Z-Transform Properties for LTI System Analysis

Hello! Welcome to your next lesson on discrete-time systems.

In our last session, we established the crucial geometric link between the z-transform and the DTFT, seeing that the frequency response of a system is simply its z-transform evaluated on the unit circle. This gave us a powerful visual intuition for how poles and zeros shape system behavior.

Today, we will build on that foundation by exploring the operational properties of the z-transform. Our goal is to apply z-transform properties to analyze discrete-time LTI systems. Just as Laplace transform properties allow us to convert continuous-time differential equations into algebraic problems, z-transform properties do the same for discrete-time difference equations. This is the core mechanism that makes the z-transform such a powerful tool for system analysis and design.

1. From Difference Equations to Transfer Functions

Many discrete-time LTI systems, especially digital filters, are described by linear constant-coefficient difference equations (LCCDEs). A general form for such an equation is:

k = 0 \sum N a_{k} y [n - k] = k = 0 \sum M b_{k} x [n - k]

where $x [n]$ is the input, $y [n]$ is the output, and $a_{k}$ and $b_{k}$ are constant coefficients. Analyzing this directly in the time domain involves complex recursive calculations.

The power of the z-transform lies in its ability to convert this equation into a simple algebraic one. By taking the z-transform of both sides and applying the time-shifting property, we can solve for the ratio of the output transform $Y (z)$ to the input transform $X (z)$ . This ratio is the system's transfer function, $H (z)$ .

The following video provides an excellent introduction to this process, showing how a difference equation is transformed into its equivalent transfer function representation.

z-Transform Analysis of LTI Systems

The video 'z-Transform Analysis of LTI Systems' by Barry Van Veen demonstrates how the z-transform converts difference equations into algebraic transfer functions, which characterize the system in terms of poles and zeros.

Please watch from the beginning to 05:08. Pay close attention to: The general form of a difference equation. How the time-shift property is used to transform the difference equation into an algebraic equation in Z. The definition of the transfer function H(z) and how it's represented by the coefficients of the difference equation.

As the video showed, the transfer function becomes:

H (z) = \frac{Y ( z )}{X ( z )} = \frac{\sum _{k = 0}^{M} b _{k} z ^{- k}}{\sum _{k = 0}^{N} a _{k} z ^{- k}}

This elegant expression encapsulates the system's input-output relationship. To master this transformation, we must first master the properties that enable it.

2. Key Properties of the Z-Transform

The properties of the z-transform are analogous to those of the Laplace and Fourier transforms. They provide the rules for manipulating signals and systems in the z-domain. The textbook chapter below provides detailed derivations, which will appeal to your interest in a deeper mathematical treatment.

Discrete-Time System Analysis Using the Z-Transform

The chapter 'Discrete-Time System Analysis Using the Z-Transform' from Lathi's textbook is a comprehensive resource. We will use it to formally define and prove the key properties.

Please read Section 11.2, 'Some properties of the Z-Transform' (pages 681-685). Focus on understanding the formulas and the logic behind the proofs for the following properties: Right Shift (Delay): Note the important distinction between f[k-m]u[k-m] (a shifted signal) and f[k-m]u[k] (a shifted signal viewed within the causal window, which brings in initial conditions). Left Shift (Advance): Understand how this involves initial values like f[0], f[1], etc. Convolution: This is the most critical property for LTI systems. Multiplication by γᵏ (Scaling in z-domain) Multiplication by k (Differentiation in z-domain)

Let's summarize the most important properties and their significance.

Property	Time Domain Signal $x [n]$	Z-Transform $X (z)$	Region of Convergence
Linearity	$a_{1} x_{1} [n] + a_{2} x_{2} [n]$	$a_{1} X_{1} (z) + a_{2} X_{2} (z)$	At least $R_{1} \cap R_{2}$
Time Shift	$x [n - n_{0}]$	$z^{- n_{0}} X (z)$	$R_{x}$ (possible addition/removal of $z = 0$ or $z = \infty$ )
Convolution	$x_{1} [n] * x_{2} [n]$	$X_{1} (z) X_{2} (z)$	At least $R_{1} \cap R_{2}$
Scaling in z-domain	$a^{n} x [n]$	$X (z / a)$	Scaled ROC: $
Differentiation in z	$n x [n]$	$- z \frac{d X ( z )}{d z}$	$R_{x}$

Table adapted from 'Discrete–Time Linear, Time Invariant Systems & z–Transforms' and Lathi's text.

