17.2 Linear time-invariant systems

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: linearity and time-invariance. 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 $x_1(t)$ produces an output $y_1(t)$ and an input $x_2(t)$ produces an output $y_2(t)$, then the input $ax_1(t) + bx_2(t)$ will produce the output $ay_1(t) + by_2(t)$ for any constants $a$ and $b$
  • Enables the use of superposition 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 $x(t)$ produces an output $y(t)$, then the input $x(t-t_0)$ will produce the output $y(t-t_0)$ for any time shift $t_0$
  • Allows for the analysis of the system's behavior independently of the specific timing of the input signals

Additional Properties of LTI Systems

• Stability ensures that the system's output remains bounded for any bounded input signal
  • A system is stable if and only if its impulse response is absolutely integrable, i.e., $\int_{-\infty}^{\infty} |h(t)| dt < \infty$
  • Guarantees that the system will not produce an unbounded output or oscillate indefinitely in response to a finite input
• Causality 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 $h(t) = 0$ for all $t < 0$
  • 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 $h(t)$ describes the system's output when the input is a unit impulse function $\delta(t)$
  • 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 convolution integral: $y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau$
• Step response characterizes the system's output when the input is a unit step function $u(t)$
  • Provides insights into the system's transient behavior, such as rise time, settling time, and overshoot
  • Can be obtained by integrating the impulse response: $s(t) = \int_{-\infty}^{t} h(\tau) d\tau$

Frequency-Domain Analysis

• Frequency response $H(j\omega)$ represents the system's steady-state response to sinusoidal inputs of different frequencies
  • Obtained by evaluating the transfer function $H(s)$ at $s = j\omega$, where $\omega$ 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 $H(s)$ is the Laplace transform of the impulse response $h(t)$
  • 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