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?

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

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

<p>$X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$ (B)</p> 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?

<p>Region of Convergence (ROC) (D)</p> Signup and view all the answers

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

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

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

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

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

<p>When $x[n]r^{-n}$ is absolutely summable (C)</p> 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?

<p>$r$ must be small enough (B)</p> Signup and view all the answers

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

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

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

<p>To ensure that sum of the second term does not vanish (B)</p> 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 $\infty$?

<p>It will diverge to infinity (C)</p> Signup and view all the answers

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

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

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

<p>A smaller $r$ is required for convergence in the first term and a larger $r$ is required for the second term. (A)</p> 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 $X(z)$?

<p>$X(z)$ does not converge (A)</p> Signup and view all the answers

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

<p>Its Region of Convergence (ROC). (D)</p> Signup and view all the answers

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

<p>It is the entire z-plane, excluding possibly z = 0 and/or z = ∞. (A)</p> 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?

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

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

<p>The summation of the z-transform series can be simplified to a formula. (D)</p> 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?

<p>The radius of z. (C)</p> Signup and view all the answers

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

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

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

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

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

<p>$1$ (C)</p> 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)?

<p>$z \ne 0$ (D)</p> 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 $x[n-k]$?

<p>$z^{-k} X(z)$ (A)</p> 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 $x[-n]$?

<p>$1/r_1 &lt; |z| &lt; 1/r_2$ (B)</p> Signup and view all the answers

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

<p>$-z \frac{dX(z)}{dz}$ (C)</p> 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 $x_1[n] * x_2[n]$?

<p>$X_1(z) \cdot X_2(z)$ (C)</p> Signup and view all the answers

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

<p>1 (C)</p> Signup and view all the answers

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

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

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

<p>$|z| &gt; |a|$ (C)</p> Signup and view all the answers

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

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

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

<p>Inside of a circle $|z| &lt; r$ (B)</p> Signup and view all the answers

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

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

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

<p>$X(a^{-1}z)$ (B)</p> 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?

<p>The intersection of both ROCs (C)</p> 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.

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Complex Variable z in Polar Form

A complex variable represented in polar form as z = re^(jθ), where r is the magnitude and θ is the angle.

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Z-Transform in Polar Form

The z-transform expressed in terms of the polar form of the complex variable 'z'.

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Value of z for Convergence

The value of the complex variable 'z' that makes the z-transform converge to a finite value.

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Z-Transform and its Applications

The z-transform is essential for analyzing and manipulating discrete-time signals in various engineering applications.

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z

A complex variable in the Z-transform, representing the magnitude and phase of a complex exponential.

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x[n]

A discrete-time signal, represented as a sequence of values indexed by 'n' (often representing time).

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X(z)

A complex-valued function that represents a discrete-time signal in the z-domain.

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r

A real-valued scalar that determines the rate of exponential growth or decay of the signal in the time domain.

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Absolutely Summable

A condition for a signal to have a finite Z-transform. It means the sum of the absolute values of the signal's terms, multiplied by r^-n, must be finite.

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Discrete-Time Linear Time-Invariant (DT-LTI) System

A system that processes discrete-time signals and is time-invariant, meaning its behavior doesn't change over time.

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Direct Z-transform Formula

The expression X(z) = ∑from n=-∞ to ∞ represents the Z-transform of the discrete-time signal x(n). It involves summing terms of x(n) multiplied by z^(-n), over all values of n.

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Z-transform as a Complex Variable

The Z-transform is considered as a function of a complex variable z. It's not just a single value but rather a function defined for various complex values in the ROC.

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Characteristic Families of Signals

The Z-transform can be applied to different types of signals, each having a characteristic ROC. For example, a finite-duration signal will have a wider ROC compared to an infinite-duration signal.

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Properties of Z-transform

Properties like linearity, time-invariance, and shifting in the time domain have corresponding properties in the z-domain.

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Z-transforms of Common Functions

The Z-transform of commonly encountered signals, such as unit step, unit impulse, exponential signals, etc., are already known. These transforms are widely used in analyzing and solving discrete-time systems.

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Z-Transform and DT-LTI System Analysis

The Z-transform is a powerful tool used in the analysis of discrete-time linear time-invariant (DT-LTI) systems. It helps us understand the behavior of these systems in the z-domain and transform complicated operations into simpler algebraic manipulations.

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What is the Z-transform?

The Z-transform is a mathematical tool for converting a discrete-time signal into a complex function of a complex variable 'z'.

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Amplitude Scaling in the Z-domain

If the Z-transform of a discrete-time signal x(n) is X(z), then the Z-transform of a scaled version of the signal, 'a' multiplied by x(n), is given by X(az).

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Time Reversal (Folding) in the Z-Domain

If the Z-transform of x(n) is X(z) with ROC r1 < |z| < r2, then the Z-transform of the time-reversed signal, x(-n), is X(1/z) with ROC 1/r2 < |z| < 1/r1.

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Time Shifting in the Z-Domain

If the Z-transform of a signal x(n) is X(z) with ROC r1 < |z| < r2, then the Z-transform of the time-shifted signal, x(n - k), is z^(-k)X(z) with the same ROC.

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Infinite-Duration Signal

A sequence where an infinite number of samples are non-zero.

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Unilateral, Causal Signal

A discrete-time signal that is non-zero only for n ≥ 0.

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Unilateral, Anti-Causal Signal

A discrete-time signal that is non-zero only for n ≤ 0.

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Bilateral Signal

A discrete-time signal that is non-zero for both positive and negative values of n.

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Linearity Property of Z-transform

The Z-transform of a linear combination of two signals is equal to the same linear combination of their individual Z-transforms.

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Convolution Property in Z-transform

If the Z-transforms of two signals are X1(z) and X2(z), then the Z-transform of their convolution is the product of their individual Z-transforms: X(z) = X1(z) * X2(z).

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Differentiation Property in Z-transform

The Z-transform of the derivative of a signal is equal to -z times the derivative of its Z-transform.

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Multiplication by 'n' Property in Z-transform

If the Z-transform of a signal x(n) is X(z), then the Z-transform of the signal multiplied by 'n' is -z times the derivative of X(z).

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How to find the ROC for a signal with multiple terms?

The ROC is the area common among the ROCs of all the terms in the Z-transform.

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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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ECE04 Lesson 08 Z-Transform and the DT-LTI System (2) PDF

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