📡Bioengineering Signals and Systems Unit 3 – Signal Basics: Types, Properties & Operations

Introduction to Signals

• Signals convey information about the behavior or attributes of a phenomenon
• Can be represented mathematically as a function of one or more independent variables (time, space, or frequency)
• Classified into different categories based on their properties and characteristics
• Play a crucial role in various fields, including bioengineering, telecommunications, and signal processing
• Understanding signals is essential for analyzing, interpreting, and processing data in bioengineering applications
  • Helps in extracting meaningful information from biological systems
  • Enables the development of diagnostic and therapeutic tools

Types of Signals

• Continuous-time signals are defined for all values of the independent variable (time)
  • Examples include electrocardiogram (ECG) and electroencephalogram (EEG) signals
• Discrete-time signals are defined only at specific values of the independent variable
  • Often obtained by sampling continuous-time signals at regular intervals
• Analog signals have continuous amplitudes and can take on any value within a range
  • Directly generated by physical phenomena (blood pressure, temperature)
• Digital signals have discrete amplitudes and can only take on a finite set of values
  • Obtained by quantizing analog signals or generated by digital devices
• Deterministic signals can be described by a mathematical function or rule
  • Examples include sinusoidal signals and exponential signals
• Random signals have unpredictable values and require statistical methods for analysis
  • Examples include noise signals and biological signals with inherent variability

Signal Properties

• Amplitude represents the magnitude or intensity of a signal at a given point
  • Measured in units specific to the signal (volts for electrical signals, pascals for acoustic signals)
• Frequency indicates the number of cycles or oscillations per unit time
  • Measured in hertz (Hz) and determines the signal's periodicity and spectral content
• Phase describes the relative position or shift of a signal with respect to a reference
  • Measured in radians or degrees and affects the alignment and synchronization of signals
• Bandwidth is the range of frequencies present in a signal
  • Determines the signal's information-carrying capacity and required processing resources
• Energy and power quantify the signal's strength and its distribution over time
  • Energy is the total signal content, while power is the average energy per unit time
• Symmetry properties, such as even and odd symmetry, describe the signal's behavior under certain transformations
  • Even symmetry: $f(t) = f(-t)$, odd symmetry: $f(t) = -f(-t)$

Time Domain Analysis

• Time domain analysis studies signals as a function of time
• Allows for the examination of signal characteristics, such as amplitude, duration, and shape
• Temporal features, including rise time, fall time, and settling time, provide insights into the signal's dynamics
• Statistical measures, such as mean, variance, and standard deviation, describe the signal's central tendency and dispersion
  • Mean: $\mu = \frac{1}{N} \sum_{i=1}^{N} x_i$, variance: $\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2$
• Correlation and cross-correlation quantify the similarity and temporal relationship between signals
  • Autocorrelation: $R_{xx}(\tau) = \int_{-\infty}^{\infty} x(t) x(t+\tau) dt$, cross-correlation: $R_{xy}(\tau) = \int_{-\infty}^{\infty} x(t) y(t+\tau) dt$
• Time-frequency analysis techniques, such as short-time Fourier transform (STFT) and wavelet transform, provide localized information in both time and frequency domains

Frequency Domain Analysis

• Frequency domain analysis studies signals as a function of frequency
• Fourier transform decomposes a signal into its constituent frequencies
  • Continuous-time Fourier transform (CTFT): $X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt$
  • Discrete-time Fourier transform (DTFT): $X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}$
• Spectrum represents the distribution of signal energy across different frequencies
  • Magnitude spectrum: $|X(f)|$ or $|X(e^{j\omega})|$, phase spectrum: $\angle X(f)$ or $\angle X(e^{j\omega})$
• Spectral analysis helps identify dominant frequencies, harmonics, and bandwidth
• Power spectral density (PSD) describes the power distribution of a signal over frequency
  • Periodogram: $P(f) = \frac{1}{N} |X(f)|^2$, where $X(f)$ is the Fourier transform of the signal
• Frequency domain filtering allows for the selective attenuation or amplification of specific frequency components
  • Low-pass, high-pass, band-pass, and band-stop filters

Signal Operations

• Amplitude scaling multiplies the signal by a constant factor, changing its magnitude
  • $y(t) = a \cdot x(t)$, where $a$ is the scaling factor
• Time shifting translates the signal along the time axis, changing its temporal position
  • $y(t) = x(t-t_0)$, where $t_0$ is the time shift
• Time scaling compresses or expands the signal in time, affecting its duration and frequency content
  • $y(t) = x(at)$, where $a$ is the scaling factor
• Addition and subtraction combine signals point-by-point, allowing for signal mixing and interference analysis
  • $y(t) = x_1(t) \pm x_2(t)$
• Multiplication and convolution perform point-by-point and sliding window operations, respectively
  • Multiplication: $y(t) = x_1(t) \cdot x_2(t)$, convolution: $y(t) = x_1(t) * x_2(t) = \int_{-\infty}^{\infty} x_1(\tau) x_2(t-\tau) d\tau$
• Differentiation and integration compute the rate of change and accumulation of signals, respectively
  • Differentiation: $y(t) = \frac{d}{dt} x(t)$, integration: $y(t) = \int_{-\infty}^{t} x(\tau) d\tau$

Applications in Bioengineering

• Biomedical signal processing analyzes physiological signals for diagnosis and monitoring
  • ECG for cardiac activity, EEG for brain activity, EMG for muscle activity
• Medical imaging utilizes signal processing techniques for image reconstruction and enhancement
  • Computed tomography (CT), magnetic resonance imaging (MRI), ultrasound imaging
• Biosensors and wearable devices rely on signal acquisition, conditioning, and transmission
  • Glucose monitoring, pulse oximetry, accelerometry for activity tracking
• Assistive technologies employ signal processing for improved functionality and user experience
  • Cochlear implants for hearing restoration, brain-computer interfaces for neural control
• Bioinformatics and genomic signal processing analyze biological sequences and data
  • DNA sequence analysis, gene expression profiling, protein structure prediction
• Physiological modeling and simulation use signals to represent and study biological systems
  • Computational modeling of cardiac electrophysiology, neuromuscular systems, and metabolic processes

Key Takeaways and Review

• Signals are fundamental to bioengineering and play a vital role in various applications
• Understanding signal types, properties, and operations is essential for effective signal analysis and processing
• Time domain analysis examines signals as a function of time, focusing on temporal characteristics and statistical measures
• Frequency domain analysis studies signals as a function of frequency, using Fourier transform and spectral analysis techniques
• Signal operations, such as scaling, shifting, addition, and convolution, allow for signal manipulation and transformation
• Bioengineering applications leverage signal processing techniques for diagnosis, monitoring, imaging, and assistive technologies
• Continuous learning and exploration of advanced signal processing methods are crucial for staying updated in the field