11.3 Discrete-time systems

Discrete-time systems are mathematical models that describe systems where variables change at specific time points. They're crucial in digital signal processing, control systems, and communications, using difference equations to relate current outputs to past inputs and outputs.

Z-transforms and transfer functions are key tools for analyzing discrete-time systems. They help convert time-domain signals to frequency-domain representations, making it easier to understand system behavior and design effective controllers for various applications.

Definition of discrete-time systems

• Discrete-time systems are mathematical models that describe the behavior of systems where the variables change at discrete time instants
• These systems are characterized by difference equations, which relate the current output to past inputs and outputs
• Discrete-time systems are commonly used in digital signal processing, control systems, and communication systems where signals are sampled and processed at regular intervals

Difference equations

• Difference equations are mathematical equations that describe the relationship between the current output and past inputs and outputs of a discrete-time system
• They are the discrete-time counterpart of differential equations used in continuous-time systems

Linear difference equations

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• Linear difference equations are a type of difference equation where the output is a linear combination of past inputs and outputs
• The general form of a linear difference equation is: $y[n] = \sum_{k=0}^{N-1} a_k y[n-k] + \sum_{k=0}^{M-1} b_k x[n-k]$ where $y[n]$ is the output, $x[n]$ is the input, and $a_k$ and $b_k$ are constant coefficients
• Example: A simple first-order linear difference equation is $y[n] = 0.5y[n-1] + x[n]$, where the current output depends on the previous output and current input

Nonlinear difference equations

• Nonlinear difference equations are difference equations where the output is a nonlinear function of past inputs and outputs
• They can exhibit complex behaviors such as chaos and bifurcations
• Example: The logistic map, $x[n+1] = rx[n](1-x[n])$, is a nonlinear difference equation used to model population growth and chaos

Z-transform

• The Z-transform is a mathematical tool used to analyze and solve discrete-time systems
• It converts a discrete-time signal or difference equation into a complex frequency-domain representation

Definition of Z-transform

• The Z-transform of a discrete-time signal $x[n]$ is defined as: $X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$ where $z$ is a complex variable
• The Z-transform maps a sequence of numbers (the discrete-time signal) to a function of a complex variable

Properties of Z-transform

• The Z-transform has several useful properties, such as linearity, time-shifting, and convolution
• Linearity: $\mathcal{Z}\{ax_1[n] + bx_2[n]\} = aX_1(z) + bX_2(z)$
• Time-shifting: $\mathcal{Z}\{x[n-k]\} = z^{-k}X(z)$
• Convolution: $\mathcal{Z}\{x_1[n] * x_2[n]\} = X_1(z)X_2(z)$, where $*$ denotes convolution

Inverse Z-transform

• The inverse Z-transform is the process of recovering the discrete-time signal from its Z-transform
• It can be performed using partial fraction expansion, power series expansion, or contour integration
• Example: The inverse Z-transform of $X(z) = \frac{1}{1-0.5z^{-1}}$ is $x[n] = (0.5)^n u[n]$, where $u[n]$ is the unit step function

Transfer functions

• Transfer functions are mathematical models that describe the input-output relationship of a discrete-time system in the Z-domain
• They are obtained by taking the Z-transform of the difference equation and expressing the output in terms of the input

Definition of transfer function

• The transfer function of a discrete-time system is defined as the ratio of the Z-transform of the output to the Z-transform of the input, assuming zero initial conditions: $H(z) = \frac{Y(z)}{X(z)}$
• It represents the system's response to an impulse input

Poles and zeros

• Poles and zeros are the roots of the denominator and numerator polynomials of the transfer function, respectively
• Poles determine the stability and transient response of the system, while zeros affect the steady-state response and can introduce cancellations with poles
• Example: A transfer function $H(z) = \frac{1+0.5z^{-1}}{1-0.8z^{-1}}$ has a zero at $z=-0.5$ and a pole at $z=0.8$

Stability of discrete-time systems

• A discrete-time system is stable if its output remains bounded for any bounded input
• For a system to be stable, all its poles must lie within the unit circle in the Z-plane (i.e., have a magnitude less than 1)
• Example: The system with transfer function $H(z) = \frac{1}{1-0.5z^{-1}}$ is stable because its pole is located at $z=0.5$, which is inside the unit circle

State-space representation

• State-space representation is an alternative way to describe discrete-time systems using state variables and state equations
• It provides a compact and convenient way to analyze and design multi-input, multi-output (MIMO) systems

