11.2 Z-transforms

Z-transforms are a key tool in Control Theory for analyzing discrete-time systems. They convert discrete-time signals into complex frequency domain representations, similar to how Laplace transforms work for continuous-time systems.

Z-transforms enable engineers to study system stability, design controllers, and analyze frequency responses. They're especially useful for digital filters, discrete-time controllers, and other applications where signals are sampled at regular intervals.

Definition of Z-transforms

• Z-transforms are a powerful mathematical tool used in Control Theory for analyzing and designing discrete-time systems
• Provide a way to represent discrete-time signals and systems in the complex frequency domain, similar to how Laplace transforms are used for continuous-time systems

Discrete-time signals

Top images from around the web for Discrete-time signals

• 

  What is Discrete Fourier Transform(DFT) | ee-diary View original

  Is this image relevant?

• 

  Analog To Digital Conversion - Sampling and Quantization - Electronics-Lab.com View original

  Is this image relevant?

• 

  Computing the Z-transform in MATLAB | ee-diary View original

  Is this image relevant?

• 

  What is Discrete Fourier Transform(DFT) | ee-diary View original

  Is this image relevant?

• 

  Analog To Digital Conversion - Sampling and Quantization - Electronics-Lab.com View original

  Is this image relevant?

1 of 3

Top images from around the web for Discrete-time signals

• 

  What is Discrete Fourier Transform(DFT) | ee-diary View original

  Is this image relevant?

• 

  Analog To Digital Conversion - Sampling and Quantization - Electronics-Lab.com View original

  Is this image relevant?

• 

  Computing the Z-transform in MATLAB | ee-diary View original

  Is this image relevant?

• 

  What is Discrete Fourier Transform(DFT) | ee-diary View original

  Is this image relevant?

• 

  Analog To Digital Conversion - Sampling and Quantization - Electronics-Lab.com View original

  Is this image relevant?

1 of 3

• Discrete-time signals are sequences of values defined at discrete time instants, usually represented as $x[n]$, where $n$ is an integer representing the time index
• Can be obtained by sampling continuous-time signals at regular intervals or generated directly in digital systems
• Examples of discrete-time signals include:
  • Audio samples in digital signal processing (speech, music)
  • Image pixels in digital image processing
  • Sensor readings in digital control systems (temperature, pressure)

Bilateral vs unilateral Z-transforms

• Bilateral Z-transform considers the entire time axis, from $-\infty$ to $+\infty$, and is defined as: $X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}$
• Unilateral Z-transform considers only the positive time axis, from $0$ to $+\infty$, and is defined as: $X(z) = \sum_{n=0}^{+\infty} x[n] z^{-n}$
• Unilateral Z-transform is more commonly used in practice, as it is more suitable for causal systems and easier to compute

Region of convergence

• The region of convergence (ROC) is the set of complex numbers $z$ for which the Z-transform converges
• Determines the uniqueness of the Z-transform and provides information about the stability and causality of the system
• ROC depends on the location of the poles of the Z-transform and can be:
  • Outside a circle (stable and causal systems)
  • Inside a circle (stable and anticausal systems)
  • Annular region (stable systems with mixed causality)

Properties of Z-transforms

• Z-transforms have several important properties that facilitate the analysis and design of discrete-time systems in Control Theory
• These properties allow for the manipulation of Z-transforms to simplify calculations and gain insights into system behavior

Linearity

• Z-transform is a linear operator, which means that for any two discrete-time signals $x_1[n]$ and $x_2[n]$ and any constants $a$ and $b$: $\mathcal{Z}\{a x_1[n] + b x_2[n]\} = a X_1(z) + b X_2(z)$
• Linearity property allows for the superposition of signals and systems in the Z-domain

Time shifting

• Time shifting a discrete-time signal $x[n]$ by $k$ samples results in a multiplication of its Z-transform by $z^{-k}$: $\mathcal{Z}\{x[n-k]\} = z^{-k} X(z)$
• Positive shifts (delay) correspond to multiplication by negative powers of $z$, while negative shifts (advance) correspond to multiplication by positive powers of $z$

Scaling in Z-domain

• Multiplying a discrete-time signal $x[n]$ by $a^n$ results in a substitution of $z$ by $z/a$ in its Z-transform: $\mathcal{Z}\{a^n x[n]\} = X(z/a)$
• Scaling property is useful for analyzing systems with exponential factors, such as damped sinusoids

Time reversal

• Time reversing a discrete-time signal $x[n]$ results in a substitution of $z$ by $1/z$ in its Z-transform: $\mathcal{Z}\{x[-n]\} = X(1/z)$
• Time reversal property is useful for analyzing systems with symmetric or antisymmetric impulse responses

Convolution in Z-domain

• Convolution of two discrete-time signals $x_1[n]$ and $x_2[n]$ in the time domain corresponds to multiplication of their Z-transforms in the Z-domain: $\mathcal{Z}\{x_1[n] * x_2[n]\} = X_1(z) X_2(z)$
• Convolution property simplifies the analysis of cascaded systems and the design of digital filters

