12.2 Z-Transform and Its Properties
4 min read•july 30, 2024
The is a powerful tool for analyzing discrete-time signals and systems. It converts time-domain signals into complex frequency-domain representations, simplifying the analysis of linear time-invariant systems. Understanding its properties and techniques is crucial for working with sampled-data systems.
This section covers the Z-transform definition, common signal transforms, and key properties like and . It also explores techniques, including partial fraction expansion and power series methods, essential for converting frequency-domain solutions back to the time domain.
Z-Transform Definition
Definition and Region of Convergence
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- The z-transform converts a discrete-time signal into a complex frequency-domain representation
- Defined as X(z) = Σ x[n] * z^(-n), where the sum is taken from n = -∞ to +∞
- The (ROC) is the set of complex numbers (z) for which the z-transform summation converges
- Determines the uniqueness and stability of the z-transform
- Depends on the location of the poles of the z-transform and the boundedness of the signal
ROC for Different Signal Types
- For a right-sided signal (x[n] = 0 for n < 0), the ROC is the region outside the outermost pole
- For a left-sided signal (x[n] = 0 for n > 0), the ROC is the region inside the innermost pole
- For a two-sided signal, the ROC is the region between the outermost poles on both sides
- The ROC must include the unit circle (|z| = 1) for a stable and causal system
Z-Transform of Signals
Common Discrete-Time Signals
- Unit impulse signal δ[n]: X(z) = 1 for all z
- Unit step signal u[n]: X(z) = z / (z - 1) for |z| > 1
- Exponential signal a^n * u[n]: X(z) = z / (z - a) for |z| > |a|
- Example: 2^n * u[n] has a z-transform X(z) = z / (z - 2) for |z| > 2
- Sinusoidal signal cos(ω0 * n) * u[n]: X(z) = (z * cos(ω0)) / (z^2 - 2z * cos(ω0) + 1) for |z| > 1
- Sinusoidal signal sin(ω0 * n) * u[n]: X(z) = (z * sin(ω0)) / (z^2 - 2z * cos(ω0) + 1) for |z| > 1
Calculating Z-Transforms
- Use the definition of the z-transform to calculate the z-transform of a given discrete-time signal
- Identify the ROC based on the signal type and the location of the poles
- Example: For x[n] = (1/2)^n * u[n], X(z) = z / (z - 1/2) for |z| > 1/2
Z-Transform Properties
Linearity and Time-Shifting
- Linearity: The z-transform of a linear combination of signals equals the linear combination of their individual z-transforms
- Example: If X1(z) and X2(z) are the z-transforms of x1[n] and x2[n], then a * x1[n] + b * x2[n] has a z-transform a * X1(z) + b * X2(z)
- Time-shifting: If x[n] has a z-transform X(z), then x[n-k] has a z-transform z^(-k) * X(z)
- Example: If X(z) is the z-transform of x[n], then x[n-3] has a z-transform z^(-3) * X(z)
Scaling, Differentiation, and Convolution
- Scaling in the z-domain: If x[n] has a z-transform X(z), then a^n * x[n] has a z-transform X(z/a)
- Differentiation in the z-domain: If x[n] has a z-transform X(z), then n * x[n] has a z-transform -z * (dX(z)/dz)
- in the time-domain: If x[n] and h[n] have z-transforms X(z) and H(z), then their convolution y[n] = x[n] * h[n] has a z-transform Y(z) = X(z) * H(z)
- Example: If X(z) = z / (z - 1) and H(z) = z / (z - 2), then Y(z) = X(z) * H(z) = z^2 / ((z - 1)(z - 2))
Initial and Final Value Theorems
- : The initial value of a signal x[n] can be found using lim(z→∞) X(z), provided that the limit exists
- : The final value of a signal x[n] can be found using lim(z→1) (z-1) * X(z), provided that the limit exists and the poles of (z-1) * X(z) are inside the unit circle
- These theorems are useful for determining the steady-state behavior of a system or signal
Inverse Z-Transform Techniques
Partial Fraction Expansion
- Partial fraction expansion decomposes a rational z-transform into a sum of simpler terms
- Each term can be individually inverse transformed using z-transform tables or properties
- Factor the denominator of the z-transform and express it as a sum of terms with simple poles or poles of higher multiplicity
- For distinct poles, the partial fraction expansion takes the form X(z) = Σ (Ai / (z - pi)), where Ai is the residue at pole pi
- The residue at a simple pole pi can be calculated using Ai = lim(z→pi) ((z - pi) * X(z))
- For repeated poles, the partial fraction expansion takes the form X(z) = Σ (Aik / (z - pi)^k), where Aik is the coefficient of the kth term in the expansion of the pole pi
Other Inverse Z-Transform Techniques
- Power series expansion: Expand the z-transform into a power series and match the coefficients with the time-domain signal
- Long division: Divide the numerator by the denominator to obtain a power series expansion of the z-transform
- Contour integration: Use complex integration techniques to evaluate the inverse z-transform integral
- Requires knowledge of complex analysis and residue theory
- These techniques can be used when partial fraction expansion is not feasible or when dealing with non-rational z-transforms
Key Terms to Review (17)
Bilateral z-transform: The bilateral z-transform is a mathematical tool used to analyze discrete-time signals and systems by transforming a sequence of values into a complex frequency domain representation. This transform is defined for all time indices, allowing it to handle both causal and non-causal signals effectively. The bilateral z-transform is essential for understanding system stability, frequency response, and the properties of linear time-invariant systems.