State variables and state equations

• State variables are a set of variables that completely describe the internal state of a system at a given time instant
• State equations are a set of first-order difference equations that relate the current state and input to the next state and output: $\mathbf{x}[n+1] = \mathbf{A}\mathbf{x}[n] + \mathbf{B}\mathbf{u}[n]$ $\mathbf{y}[n] = \mathbf{C}\mathbf{x}[n] + \mathbf{D}\mathbf{u}[n]$ where $\mathbf{x}$ is the state vector, $\mathbf{u}$ is the input vector, $\mathbf{y}$ is the output vector, and $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, and $\mathbf{D}$ are constant matrices

State transition matrix

• The state transition matrix, denoted as $\mathbf{\Phi}[n]$, relates the state at time $n$ to the initial state: $\mathbf{x}[n] = \mathbf{\Phi}[n]\mathbf{x}[0]$
• It can be computed using the matrix exponential: $\mathbf{\Phi}[n] = \mathbf{A}^n$
• The state transition matrix plays a crucial role in analyzing the stability and transient response of the system

Controllability and observability

• Controllability is the ability to steer the system from any initial state to any desired final state in a finite number of steps by applying appropriate inputs
• Observability is the ability to determine the initial state of the system by observing its outputs over a finite number of steps
• Both controllability and observability are important properties for the design of state feedback controllers and observers

Time-domain analysis

• Time-domain analysis involves studying the response of a discrete-time system to various inputs, such as impulses, steps, and sinusoids
• It provides insights into the system's transient and steady-state behavior

Impulse response

• The impulse response is the output of a system when the input is an impulse (a single non-zero value at time zero)
• It characterizes the system's response to a brief input and is denoted as $h[n]$
• The impulse response can be obtained by setting the input $x[n] = \delta[n]$ (unit impulse) and solving the difference equation or using the inverse Z-transform of the transfer function

Step response

• The step response is the output of a system when the input is a step function (a constant value applied at time zero and held thereafter)
• It shows how the system responds to a sudden change in input and is useful for analyzing the system's rise time, settling time, and steady-state value
• The step response can be obtained by setting the input $x[n] = u[n]$ (unit step) and solving the difference equation or using the Z-transform and partial fraction expansion

Transient and steady-state response

• The transient response is the initial part of the system's response that depends on the system's poles and initial conditions
• It typically exhibits oscillations or exponential decay and lasts until the system reaches a steady state
• The steady-state response is the long-term behavior of the system after the transient has died out
• It depends on the system's poles, zeros, and input signal
• Example: For a stable system with a pole at $z=0.5$ and a step input, the transient response will decay exponentially, while the steady-state response will be a constant value

Frequency-domain analysis

• Frequency-domain analysis involves studying the response of a discrete-time system to sinusoidal inputs of different frequencies
• It provides insights into the system's frequency selectivity, stability, and resonance properties

Frequency response

• The frequency response is the steady-state output of a system when the input is a sinusoid of a given frequency
• It is obtained by evaluating the transfer function $H(z)$ on the unit circle, i.e., setting $z=e^{j\omega}$, where $\omega$ is the normalized frequency in radians per sample
• The frequency response can be represented by its magnitude $|H(e^{j\omega})|$ and phase $\angle H(e^{j\omega})$ as functions of frequency

Bode plots

• Bode plots are graphical representations of the frequency response, consisting of two separate plots: magnitude (in decibels) vs. frequency and phase (in degrees) vs. frequency, both using logarithmic frequency scales
• They provide a clear visualization of the system's gain and phase characteristics, such as passband, stopband, rolloff, and phase margin
• Example: A low-pass filter will have a Bode magnitude plot that is flat in the passband and decreases with a certain rolloff rate (e.g., -20 dB/decade) in the stopband

Nyquist plots

• Nyquist plots are polar plots of the frequency response, obtained by plotting the real part of $H(e^{j\omega})$ against its imaginary part as $\omega$ varies from $0$ to $2\pi$
• They are useful for analyzing the stability of closed-loop systems using the Nyquist stability criterion
• The Nyquist plot can also reveal the presence of resonance (large loops) and the system's gain and phase margins

Sampling and reconstruction

• Sampling is the process of converting a continuous-time signal into a discrete-time signal by measuring its value at regular time intervals
• Reconstruction is the process of recovering the original continuous-time signal from its samples

Sampling theorem

• The sampling theorem (also known as the Nyquist-Shannon sampling theorem) states that a continuous-time signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the highest frequency component in the signal
• The minimum sampling frequency required to avoid aliasing is called the Nyquist rate
• If the sampling theorem is not satisfied, aliasing occurs, where high-frequency components are mistaken for low-frequency components in the sampled signal