Differentiation in Z-domain

• Differentiation of a discrete-time signal $x[n]$ in the time domain corresponds to multiplication of its Z-transform by $nz^{-1}$: $\mathcal{Z}\{nx[n]\} = -z \frac{d}{dz} X(z)$
• Differentiation property is useful for analyzing systems with integrators or differentiators

Initial & final value theorems

• Initial value theorem allows for the calculation of the initial value of a discrete-time signal from its Z-transform: $x[0] = \lim_{z \to \infty} X(z)$
• Final value theorem allows for the calculation of the steady-state value of a discrete-time signal from its Z-transform: $\lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1) X(z)$
• These theorems are useful for analyzing the transient and steady-state behavior of discrete-time systems

Z-transforms of common signals

• Knowing the Z-transforms of common discrete-time signals is essential for analyzing and designing discrete-time systems in Control Theory
• These Z-transforms serve as building blocks for more complex signals and systems

Unit impulse

• The unit impulse signal, also known as the Kronecker delta function, is defined as: $\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \neq 0 \end{cases}$
• The Z-transform of the unit impulse signal is: $\mathcal{Z}\{\delta[n]\} = 1$
• The unit impulse is used to represent instantaneous events or to sample continuous-time signals

Unit step

• The unit step signal, also known as the Heaviside function, is defined as: $u[n] = \begin{cases} 1, & n \geq 0 \\ 0, & n < 0 \end{cases}$
• The Z-transform of the unit step signal is: $\mathcal{Z}\{u[n]\} = \frac{1}{1-z^{-1}}, \quad |z| > 1$
• The unit step is used to represent sudden changes or to model systems with constant inputs

Exponential

• The exponential signal is defined as: $x[n] = a^n u[n], \quad a \in \mathbb{C}$
• The Z-transform of the exponential signal is: $\mathcal{Z}\{a^n u[n]\} = \frac{1}{1-az^{-1}}, \quad |z| > |a|$
• Exponential signals are used to model growth, decay, or damping in discrete-time systems

Sinusoidal

• The sinusoidal signal is defined as: $x[n] = \cos(\omega n) u[n], \quad \omega \in \mathbb{R}$
• The Z-transform of the sinusoidal signal is: $\mathcal{Z}\{\cos(\omega n) u[n]\} = \frac{z(z-\cos\omega)}{z^2-2z\cos\omega+1}, \quad |z| > 1$
• Sinusoidal signals are used to represent oscillations or periodic phenomena in discrete-time systems

Damped sinusoidal

• The damped sinusoidal signal is defined as: $x[n] = a^n \cos(\omega n) u[n], \quad a \in \mathbb{R}, \omega \in \mathbb{R}$
• The Z-transform of the damped sinusoidal signal is: $\mathcal{Z}\{a^n \cos(\omega n) u[n]\} = \frac{z(z-a\cos\omega)}{z^2-2az\cos\omega+a^2}, \quad |z| > |a|$
• Damped sinusoidal signals are used to model oscillations with exponential decay in discrete-time systems

Inverse Z-transforms

• Inverse Z-transform is the process of converting a Z-transform back to its corresponding discrete-time signal
• Essential for obtaining time-domain solutions and implementing discrete-time systems in Control Theory
• Several methods exist for computing inverse Z-transforms, each with its own advantages and limitations

Partial fraction expansion

• Partial fraction expansion decomposes a rational Z-transform into a sum of simpler fractions
• The resulting fractions can be easily inverse Z-transformed using a lookup table or by inspection
• Steps for partial fraction expansion:
  1. Factor the denominator of the Z-transform
  2. Determine the form of the partial fractions based on the factors (distinct poles, repeated poles, or complex conjugate poles)
  3. Solve for the coefficients of the partial fractions using the cover-up method or by equating coefficients
  4. Inverse Z-transform each partial fraction separately and combine the results
• Suitable for rational Z-transforms with a manageable number of poles

Residue method

• The residue method is based on the Cauchy residue theorem from complex analysis
• Expresses the inverse Z-transform as a sum of residues of the Z-transform multiplied by $z^{n-1}$
• The residue at a pole $z_k$ of a Z-transform $X(z)$ is given by: $\text{Res}(X(z), z_k) = \lim_{z \to z_k} (z-z_k) X(z) z^{n-1}$
• The inverse Z-transform is then: $x[n] = \sum_k \text{Res}(X(z), z_k)$
• Suitable for rational Z-transforms with simple poles

Power series expansion

• Power series expansion represents the Z-transform as an infinite series in powers of $z^{-1}$
• The coefficients of the power series correspond to the values of the discrete-time signal
• To obtain the power series expansion, perform long division of the numerator by the denominator of the Z-transform
• The resulting coefficients are the values of the discrete-time signal: $X(z) = \sum_{n=0}^{\infty} x[n] z^{-n}$
• Suitable for Z-transforms with a region of convergence that includes the unit circle

Contour integration

• Contour integration is based on the Cauchy integral formula from complex analysis
• Expresses the inverse Z-transform as a contour integral of the Z-transform multiplied by $z^{n-1}$
• The inverse Z-transform is given by: $x[n] = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} dz$
• The contour $C$ must enclose all the poles of $X(z)$ and lie within the region of convergence
• Suitable for rational and irrational Z-transforms, but requires knowledge of complex analysis

