1.4 Linear Time-Invariant (LTI) Systems

Linear Time-Invariant (LTI) systems are the backbone of signal processing. They're predictable and easy to work with because they follow two simple rules: linearity and time-invariance. These properties make LTI systems super useful for analyzing and designing all sorts of signal processing applications.

Understanding LTI systems is key to grasping how signals are processed and transformed. We'll look at their properties, how they respond to different inputs, and how we can describe them mathematically. This stuff is crucial for anyone wanting to dive deeper into signal processing and system analysis.

Properties of LTI Systems

Linearity and Time-Invariance

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• LTI systems have two key properties: linearity and time-invariance
  • Linearity means the system obeys the properties of superposition and scaling ($y(t) = a_1x_1(t) + a_2x_2(t)$ for inputs $x_1(t)$ and $x_2(t)$ and constants $a_1$ and $a_2$)
  • Time-invariance means the system's response to an input does not depend on when the input is applied ($y(t-t_0) = x(t-t_0) * h(t)$ for a time shift $t_0$)
• The output of an LTI system can be expressed as a linear combination of scaled and shifted impulse responses, which forms the basis for convolution

Impulse Response and System Characterization

• LTI systems can be fully characterized by their impulse response, which is the output of the system when the input is a unit impulse (delta function, $\delta(t)$)
• The impulse response, denoted as $h(t)$, completely describes the system's behavior and can be used to determine the output for any input through convolution
• Stability and causality are important properties of LTI systems
  • A stable system produces a bounded output for any bounded input ($\int_{-\infty}^{\infty} |h(t)| dt < \infty$)
  • A causal system's output depends only on current and past inputs ($h(t) = 0$ for $t < 0$)

Impulse and Step Responses

Impulse Response

• The impulse response, denoted as $h(t)$, is the output of an LTI system when the input is a unit impulse (delta function, $\delta(t)$)
• For discrete-time LTI systems, the impulse response is the output sequence when the input is the unit impulse sequence $\delta[n]$
• The impulse response completely characterizes the system's behavior and can be used to determine the output for any input through convolution

Step Response

• The step response, denoted as $s(t)$, is the output of an LTI system when the input is a unit step function $u(t)$ (Heaviside function)
• For discrete-time LTI systems, the step response is the output sequence when the input is the unit step sequence $u[n]$
• The step response can be obtained by integrating the impulse response: $s(t) = \int_{0}^{t} h(\tau) d\tau$
• The impulse response and step response are related by the fundamental theorem of calculus: $h(t) = \frac{ds(t)}{dt}$

Convolution in Time Domain

Convolution Operation

• Convolution is a mathematical operation that describes the output of an LTI system as a function of its input and impulse response, denoted by the $*$ symbol
• For continuous-time LTI systems, the convolution integral is given by: $y(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau = x(t) * h(t)$, where $x(t)$ is the input signal and $h(t)$ is the impulse response
• For discrete-time LTI systems, the convolution sum is given by: $y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] = x[n] * h[n]$, where $x[n]$ is the input sequence and $h[n]$ is the impulse response sequence

Properties and Interpretation

• Convolution satisfies several properties, including commutativity ($x(t) * h(t) = h(t) * x(t)$), associativity ($(x(t) * h_1(t)) * h_2(t) = x(t) * (h_1(t) * h_2(t))$), and distributivity ($x(t) * (h_1(t) + h_2(t)) = x(t) * h_1(t) + x(t) * h_2(t)$), which can be used to simplify calculations
• The convolution operation can be interpreted as a sliding and flipping of the impulse response over the input signal, followed by a point-wise multiplication and summation (or integration)
• Convolution can be used to determine the output of an LTI system for any input, given the system's impulse response

Transfer Functions in Frequency Domain

Definition and Properties

• The transfer function is a representation of an LTI system in the frequency domain, denoted as $H(s)$ for continuous-time systems and $H(z)$ for discrete-time systems
• For continuous-time LTI systems, the transfer function is the Laplace transform of the impulse response: $H(s) = \mathcal{L}\{h(t)\}$, where $s$ is the complex frequency variable
• For discrete-time LTI systems, the transfer function is the Z-transform of the impulse response: $H(z) = \mathcal{Z}\{h[n]\}$, where $z$ is the complex frequency variable
• The transfer function relates the output $Y(s)$ or $Y(z)$ to the input $X(s)$ or $X(z)$ in the frequency domain: $Y(s) = H(s) \cdot X(s)$ for continuous-time systems and $Y(z) = H(z) \cdot X(z)$ for discrete-time systems

Frequency Response and Bode Plots

• The poles and zeros of the transfer function provide insight into the system's stability and frequency response characteristics
  • Poles in the right-half $s$-plane or outside the unit circle in the $z$-plane indicate instability
  • Zeros in the right-half $s$-plane or outside the unit circle in the $z$-plane indicate non-minimum phase behavior
• The frequency response of an LTI system can be obtained by evaluating the transfer function along the imaginary axis ($s = j\omega$ for continuous-time systems) or the unit circle ($z = e^{j\omega}$ for discrete-time systems)
• Bode plots display the magnitude and phase of the frequency response on logarithmic scales and are useful for analyzing the behavior of LTI systems in the frequency domain
  • The magnitude plot shows the gain of the system as a function of frequency (in decibels, $20\log_{10}|H(j\omega)|$)
  • The phase plot shows the phase shift introduced by the system as a function of frequency (in degrees or radians)