18.1 Time-domain analysis of continuous-time systems

Time-domain analysis is crucial for understanding continuous-time systems. It lets us see how signals change over time and how systems respond to different inputs. This approach is key for grasping the basics of signal processing and control systems.

We'll look at important concepts like impulse and step responses, convolution, and differential equations. These tools help us predict system behavior, design better systems, and solve real-world engineering problems.

Continuous-Time Signals and Systems

Signal and System Properties

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• Continuous-time signals are defined over a continuous range of time values
  • Represented mathematically as functions of a continuous variable, typically denoted as $x(t)$
  • Examples include audio signals, voltage signals, and temperature variations over time
• Linear time-invariant (LTI) systems exhibit the properties of linearity and time-invariance
  • Linearity: The system's response to a weighted sum of inputs is equal to the weighted sum of the individual responses
  • Time-invariance: The system's behavior does not change with time; shifting the input in time results in an equal shift of the output
• Causality refers to the property of a system where the output at any given time depends only on the current and past inputs
  • A causal system does not respond to future inputs
  • Most physical systems are causal, as they cannot predict or respond to future events
• Stability is the property of a system where bounded inputs always produce bounded outputs
  • A stable system's output remains finite for any finite input
  • Unstable systems may produce unbounded outputs even for bounded inputs, leading to undesirable behavior or system failure

System Characteristics and Behavior

• The impulse response, denoted as $h(t)$, characterizes the system's response to a unit impulse input, $\delta(t)$
  • The impulse response completely describes the behavior of an LTI system
  • Convolving the impulse response with any input signal yields the corresponding output signal
• The step response represents the system's response to a unit step input, $u(t)$
  • The step response provides insights into the system's transient behavior and steady-state value
  • It helps analyze the system's rise time, settling time, and overshoot
• Differential equations are mathematical models that describe the relationship between a system's input, output, and their derivatives
  • Linear constant-coefficient differential equations are commonly used to model LTI systems
  • The order of the differential equation determines the complexity and behavior of the system (e.g., first-order, second-order systems)

System Analysis Tools

Convolution and Impulse Response

• The convolution integral is a mathematical operation that determines the output of an LTI system given its input and impulse response
  • Denoted as $y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau$
  • Convolution involves multiplying the input signal with a time-reversed and shifted version of the impulse response and integrating the result
• The impulse response, $h(t)$, is the output of the system when the input is a unit impulse, $\delta(t)$
  • The impulse response fully characterizes the behavior of an LTI system
  • Knowing the impulse response allows for the determination of the system's output for any given input through convolution

Step Response and Differential Equations

• The step response is the system's output when the input is a unit step function, $u(t)$
  • The step response provides information about the system's transient and steady-state behavior
  • Key characteristics of the step response include rise time, settling time, overshoot, and steady-state value
  • Rise time: The time required for the output to rise from 10% to 90% of its final value
  • Settling time: The time it takes for the output to settle within a specified percentage (e.g., 2%) of its final value
• Differential equations describe the relationship between a system's input, output, and their derivatives
  • Linear constant-coefficient differential equations are commonly used to model LTI systems
  • The general form of a linear constant-coefficient differential equation is: $a_n\frac{d^ny(t)}{dt^n} + a_{n-1}\frac{d^{n-1}y(t)}{dt^{n-1}} + ... + a_1\frac{dy(t)}{dt} + a_0y(t) = b_m\frac{d^mx(t)}{dt^m} + b_{m-1}\frac{d^{m-1}x(t)}{dt^{m-1}} + ... + b_1\frac{dx(t)}{dt} + b_0x(t)$
  • The coefficients $a_i$ and $b_i$ determine the system's characteristics and behavior
  • Solving differential equations helps analyze the system's response and stability