Signals and Systems

10.7 LTI System Analysis with Z-Transforms

Imron Rosyadi

10.7 Analysis and Characterization of LTI Systems Using Z-Transforms

The Z-transform is not just a mathematical tool; it’s a powerful framework for analyzing discrete-time Linear Time-Invariant (LTI) systems. By transforming difference equations into algebraic expressions, we can gain deep insights into system properties like causality, stability, and frequency response directly from the system function, \(H(z)\). This section ties together many concepts we’ve discussed so far.

The System Function \(H(z)\)

Core Relationship:

For an LTI system, the output \(Y(z)\) is the product of the input \(X(z)\) and the system function \(H(z)\): \[ Y(z) = H(z)X(z) \]

• \(H(z)\) is the Z-transform of the system’s impulse response \(h[n]\).
• It’s also known as the transfer function.

Connection to Frequency Response:

• If the ROC of \(H(z)\) includes the unit circle, then the frequency response \(H(e^{j\omega})\) is obtained by evaluating \(H(z)\) at \(z=e^{j\omega}\).
• This means \(H(z)\) contains all information about the system’s frequency response.

The System Function \(H(z)\)

Eigenfunctions:

• Complex exponentials \(z^n\) are eigenfunctions of LTI systems.
• If \(x[n]=z^n\), then \(y[n]=H(z)z^n\).

Key Aspects of \(H(z)\):

1. Algebraic Expression: The form of \(H(z)\) (e.g., rational function, poles, zeros).
2. Region of Convergence (ROC): Crucial for determining causality and stability.

Pole-Zero Plot of \(H(z)\):

• The locations of poles and zeros in the \(z\)-plane provide a visual representation of the system’s characteristics.
• Combined with the ROC, this plot is a complete characterization of an LTI system in the Z-domain.

At the heart of LTI system analysis with Z-transforms is the system function, \(H(z)\). It’s essentially the Z-transform of the system’s impulse response. The convolution property tells us that in the Z-domain, the output is simply the product of the input and the system function.

This algebraic relationship is incredibly powerful. The system function, along with its ROC, encapsulates all the information about an LTI system, including its frequency response and crucial properties like causality and stability. The pole-zero plot then gives us a visual map of these characteristics.

10.7.1 Causality

Definition:

A discrete-time LTI system is causal if its impulse response \(h[n]\) is zero for \(n<0\).

This means \(h[n]\) is a right-sided sequence.

Z-Domain Condition for Causality:

An LTI system is causal if and only if the ROC of its system function \(H(z)\) is the exterior of a circle, including infinity.

For Rational \(H(z)\):

A causal LTI system with rational \(H(z)\) has two conditions:

1. The ROC is the exterior of a circle outside the outermost pole.
2. When \(H(z)\) is expressed as a ratio of polynomials in \(z\), the order of the numerator cannot be greater than the order of the denominator. (This ensures that \(H(z)\) is finite at \(z=\infty\)).

10.7.1 Causality

Example 10.20 (Non-Causal):

\(H(z)=\frac{z^{3}-2 z^{2}+z}{z^{2}+\frac{1}{4} z+\frac{1}{8}}\)

• Numerator order (3) > Denominator order (2).
• Therefore, \(H(z)\) is not finite at \(z=\infty\).
• Conclusion: System is NOT causal.

Example 10.21 (Causal):

\(H(z)=\frac{1}{1-\frac{1}{2} z^{-1}}+\frac{1}{1-2 z^{-1}}\) with ROC \(|z|>2\).

Combining terms: \(H(z)=\frac{2-\frac{5}{2} z^{-1}}{\left(1-\frac{1}{2} z^{-1}\right)\left(1-2 z^{-1}\right)} = \frac{2z^2 - \frac{5}{2}z}{z^2 - \frac{5}{2}z + 1}\)

• ROC is \(|z|>2\) (exterior of outermost pole).
• Numerator order (2) = Denominator order (2).
• Conclusion: System IS causal.

Causality is a crucial property for physically realizable systems; a causal system’s output cannot depend on future inputs. In the Z-domain, this translates directly to the properties of the ROC.

For a general system, causality means the ROC must be an exterior region and include infinity. For rational system functions, this simplifies to two conditions: the ROC is outside the outermost pole, and the degree of the numerator polynomial is not greater than the degree of the denominator polynomial (when expressed in positive powers of \(z\)). These conditions allow us to quickly determine causality by inspecting \(H(z)\).

10.7.2 Stability

Definition:

An LTI system is stable if its impulse response \(h[n]\) is absolutely summable (\(\sum_{n=-\infty}^{\infty} |h[n]| < \infty\)).