The Convolution Property is paramount. For an LTI system with impulse response $h [n]$ and input $x [n]$ , the output is $y [n] = x [n] * h [n]$ . Applying the z-transform, we get the simple multiplicative relationship:
$Y (z) = X (z) H (z)$
This confirms our definition of the transfer function $H (z)$ as the z-transform of the impulse response, $h [n]$ .

Test your understanding!

A signal is defined as $y [n] = n a^{n} u [n]$ . Use the z-transform properties to find its z-transform, $Y (z)$ .

Hint: Start with the known transform of $u [n]$ and apply two properties in sequence.

Show answer

Start with the base signal $x [n] = u [n]$ . Its z-transform is $X (z) = \frac{z}{z - 1}$ .
Apply the Scaling in z-domain property. Let $x_{1} [n] = a^{n} x [n] = a^{n} u [n]$ . Its transform is $X_{1} (z) = X (z / a)$ . $X_{1} (z) = \frac{z / a}{( z / a ) - 1} = \frac{z}{z - a}$ This is the known transform for $a^{n} u [n]$ .
Now apply the Differentiation in z-domain property to $x_{1} [n]$ . Let $y [n] = n x_{1} [n] = n a^{n} u [n]$ . Its transform is $Y (z) = - z \frac{d X _{1} ( z )}{d z}$ . $\frac{d X _{1} ( z )}{d z} = \frac{d}{d z} (\frac{z}{z - a}) = \frac{( 1 ) ( z - a ) - ( z ) ( 1 )}{( z - a ) ^{2}} = \frac{- a}{( z - a ) ^{2}}$ $Y (z) = - z (\frac{- a}{( z - a ) ^{2}}) = \frac{a z}{( z - a ) ^{2}}$ This is a standard z-transform pair, and we have derived it using only the basic properties.

3. Application: Solving Difference Equations

With these properties, we can now solve LCCDEs. The process involves three main steps:

Transform the entire difference equation from the time domain to the z-domain using the linearity and time-shift properties. This step requires careful handling of initial conditions.
Algebraically solve for the z-transform of the output, $Y (z)$ .
Find the output signal $y [n]$ by calculating the inverse z-transform of $Y (z)$ , typically using partial fraction expansion and a table of z-transform pairs.

This method allows us to find the complete response of a system, which can be further separated into its zero-input response (due to initial conditions alone) and zero-state response (due to the input signal alone).

The following reading walks through a detailed example that puts all these pieces together.

Discrete-Time System Analysis Using the Z-Transform

Let's return to the Lathi textbook chapter to see these properties in action. This section provides a worked example of solving a difference equation with non-zero initial conditions.

Please read Section 11.3, 'Z-Transform Solution of Linear Difference Equations', focusing on Example 11.5 (pages 685-688). Then, read the subsection 'Zero-Input and Zero-State Components' (page 689) which decomposes the solution from that same example. Follow the example closely to see how: The right-shift property is applied to terms like y[k-1] and y[k-2], introducing terms for the initial conditions y[-1] and y[-2]. The terms in the resulting expression for Y(z) are grouped according to whether they came from the initial conditions (zero-input) or the input signal f[k] (zero-state).

The key insight from the example is how the unilateral z-transform property for a right-shifted signal,

Z {f [k - 1] u [k]} = z^{- 1} F (z) + f [- 1]

naturally incorporates the system's state (its initial conditions) into the algebraic equation. This is what makes the z-transform a complete tool for analyzing LTI systems, not just for finding the transfer function in a zero-state scenario.

Conclusion

In this lesson, we transitioned from understanding what the z-transform is to how it is used. By mastering its properties, you have unlocked the primary method for analyzing discrete-time LTI systems described by difference equations.

Key Takeaways:

The z-transform converts linear constant-coefficient difference equations (time domain) into algebraic equations (z-domain).
The linearity, time-shift, and convolution properties are fundamental to this process.
The system's transfer function $H (z)$ is the z-transform of its impulse response $h [n]$ and can be found directly from the coefficients of its difference equation.
The relationship $Y (z) = H (z) X (z)$ is the cornerstone of LTI system analysis in the z-domain.
The z-transform method can solve for the complete system response, including contributions from both the input (zero-state response) and the initial conditions (zero-input response).

Preview of the Next Lesson:

We have now defined the transfer function $H (z)$ and seen how it arises from a system's difference equation. The rational function $H (z)$ is characterized by its poles and zeros. In our next lesson, we will focus entirely on how the locations of these poles and zeros in the z-plane allow us to determine critical system attributes like stability and causality without having to transform back to the time domain. This will complete our foundational understanding of z-domain analysis.

Recommended resources

1. From Difference Equations to Transfer Functions

2. Key Properties of the Z-Transform

3. Application: Solving Difference Equations

Conclusion