Glossary

17.2 Linear time-invariant systems

3 min read•august 6, 2024

Linear time-invariant systems are the backbone of signal processing. They're predictable and consistent, making them super useful for analyzing and designing all sorts of systems. Think of them as the reliable friend who always responds the same way, no matter when you ask.

These systems have two key features: and . Linearity means the output is proportional to the input, while time-invariance means the system's behavior doesn't change over time. This makes them easier to understand and work with in real-world applications.

Properties of Linear Time-Invariant Systems

Fundamental Characteristics of LTI Systems

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  • Linearity property states that the output of a system is directly proportional to its input
    • If an input x1(t)x_1(t) produces an output y1(t)y_1(t) and an input x2(t)x_2(t) produces an output y2(t)y_2(t), then the input ax1(t)+bx2(t)ax_1(t) + bx_2(t) will produce the output ay1(t)+by2(t)ay_1(t) + by_2(t) for any constants aa and bb
    • Enables the use of principle to analyze complex systems by breaking them down into simpler components
  • Time-invariance property implies that the system's response to an input does not depend on the absolute time at which the input is applied
    • If an input x(t)x(t) produces an output y(t)y(t), then the input x(t−t0)x(t-t_0) will produce the output y(t−t0)y(t-t_0) for any time shift t0t_0
    • Allows for the analysis of the system's behavior independently of the specific timing of the input signals

Additional Properties of LTI Systems

  • ensures that the system's output remains bounded for any bounded input signal
    • A system is stable if and only if its is absolutely integrable, i.e., ∫−∞∞∣h(t)∣dt<∞\int_{-\infty}^{\infty} |h(t)| dt < \infty
    • Guarantees that the system will not produce an unbounded output or oscillate indefinitely in response to a finite input
  • requires that the system's output at any given time depends only on the input up to that time
    • A system is causal if its impulse response h(t)=0h(t) = 0 for all t<0t < 0
    • Ensures that the system does not respond to future inputs, which is essential for physical realizability and real-time processing applications

System Responses and Characterization

Time-Domain Analysis

  • Impulse response h(t)h(t) describes the system's output when the input is a unit impulse function δ(t)\delta(t)
    • Serves as a fundamental building block for analyzing the system's response to any input signal
    • Enables the computation of the system's output through the integral: y(t)=∫−∞∞x(τ)h(t−τ)dτy(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau
  • characterizes the system's output when the input is a unit step function u(t)u(t)
    • Provides insights into the system's transient behavior, such as rise time, settling time, and overshoot
    • Can be obtained by integrating the impulse response: s(t)=∫−∞th(τ)dτs(t) = \int_{-\infty}^{t} h(\tau) d\tau

Frequency-Domain Analysis

  • H(jω)H(j\omega) represents the system's steady-state response to sinusoidal inputs of different frequencies
    • Obtained by evaluating the H(s)H(s) at s=jωs = j\omega, where ω\omega is the angular frequency
    • Allows for the analysis of the system's behavior in terms of gain and phase shift as a function of frequency (Bode plots)
  • Transfer function H(s)H(s) is the of the impulse response h(t)h(t)
    • Provides a compact representation of the system's input-output relationship in the complex frequency domain
    • Enables the use of algebraic techniques for system analysis and design, such as pole-zero analysis and stability assessment

Key Terms to Review (11)

Causality: Causality refers to the relationship between cause and effect, indicating that the output of a system at any given time depends only on the input at that time and possibly past inputs, not future inputs. In systems analysis, this concept is crucial as it helps to ensure that the behavior of systems aligns with physical reality, where outputs cannot occur before their corresponding inputs. Understanding causality is vital for analyzing the dynamic behavior of systems over time.
Convolution: Convolution is a mathematical operation that combines two functions to produce a third function, representing how the shape of one is modified by the other. It plays a crucial role in system analysis, particularly in understanding how input signals interact with system responses over time. This operation is key in areas like signal processing, where it helps to analyze and design linear time-invariant systems.
Frequency response: Frequency response is a measure of a system's output spectrum in response to an input signal of varying frequency, essentially describing how a system reacts at different frequencies. It helps in understanding how systems behave in terms of gain and phase shift across a range of frequencies, providing insight into their dynamic characteristics and stability.
Impulse Response: Impulse response refers to the output of a system when an impulse function, or Dirac delta function, is applied as the input. This characteristic is crucial for understanding how systems react to various inputs and forms the basis for analyzing linear time-invariant systems, connecting time-domain analysis with convolution, correlation, and discrete-time signal processing.
Laplace Transform: The Laplace Transform is a mathematical technique that transforms a time-domain function into a complex frequency-domain representation, making it easier to analyze linear time-invariant systems. This transformation helps in solving differential equations and analyzing system behavior, particularly in control systems and signal processing.
Linearity: Linearity refers to the property of a system or function where the output is directly proportional to the input, allowing for the principle of superposition to apply. This concept is fundamental in analyzing various electrical devices and signals, as it simplifies their behavior into manageable mathematical relationships, making it easier to predict and control their responses.
Stability: Stability refers to the ability of a system to return to its equilibrium state after a disturbance. In engineering, particularly in the context of control systems, it is essential for ensuring that the system performs predictably and does not diverge into chaotic behavior over time.
Step response: Step response refers to the output behavior of a system when subjected to a sudden change in input, typically modeled as a step function. It provides insights into how quickly and accurately a system can respond to changes, showcasing characteristics like stability, transient response, and steady-state behavior. Understanding the step response is crucial for analyzing the performance of various systems in both continuous and discrete time domains.
Superposition: Superposition is the principle that states that in a linear system, the total response at any given time or point is the sum of the responses caused by each individual input acting alone. This principle is fundamental in analyzing electrical circuits and systems, as it allows for the simplification of complex problems by breaking them down into smaller, manageable parts. Understanding superposition enables the analysis of circuits and systems more effectively, especially when dealing with multiple sources and inputs.
Time-invariance: Time-invariance refers to a property of systems where the behavior and characteristics of the system remain unchanged over time. This means that if an input signal is applied to the system at one point in time, the output will be the same as if the same input were applied at any other point in time. Time-invariance is essential in understanding how systems respond consistently regardless of when inputs are applied, which is crucial in the analysis of linear time-invariant systems.
Transfer Function: A transfer function is a mathematical representation that relates the output of a system to its input using the Laplace transform. It provides insights into the dynamic behavior of linear time-invariant systems, enabling the analysis of stability, frequency response, and system performance across various domains.
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