Aliasing and anti-aliasing filters

• Aliasing is the distortion that occurs when the sampling theorem is violated, causing high-frequency components to appear as lower frequencies in the sampled signal
• To prevent aliasing, an anti-aliasing filter (a low-pass filter) is used before sampling to remove frequency components above the Nyquist rate
• The anti-aliasing filter ensures that the continuous-time signal is bandlimited and can be properly reconstructed from its samples

Zero-order hold and other reconstruction methods

• Zero-order hold (ZOH) is a simple reconstruction method that holds each sample value constant until the next sample arrives, resulting in a staircase-like approximation of the original signal
• Other reconstruction methods include first-order hold (linear interpolation between samples), sinc interpolation (ideal low-pass filtering), and spline interpolation (using smooth polynomial curves)
• The choice of reconstruction method depends on the desired accuracy, complexity, and the properties of the original signal

Digital filters

• Digital filters are discrete-time systems designed to process and modify the frequency content of signals
• They are widely used in signal processing applications, such as audio and video processing, communications, and control systems

FIR vs IIR filters

• Finite Impulse Response (FIR) filters are digital filters whose output depends only on the current and past input samples
• FIR filters are always stable, have linear phase response if designed properly, and can be easily implemented using convolution
• Infinite Impulse Response (IIR) filters are digital filters whose output depends on both the current and past input samples, as well as past output samples
• IIR filters can achieve a given frequency response with lower order compared to FIR filters but may have stability issues and nonlinear phase response

Filter design techniques

• Filter design techniques are methods used to determine the filter coefficients that achieve a desired frequency response
• Common FIR filter design techniques include the window method (e.g., Hamming, Kaiser), frequency sampling method, and optimal design methods (e.g., Parks-McClellan algorithm)
• Common IIR filter design techniques include the bilinear transform method (based on analog filter prototypes), impulse invariance method, and optimization-based methods (e.g., Yule-Walker equations)

Realization structures

• Realization structures are block diagrams that represent the implementation of digital filters using basic building blocks such as delays, adders, and multipliers
• Common realization structures for FIR filters include direct form, transposed direct form, and cascade form
• Common realization structures for IIR filters include direct form I and II, transposed direct form I and II, cascade form, and parallel form
• The choice of realization structure affects the filter's computational complexity, numerical stability, and quantization noise performance

Discrete-time control systems

• Discrete-time control systems are control systems where the controller is implemented using a digital computer or microcontroller, and the control actions are applied at discrete time instants
• They are widely used in industrial automation, robotics, and automotive systems due to their flexibility, programmability, and ability to handle complex control algorithms

Discrete-time PID control

• Discrete-time PID (Proportional-Integral-Derivative) control is a digital implementation of the classic PID controller, which calculates the control action based on the error between the desired setpoint and the measured process variable
• The discrete-time PID controller is described by the difference equation: $u[n] = K_p e[n] + K_i \sum_{k=0}^{n} e[k] + K_d (e[n] - e[n-1])$ where $u[n]$ is the control action, $e[n]$ is the error, and $K_p$, $K_i$, and $K_d$ are the proportional, integral, and derivative gains, respectively
• The gains are tuned to achieve the desired closed-loop performance, such as fast response, small overshoot, and good disturbance rejection

Deadbeat control

• Deadbeat control is a discrete-time control strategy that aims to bring the system output to the desired setpoint in the minimum number of time steps
• It is achieved by designing a controller that cancels the system's poles and replaces them with poles at the origin of the Z-plane
• Deadbeat control provides the fastest possible response but is sensitive to model uncertainties and measurement noise
• Example: For a simple first-order system with a pole at $z=0.5$, a deadbeat controller would have a transfer function $G_c(z) = \frac{2}{z}$, resulting in a closed-loop transfer function $\frac{Y(z)}{R(z)} = z^{-1}$, which means the output reaches the setpoint in one time step

Optimal control

• Optimal control is a control strategy that minimizes a given performance index, such as the quadratic cost function: $J = \sum_{k=0}^{N-1} (x^T[k]Qx[k] + u^T[k]Ru[k]) + x^T[N]P_Nx[N]$ where $x[k]$ is the state vector, $u[k]$ is the control input, $Q$ and $R$ are weighting matrices, and $P_N$ is the terminal cost matrix
• The optimal control law is obtained by solving the discrete-time algebraic Riccati equation (DARE) and is given by a state feedback law: $u[k] = -Kx[k]$
• Example: The Linear Quadratic Regulator (LQR) is a well-known optimal control method that balances the control effort and the state deviation to achieve a desired closed-loop performance

Applications of discrete-time systems

• Discrete-time systems have a wide range of applications in various fields, such as signal processing, control, and communication systems
• Some notable applications include:

Digital signal processing

• Digital signal processing (DSP) deals with the representation