]$ with Z-transform $X(z)$, what is the Z-transform of the time-shifted signal $x[n-k]$?

$z^{-k} X(z)$ (A) Signup and view all the answers

If $x[n]$ has a Z-transform $X(z)$ with ROC $r_1 < |z| < r_2$, what is the ROC for $x[-n]$?

$1/r_1 < |z| < 1/r_2$ (B) Signup and view all the answers

If $x[n]$ has the Z-tranform $X(z)$, what is the Z-transform of $nx[n]$?

$-z \frac{dX(z)}{dz}$ (C) Signup and view all the answers

Given two sequences $x_1[n]$ and $x_2[n]$ with Z-transforms $X_1(z)$ and $X_2(z)$ respectively, what is the Z-transform of their convolution $x_1[n] * x_2[n]$?

$X_1(z) \cdot X_2(z)$ (C) Signup and view all the answers

What is the Z-transform of the unit impulse signal $\delta[n]$?

1 (C) Signup and view all the answers

What is the Z-transform of the unit step function $u[n]$?

$\frac{1}{1-z^{-1}}$ (D) Signup and view all the answers

For a causal signal $x[n]$ and $a<1$, what is the ROC for the signal $a^n u[n]$?

$|z| > |a|$ (C) Signup and view all the answers

What is the Z-transform of $(cos(\omega_0 n))u[n]$?

$\frac{1 - z^{-1}cos(\omega_0)}{1 - 2z^{-1}cos(\omega_0)+z^{-2}}$ (C) Signup and view all the answers

For an anti-causal signal, what is the general form of the ROC?

Inside of a circle $|z| < r$ (B) Signup and view all the answers

What is the z-transform of $na^n u[n]$?

$\frac{az^{-1}}{(1-az^{-1})^2}$ (C) Signup and view all the answers

If $x[n]$ has a z-transform $X(z)$, what is the z-transform of $a^n x[n]$?

$X(a^{-1}z)$ (B) Signup and view all the answers

If a signal has a z-transform that sums to two terms, one with an ROC outside a circle of radius $r_1$ and the other with ROC inside a circle of radius $r_2$, what is the overall ROC for the signal?

The intersection of both ROCs (C) Signup and view all the answers

Flashcards

Region of Convergence (ROC)

The range of values for the complex variable 'z' for which the z-transform converges.

Z-Transform

A mathematical tool that transforms a discrete-time signal into a function of a complex variable 'z'.

Finite Duration Signal

A discrete-time signal that has non-zero values only for a finite number of time instants.

ROC of Finite Duration Signals

The z-transform of a finite duration signal has an ROC that includes the entire complex plane except for possibly z = 0 or z = infinity.