1.3 Basic concepts of signal processing and system analysis

Signal processing is crucial in bioengineering. It involves sampling, quantization, and filtering to convert and modify signals. These techniques help extract meaningful information from complex biomedical data, enabling accurate analysis and interpretation of physiological phenomena.

System analysis principles and mathematical tools are essential for understanding biomedical systems. Concepts like linearity, time-invariance, and stability, along with Fourier and Laplace transforms, provide powerful methods for analyzing and interpreting signals from various sources like ECG, EEG, and EMG.

Fundamental Concepts of Signal Processing

Fundamentals of signal processing

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• Sampling converts a continuous-time signal into a discrete-time signal by capturing values at regular intervals (sampling rate, $f_s$) determines the number of samples per second
  • Nyquist-Shannon sampling theorem states that the sampling rate must be at least twice the maximum frequency of the signal ($f_s \geq 2f_{max}$) to avoid aliasing (distortion caused by insufficient sampling)
• Quantization represents a continuous-amplitude signal using a finite set of discrete values
  • Quantization levels determine the resolution (precision) of the digital signal
  • Quantization error is the difference between the original signal and the quantized signal, which introduces noise
• Filtering modifies a signal to emphasize or suppress specific frequency components
  • Low-pass filters attenuate (reduce) high-frequency components (noise, interference)
  • High-pass filters attenuate low-frequency components (baseline drift, motion artifacts)
  • Band-pass filters allow a specific range of frequencies to pass through (relevant signal components)
  • Band-stop filters attenuate a specific range of frequencies (power line interference, 50/60 Hz)

System Analysis Principles and Mathematical Tools

Principles of system analysis

• Linearity is a property of a system that satisfies superposition and homogeneity
  • Superposition: the output of a linear system for a sum of inputs is equal to the sum of the outputs for each input ($y[n_1 + n_2] = y[n_1] + y[n_2]$)
  • Homogeneity: scaling the input of a linear system by a constant results in the output being scaled by the same constant ($y[ax] = ay[x]$)
• Time-invariance means that a time shift in the input results in an equal time shift in the output ($y[n-k] = T\{x[n-k]\}$, where $T\{\cdot\}$ represents the system)
• Stability ensures that bounded inputs produce bounded outputs
  • For a linear time-invariant (LTI) system, stability is determined by the absolute summability of the impulse response ($\sum_{n=-\infty}^{\infty} |h[n]| < \infty$)

Mathematical tools for biomedical signals

• Fourier analysis represents a signal as a sum of sinusoidal components with different frequencies
  • Fourier series for periodic signals: $x(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}$
  • Fourier transform for aperiodic signals: $X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt$
• Laplace transforms convert a time-domain signal into a complex frequency-domain representation
  • Laplace transform: $X(s) = \int_{0}^{\infty} x(t) e^{-st} dt$
  • Useful for analyzing LTI systems, stability, and transient response
  • Inverse Laplace transform: $x(t) = \frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} X(s) e^{st} ds$

Interpretation of signal processing results

• Biomedical signal characteristics vary depending on the source
  • Electrocardiogram (ECG) represents the electrical activity of the heart
  • Electroencephalogram (EEG) captures the electrical activity of the brain
  • Electromyogram (EMG) records the electrical activity of muscles
• Applying signal processing techniques to biomedical signals
  • Filtering ECG signals removes noise (high-frequency) and baseline wander (low-frequency)
  • Fourier analysis of EEG signals identifies frequency components related to brain activity (alpha, beta, theta, delta waves)
  • Detecting and classifying EMG signals enables prosthetic control (pattern recognition, machine learning)
• Interpreting system analysis results in biomedical applications
  • Assessing the stability of a glucose regulation model helps prevent dangerous fluctuations (hypoglycemia, hyperglycemia)
  • Analyzing the frequency response of a medical imaging system determines its resolution (ability to distinguish small details) and contrast (ability to differentiate between tissues)
  • Evaluating the transient response of a drug delivery system optimizes dosage and timing (minimizing side effects, maximizing therapeutic effect)