Applications of Z-transforms

• Z-transforms have numerous applications in Control Theory, particularly in the analysis and design of discrete-time systems
• These applications leverage the properties and techniques of Z-transforms to simplify complex problems and obtain practical solutions

Discrete-time system analysis

• Z-transforms enable the analysis of discrete-time systems in the complex frequency domain
• By representing the input-output relationship of a system using Z-transforms, one can study its:
  • Stability (location of poles)
  • Transient response (partial fraction expansion)
  • Steady-state response (final value theorem)
  • Frequency response (evaluation of Z-transform on the unit circle)
• Example: Analyzing the stability and performance of a digital control system for a robotic arm

Transfer functions in Z-domain

• Transfer functions in the Z-domain describe the input-output relationship of discrete-time systems
• Obtained by taking the Z-transform of the difference equation governing the system
• Represented as a ratio of polynomials in $z^{-1}$: $H(z) = \frac{Y(z)}{X(z)} = \frac{b_0 + b_1 z^{-1} + \cdots + b_M z^{-M}}{a_0 + a_1 z^{-1} + \cdots + a_N z^{-N}}$
• Transfer functions allow for the analysis and design of discrete-time systems using algebraic techniques
• Example: Deriving the transfer function of a digital filter for audio processing

Stability analysis using Z-transforms

• Stability is a crucial property of discrete-time systems, ensuring bounded outputs for bounded inputs
• Z-transforms facilitate stability analysis by examining the location of the system's poles in the complex plane
• A discrete-time system is stable if all its poles lie within the unit circle (|z| < 1)
• Stability can be determined by:
  • Factoring the denominator of the transfer function
  • Applying the Jury stability test to the characteristic equation
  • Using the Nyquist stability criterion in the Z-domain
• Example: Assessing the stability of a digital PID controller for a temperature regulation system

Discrete-time controllers design

• Z-transforms are used to design discrete-time controllers that achieve desired performance specifications
• Common design techniques include:
  • Pole placement: Placing the closed-loop poles at desired locations to shape the system's response
  • Root locus: Graphical method for studying the effect of controller gains on the closed-loop pole locations
  • Frequency response methods: Designing controllers based on the desired frequency response characteristics (gain and phase margins)
• Discrete-time controllers are implemented using difference equations or digital hardware
• Example: Designing a discrete-time lead-lag compensator for a satellite attitude control system

Digital filters design

• Digital filters are discrete-time systems that process signals by selectively amplifying or attenuating certain frequency components
• Z-transforms are used to design and analyze digital filters, such as:
  • Finite Impulse Response (FIR) filters: Characterized by a finite-duration impulse response and a transfer function with only zeros
  • Infinite Impulse Response (IIR) filters: Characterized by an infinite-duration impulse response and a transfer function with both poles and zeros
• Filter design techniques in the Z-domain include:
  • Bilinear transformation: Mapping continuous-time filter designs to the discrete-time domain
  • Windowing: Truncating the ideal impulse response of a filter using a window function
  • Optimization methods: Minimizing the error between the desired and actual frequency responses
• Example: Designing a low-pass IIR filter for smoothing sensor data in a industrial control system

Z-transforms vs other transforms

• Z-transforms are one of several transforms used in Control Theory and signal processing, each with its own properties and applications
• Understanding the relationships and differences between Z-transforms and other transforms is essential for selecting the appropriate tool for a given problem

Z-transforms vs Laplace transforms

• Laplace transforms are used for analyzing continuous-time systems, while Z-transforms are used for discrete-time systems
• The Laplace transform variable $s$ is related to the Z-transform variable $z$ through the mapping: $z = e^{sT}$, where $T$ is the sampling period
• Laplace transforms have a continuous region of convergence, while Z-transforms have a discrete region of convergence
• Some properties and techniques are similar between the two transforms, such as linearity, time-shifting, and partial fraction expansion
• Example: Comparing the stability analysis of a continuous-time PID controller using Laplace transforms and its discrete-time equivalent using Z-transforms

Z-transforms vs Fourier transforms

• Fourier transforms are used for analyzing the frequency content of continuous-time signals, while Z-transforms are used for discrete-time signals
• The Fourier transform variable $\omega$ is related to the Z-transform variable $z$ through the mapping: $z = e^{j\omega T}$, where $T$ is the sampling period
• Fourier transforms assume an infinite-duration signal, while Z-transforms can handle finite-duration signals
• The frequency response of a discrete-time system can be obtained by evaluating its Z-transform on the unit circle ($|z| = 1$)
• Example: Comparing the frequency response analysis of an analog low-pass filter using Fourier transforms and its digital equivalent using Z-transforms

Z-transforms vs discrete Fourier transforms

• Discrete Fourier transforms (DFTs) are used for analyzing the frequency content of discrete-time signals over a finite duration
• DFTs are computed using the Fast Fourier Transform (FFT) algorithm, which is more efficient than directly evaluating the Z-transform
• DFTs assume a periodic extension