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. Signup and view all the answers

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

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

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

Its Region of Convergence (ROC). 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 = ∞. 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 $z = 0$. 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. 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. Signup and view all the answers

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

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

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

$1$ 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)?

$z \ne 0$ 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 $x[n-k]$?

$z^{-k} X(z)$ 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 $x[-n]$?

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

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

$-z \frac{dX(z)}{dz}$ 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 $x_1[n] * x_2[n]$?

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

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

1 Signup and view all the answers

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

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

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

$|z| > |a|$ Signup and view all the answers

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

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

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

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

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

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

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

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

Study Notes

ECE04: Signals, Spectra, and Signal Processing

This course covers Z-transforms and Discrete-Time Linear Time-Invariant (DT-LTI) systems.
The Z-transform is a powerful tool for analyzing DT-LTI systems.
The Z-transform of a discrete-time function x(n) is defined as X(z) = Σ (n=-∞ to ∞) x(n)z⁻ⁿ, where z is a complex variable.
The Z-transform of a signal is denoted by X(z) = Z{x(n)}.
The region of convergence (ROC) is the set of all z values for which the Z-transform converges.
The ROC of a finite-duration signal is the entire z-plane, excluding possible values of z = 0 or z=∞.
For infinite-duration signals, the ROC is typically an annulus/ring in the z-plane. A signal's ROC plays a pivotal role in determining the stability and causality of DT-LTI systems.
The ROC is important to determine the stability of a system.

Z-Transform and DT-LTI System: Points for Discussion

Direct Z-Transform: The transformation of a discrete-time signal x(n) into its corresponding Z-domain representation X(z).
Region of Convergence (ROC): The set of values for 'z' where the Z-transform converges to a finite value.
Z-Transform as a Complex Variable: Understanding how properties of a complex variable relate to the Z-transform.
Characteristic Families of Signals: A discussion about the behavior of known signals in the z-domain.
Properties of Z-Transforms: The rules that govern how Z-transforms behave under various operations on the input signals.
Z-Transforms of Common Functions: A table listing common signals and their corresponding z-transforms.

Properties of the Z-Transform

Linearity: If x₁ (n) ↔ X₁ (z) and x₂ (n) ↔ X₂ (z), then a₁ x₁ (n) + a₂ x₂ (n) ↔ a₁ X₁ (z) + a₂ X₂ (z).
Time Shifting: If x(n) ↔ X(z), then x(n − k) ↔ z⁻ᵏ X(z).
Time Reversal (Folding): If x(n) ↔ X(z), then x(-n) ↔ X(z⁻¹).
Amplitude Scaling: If x(n) ↔ X(z), then ax(n) ↔ aX(z). Also, the ROC changes as well.
Differentiation: If x(n) ↔ X(z), then nx(n) ↔ z(dX(z)/dz).
Convolution: If x₁ (n) ↔ X₁ (z) and x₂ (n) ↔ X₂ (z), then x₁ (n) * x₂ (n) ↔ X₁ (z) * X₂ (z).

Z-Transform as a Complex Variable

In the polar form, z = reʲ⁰, where r = |z| and 0 is the angle of z.
The Z-transform X(z) expressed in the polar form is X(z)|z=reʲ⁰ = Σ(n=-∞ to ∞) x(n)(reʲ⁰)⁻ⁿ = Σ(n=-∞ to ∞) x(n)r⁻ⁿ e⁻ʲⁿθ.

Examples and Sample Problems

Provide examples and sample problems to determine the z-transform and the region of convergence for various discrete-time signals.
Include examples utilizing the convolution property and other characteristics.

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Description

This quiz covers the fundamental concepts of Z-transforms and Discrete-Time Linear Time-Invariant (DT-LTI) systems. Learn about the definition and importance of the Z-transform, the region of convergence (ROC), and how these concepts relate to system stability and causality. Test your understanding of these key topics in signal processing.