📡Bioengineering Signals and Systems Unit 4 – LTI Systems: Properties and Convolution

Introduction to LTI Systems

• LTI systems (Linear Time-Invariant) are fundamental building blocks in signal processing and system analysis
• Linearity property ensures that the system's output is proportional to its input and follows the principle of superposition
• Time-invariance property guarantees that the system's behavior remains consistent over time, meaning a delayed input will result in an equally delayed output
• LTI systems are characterized by their impulse response, which completely describes the system's behavior
• Understanding LTI systems is crucial for analyzing and designing various bioengineering applications, such as signal filtering, image processing, and physiological modeling

Key Properties of LTI Systems

• Linearity is a key property of LTI systems, which consists of two sub-properties: additivity and homogeneity
  • Additivity states that the response to a sum of inputs is equal to the sum of the responses to each individual input: $y(x_1 + x_2) = y(x_1) + y(x_2)$
  • Homogeneity implies that scaling the input by a constant factor will scale the output by the same factor: $y(ax) = ay(x)$
• Time-invariance means that shifting the input in time will result in an equal shift in the output, without changing its shape or magnitude
  • Mathematically, if $y(t)$ is the output of an LTI system for an input $x(t)$, then $y(t-t_0)$ is the output for the input $x(t-t_0)$
• Stability is another important property of LTI systems, ensuring that the system's output remains bounded for any bounded input
• Causality is a property that guarantees the system's output at any given time depends only on the current and past inputs, not on future inputs

Understanding Convolution

• Convolution is a mathematical operation that describes the relationship between the input and output of an LTI system
• The convolution of two signals, $x(t)$ and $h(t)$, is denoted as $x(t) * h(t)$ and is defined by the integral: $y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau$
• In the context of LTI systems, $x(t)$ represents the input signal, $h(t)$ is the system's impulse response, and $y(t)$ is the output signal
• Convolution can be interpreted as a sliding and flipping operation, where the impulse response is shifted and scaled by the input signal at each time instant
• The commutative property of convolution states that the order of the signals being convolved does not affect the result: $x(t) * h(t) = h(t) * x(t)$

Time Domain Analysis

• Time domain analysis focuses on studying the behavior of signals and systems as a function of time
• The impulse response, denoted as $h(t)$, is the output of an LTI system when the input is an impulse or Dirac delta function, $\delta(t)$
• The impulse response fully characterizes the behavior of an LTI system and can be used to determine the output for any given input through convolution
• Step response is another important concept in time domain analysis, which represents the system's output when the input is a unit step function, $u(t)$
• The relationship between the impulse response and step response is given by: $s(t) = \int_{-\infty}^{t} h(\tau)d\tau$, where $s(t)$ is the step response

Frequency Domain Analysis

• Frequency domain analysis involves studying the behavior of signals and systems in terms of their frequency components
• The Fourier transform is a powerful tool for converting signals from the time domain to the frequency domain, providing insight into the signal's frequency content
• For a continuous-time signal $x(t)$, the Fourier transform is defined as: $X(j\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt$
• The inverse Fourier transform allows the reconstruction of the time-domain signal from its frequency-domain representation: $x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(j\omega)e^{j\omega t}d\omega$
• In the context of LTI systems, the Fourier transform of the impulse response, known as the transfer function, provides a complete description of the system's frequency response

Impulse Response and Transfer Functions

• The impulse response, $h(t)$, is the foundation for understanding and analyzing LTI systems in the time domain
• The transfer function, denoted as $H(j\omega)$, is the frequency-domain equivalent of the impulse response and is obtained by taking the Fourier transform of $h(t)$
• The transfer function represents the system's frequency response, describing how the system amplifies or attenuates different frequency components of the input signal
• The relationship between the input and output of an LTI system in the frequency domain is given by: $Y(j\omega) = H(j\omega)X(j\omega)$, where $X(j\omega)$ and $Y(j\omega)$ are the Fourier transforms of the input and output signals, respectively
• Poles and zeros of the transfer function provide valuable insights into the system's stability and frequency response characteristics

Applications in Bioengineering

• LTI systems and convolution have numerous applications in the field of bioengineering, enabling the analysis and processing of biological signals and systems
• Filtering techniques, such as low-pass, high-pass, and band-pass filters, are used to remove noise, extract specific frequency components, or isolate desired signal features (ECG, EEG)
• Image processing algorithms, including edge detection and image enhancement, rely on convolution operations to analyze and manipulate medical images (X-ray, MRI)
• Physiological modeling often involves the use of LTI systems to describe the behavior of biological processes, such as drug absorption, distribution, and elimination in pharmacokinetics
• Signal averaging and template matching techniques, based on convolution, are employed to improve the signal-to-noise ratio and detect specific patterns in biological signals (evoked potentials)

Practice Problems and Examples

• Example 1: Determine the output of an LTI system with impulse response $h(t) = e^{-at}u(t)$ for an input signal $x(t) = e^{-bt}u(t)$, where $a$ and $b$ are positive constants, and $u(t)$ is the unit step function
• Example 2: Given an LTI system with transfer function $H(j\omega) = \frac{1}{j\omega + \alpha}$, find the corresponding impulse response $h(t)$
• Example 3: Convolve the signals $x(t) = rect(t)$ and $h(t) = e^{-|t|}$ to find the output $y(t)$, where $rect(t)$ is the rectangular function defined as $rect(t) = 1$ for $|t| \leq 0.5$ and $0$ otherwise
• Example 4: A biomedical engineer designs a low-pass filter with a cutoff frequency of 50 Hz to remove high-frequency noise from an ECG signal. Determine the filter's transfer function and impulse response
• Example 5: In a pharmacokinetic model, the drug concentration in the blood is described by an LTI system with impulse response $h(t) = Ce^{-kt}u(t)$, where $C$ and $k$ are constants. If a patient receives a bolus injection of the drug, modeled as an impulse function, find the drug concentration over time