Bilinear Transformation: The bilinear transformation is a mathematical technique used to convert continuous-time systems into discrete-time systems by mapping the s-plane into the z-plane. This method allows for the design of discrete-time filters that closely approximate their continuous-time counterparts, making it a powerful tool in digital signal processing. It preserves the stability and frequency characteristics of the original system while facilitating analysis and implementation in a digital context.
Convolution: Convolution is a mathematical operation that combines two functions to produce a third function, representing the way in which the shape of one is modified by the other. In the context of systems and signals, convolution describes how an input signal is transformed by a system's impulse response, allowing for the analysis of linear time-invariant systems. This operation is essential in understanding the behavior of systems in the Z-Transform framework, as it provides insight into output responses based on input signals.
Discrete-Time Systems: Discrete-time systems are systems that operate on discrete-time signals, meaning that the input and output signals are defined only at distinct intervals, rather than continuously over time. This makes them suitable for digital processing, where information is represented in binary form and computations are performed at specific time steps. The behavior of these systems is often analyzed using tools like the Z-transform, which helps in understanding their stability and response characteristics.
Fast Fourier Transform: The Fast Fourier Transform (FFT) is an algorithm that efficiently computes the Discrete Fourier Transform (DFT) and its inverse, significantly speeding up the process of converting a signal from its original domain into the frequency domain. By reducing the complexity from O(N^2) to O(N log N), FFT is essential in applications that require analyzing the frequency components of signals, making it a crucial tool in various fields such as signal processing, mechanical systems modeling, and control theory.
Filter design: Filter design refers to the process of creating filters that selectively allow certain frequencies of signals to pass through while attenuating others. This involves understanding the frequency response of the system and the desired specifications for how the filter should perform in terms of gain, phase shift, and stability. The design process takes into account various methods and techniques to achieve optimal filtering characteristics for different applications, such as signal analysis and processing.
Final Value Theorem: The Final Value Theorem is a key concept in control theory and signal processing that provides a method to determine the steady-state value of a function as time approaches infinity. This theorem connects the Laplace and Z-transforms, allowing engineers and scientists to predict the long-term behavior of dynamic systems without solving differential equations directly. It is particularly useful in analyzing the stability and response of systems in both continuous and discrete time domains.
Initial Value Theorem: The initial value theorem is a fundamental concept in the context of the Z-transform, stating that the initial value of a discrete-time signal can be directly determined from its Z-transform. This theorem is particularly useful for analyzing dynamic systems as it provides a way to extract the behavior of a system at the start of its operation without having to revert to time-domain calculations.
Inverse z-transform: The inverse z-transform is a mathematical operation that converts a function in the z-domain back into the time-domain representation. This process is crucial for analyzing and understanding discrete-time systems, as it allows engineers and scientists to retrieve the original time-series data from its z-transform representation. The inverse z-transform utilizes various techniques such as long division, residue theorem, or power series expansion to achieve this transformation.
Linearity: Linearity refers to the property of a system or function where the principle of superposition applies, meaning that the output is directly proportional to the input. This property is fundamental in various mathematical transformations and analysis techniques, allowing complex systems to be simplified and analyzed more easily, especially when dealing with differential equations and signal processing.
LTI Systems: LTI systems, or Linear Time-Invariant systems, are a class of systems characterized by linearity and time invariance. This means that the response of the system to an input signal can be expressed as a linear combination of the inputs and that the system's behavior does not change over time. LTI systems are fundamental in signal processing and control theory due to their desirable properties, such as the ability to analyze them using tools like the Z-transform.
Partial fraction decomposition: Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions, making it easier to perform operations such as integration and finding inverse transforms. This method is particularly useful in transforming expressions into a form that can be more easily handled when applying inverse Laplace or Z-transforms, as it allows for the separation of terms based on their order and coefficients.
Pole-Zero Plot: A pole-zero plot is a graphical representation used in control theory and signal processing to illustrate the locations of the poles and zeros of a system's transfer function in the complex plane. This plot provides insights into system stability, frequency response, and transient behavior, as poles correspond to the system's natural frequencies and zeros affect the gain at those frequencies. The arrangement of poles and zeros directly influences the dynamics of a system, making this plot a fundamental tool for analyzing and designing control systems.
Region of Convergence: The region of convergence (ROC) is the set of values in the complex plane for which a given integral or series converges to a finite value. This concept is crucial for determining the stability and behavior of systems when using transforms like the Laplace and Z-transforms, as it defines where these transforms are valid and provides insights into system properties such as stability.
Stability Analysis: Stability analysis is the process of determining whether a dynamic system will return to equilibrium after a disturbance. It involves assessing how system parameters affect system behavior over time, particularly in response to changes or inputs. This concept is essential for designing systems that behave predictably and remain functional under various conditions, connecting deeply with modeling, nonlinear dynamics, feedback systems, and discrete-time analysis.
Time-shifting: Time-shifting refers to the process of changing the time reference of a function in the context of system analysis, allowing the function to be evaluated at different time intervals. This concept is crucial for understanding how signals can be manipulated, particularly in inverse transformations and discrete signal processing. By shifting a signal in time, we can analyze its behavior at various points and understand how these shifts affect the system's response.
Z-transform: The z-transform is a mathematical tool used to analyze discrete-time signals and systems by converting sequences of data into a complex frequency domain representation. This transformation allows for the manipulation of signals in a way that makes it easier to understand their behavior, particularly in the context of stability and frequency response. It serves as a bridge between time-domain representations and frequency-domain analysis, playing a crucial role in the study of discrete-time systems and transfer functions.