Z-Domain Condition for Stability:

An LTI system is stable if and only if the ROC of its system function \(H(z)\) includes the unit circle, \(|z|=1\).

For Causal Systems with Rational \(H(z)\):

If a system is causal AND has a rational system function, then stability has a simpler condition:

All poles of \(H(z)\) must lie inside the unit circle (\(|p_k| < 1\) for all poles \(p_k\)).

10.7.2 Stability

Example 10.22 (Unstable):

\(H(z)=\frac{1}{1-\frac{1}{2} z^{-1}}+\frac{1}{1-2 z^{-1}}\) with ROC \(|z|>2\).

• Poles at \(z=1/2\) and \(z=2\).
• ROC \(|z|>2\) does NOT include the unit circle.
• Conclusion: System is NOT stable.

Example 10.24 (Second-Order System):

\(H(z)=\frac{1}{1-(2 r \cos \theta) z^{-1}+r^{2} z^{-2}}\) with poles at \(z_1=re^{j\theta}, z_2=re^{-j\theta}\).

• Assuming causality, ROC is \(|z|>r\).
• For stability, the unit circle must be in the ROC, so \(1 < r\) must be false.
• Thus, for a causal system, stability requires \(r<1\).
• Conclusion: Causal system is stable if and only if \(r<1\).

Stability is another critical property for LTI systems, ensuring that bounded inputs produce bounded outputs. In the Z-domain, stability means the ROC must contain the unit circle.

For the common case of causal systems with rational system functions, stability is equivalent to all poles lying strictly inside the unit circle. This provides a very quick and visual way to check stability by looking at the pole-zero plot. The interactive example for the second-order system vividly shows how moving poles inside or outside the unit circle affects stability.

10.7.3 LTI Systems Characterized by Linear Constant-Coefficient Difference Equations

General Form of Difference Equation: \[ \sum_{k=0}^{N} a_{k} y[n-k]=\sum_{k=0}^{M} b_{k} x[n-k] \] Procedure to find \(H(z)\):

1. Apply the Z-transform to both sides.
2. Use linearity and time-shifting properties.
3. Rearrange to find \(H(z) = Y(z)/X(z)\).

Resulting System Function: \[ H(z)=\frac{Y(z)}{X(z)}=\frac{\sum_{k=0}^{M} b_{k} z^{-k}}{\sum_{k=0}^{N} a_{k} z^{-k}} \]

• The system function for such systems is always rational.
• The difference equation alone does not specify the ROC.
• Additional information (e.g., causality, stability) is needed to determine the ROC and thus the unique impulse response.

10.7.3 LTI Systems Characterized by Linear Constant-Coefficient Difference Equations

Example 10.25:

\(y[n]-\frac{1}{2} y[n-1]=x[n]+\frac{1}{3} x[n-1]\)

Z-Transforming:

\(Y(z) - \frac{1}{2}z^{-1}Y(z) = X(z) + \frac{1}{3}z^{-1}X(z)\)

\(Y(z)(1 - \frac{1}{2}z^{-1}) = X(z)(1 + \frac{1}{3}z^{-1})\)

\(H(z) = \frac{Y(z)}{X(z)} = \frac{1 + \frac{1}{3}z^{-1}}{1 - \frac{1}{2}z^{-1}}\)

Possible ROCs and their implications:

• ROC: \(|z| > 1/2\) (outside pole at \(z=1/2\))
  • Implies causal system.
  • \(h[n] = (\frac{1}{2})^n u[n] + \frac{1}{3}(\frac{1}{2})^{n-1} u[n-1]\)
  • Causal and Stable (pole at \(1/2\) is inside unit circle).
• ROC: \(|z| < 1/2\) (inside pole at \(z=1/2\))
  • Implies anti-causal system.
  • \(h[n] = -(\frac{1}{2})^n u[-n-1] - \frac{1}{3}(\frac{1}{2})^{n-1} u[-n]\)
  • Anti-causal and Unstable (ROC does not include unit circle).

Linear constant-coefficient difference equations are the workhorse of discrete-time system implementation. The Z-transform provides a systematic way to convert these time-domain equations into algebraic equations in the Z-domain, making it easy to find the system function \(H(z)\).

It’s crucial to remember that the algebraic form of \(H(z)\) alone isn’t enough; the ROC must be specified. Without it, the same \(H(z)\) could correspond to multiple different impulse responses, each with different causality and stability properties. This example clearly shows how different ROCs lead to different \(h[n]\) and different system behaviors.

10.7.4 Examples Relating System Behavior to the System Function

Example 10.26: System Identification

Given:

1. Input \(x_1[n]=(1/6)^n u[n]\) gives output \(y_1[n]=[a(1/2)^n + 10(1/3)^n] u[n]\).
2. Input \(x_2[n]=(-1)^n\) gives output \(y_2[n]=\frac{7}{4}(-1)^n\).

Goal: Determine \(H(z)\), value of \(a\), and system properties.

Steps:

• Find \(X_1(z)\) and \(Y_1(z)\).
• From \(Y_1(z) = H(z)X_1(z)\), express \(H(z)\) algebraically in terms of \(a\).
• Use property that for \(x_2[n]=(-1)^n\), \(y_2[n]=H(-1)x_2[n]\).
• Solve for \(a\) using \(H(-1) = 7/4\).
• Once \(a\) is found, determine the full \(H(z)\) and its ROC from \(X_1(z)\) and \(Y_1(z)\) ROCs.
• Infer causality and stability.

Result:

\(a=-9\), and \(H(z)=\frac{1-\frac{13}{6} z^{-1}+\frac{1}{3} z^{-2}}{1-\frac{5}{6} z^{-1}+\frac{1}{6} z^{-2}}\) ROC is \(|z|>1/2\).

• Causal (numerator/denominator order is 2, ROC outside outermost pole).
• Stable (poles at \(z=1/2, z=1/3\) are inside unit circle, ROC includes unit circle).

10.7.4 Examples Relating System Behavior to the System Function

Example 10.27: Property Deduction

Given: Stable and causal system, rational \(H(z)\), pole at \(z=1/2\), zero on unit circle.

Statements:

1. \(\mathcal{F}\left\{(1/2)^{n} h[n]\right\}\) converges. TRUE. (Corresponds to \(H(2)\), and \(z=2\) is in ROC).
2. \(H\left(e^{j \omega}\right)=0\) for some \(\omega\). TRUE. (Zero on unit circle means \(H(e^{j\omega})=0\) at that \(\omega\)).
3. \(h[n]\) has finite duration. FALSE. (Pole at \(z=1/2\) implies infinite duration).
4. \(h[n]\) is real. INSUFFICIENT INFO. (Complex poles/zeros must appear in conjugate pairs for real \(h[n]\), but we don’t know other poles/zeros).
5. \(g[n]=n[h[n] * h[n]]\) is the impulse response of a stable system. TRUE.
   • \(H(z)\) has poles inside unit circle.
   • \(H_{conv}(z) = H(z)H(z)\) also has poles inside unit circle.
   • \(G(z) = -z \frac{d}{dz} H_{conv}(z)\) has poles at same locations as \(H_{conv}(z)\) (except possibly origin).
   • Thus, \(G(z)\) has poles inside unit circle, implying stability.

These examples demonstrate the true power of the Z-transform. We can solve complex system identification problems, infer unknown parameters, and deduce system properties from limited information, all by leveraging the properties of the Z-transform and the characteristics of \(H(z)\) and its ROC.

The ability to relate time-domain operations to Z-domain manipulations, and then quickly assess causality and stability from pole-zero locations, is a hallmark of effective discrete-time system analysis.

Conclusion: LTI System Analysis

Key Concepts:

• System Function \(H(z)\): The Z-transform of the impulse response, linking input and output in the Z-domain (\(Y(z)=H(z)X(z)\)).
• Causality: ROC is an exterior region, including infinity (for rational \(H(z)\), numerator order \(\le\) denominator order).
• Stability: ROC includes the unit circle (for causal rational \(H(z)\), all poles inside unit circle).
• Difference Equations: Easily transformed into rational \(H(z)\) using Z-transform properties.

Practical Importance in ECE:

• Filter Design: Design digital filters by specifying pole and zero locations to achieve desired frequency responses while maintaining causality and stability.
• Control Systems: Analyze the stability and transient response of discrete-time control loops.
• System Modeling: Model and simulate complex systems using difference equations and their Z-transforms.
• Digital Signal Processing (DSP): Fundamental for understanding and implementing algorithms in audio, image, and communication processing.

Important

The Z-transform provides a comprehensive and elegant framework for analyzing, designing, and understanding discrete-time LTI systems, making it an indispensable tool for ECE engineers.

In conclusion, the Z-transform is a cornerstone of discrete-time signal and system analysis in ECE. It allows us to characterize LTI systems completely using their system function \(H(z)\) and its ROC. We can determine causality, stability, and even infer the frequency response from the pole-zero plot.

This mathematical elegance translates directly into practical applications, from designing sophisticated digital filters to analyzing complex control systems. A strong grasp of these concepts is absolutely essential for any engineer working in modern digital systems.