3.3 Basic signal operations and transformations

Signal operations and transformations are essential tools for manipulating and analyzing biomedical signals. These techniques allow us to shift, scale, reflect, and combine signals, providing valuable insights into physiological processes and improving diagnostic capabilities.

Time-shifting, scaling, and reflection help align and normalize signals for comparison. Addition, subtraction, multiplication, and convolution enable noise removal, signal isolation, and filtering. These operations are crucial for processing complex biomedical data like ECGs and EEGs.

Signal Operations and Transformations

Basic signal operations

• Time-shifting moves a signal forward or backward in time
  • Denoted as $x(t-t_0)$, where $t_0$ represents the amount of shift
  • Positive $t_0$ delays the signal by shifting it to the right (e.g., $x(t-2)$ delays the signal by 2 seconds)
  • Negative $t_0$ advances the signal by shifting it to the left (e.g., $x(t+1)$ advances the signal by 1 second)
• Scaling modifies the amplitude or duration of a signal
  • Amplitude scaling multiplies the signal by a constant factor $a$, represented as $ax(t)$
    • $a > 1$ amplifies the signal, increasing its amplitude (e.g., $2x(t)$ doubles the signal's amplitude)
    • $0 < a < 1$ attenuates the signal, reducing its amplitude (e.g., $0.5x(t)$ halves the signal's amplitude)
    • $a < 0$ inverts the signal, flipping it about the horizontal axis (e.g., $-x(t)$ inverts the signal)
  • Time scaling modifies the signal's duration without changing its shape, represented as $x(at)$
    • $a > 1$ compresses the signal in time, making it shorter (e.g., $x(2t)$ compresses the signal by a factor of 2)
    • $0 < a < 1$ expands the signal in time, making it longer (e.g., $x(0.5t)$ expands the signal by a factor of 2)
• Reflection flips the signal about the vertical axis, reversing its time order
  • Denoted as $x(-t)$
  • Useful for analyzing signals with symmetry or palindromic properties (e.g., certain ECG waveforms)

Time reversal and scaling

• Time reversal reverses the order of a signal in time, denoted as $x(-t)$
  • Flips the signal about the vertical axis, reversing its time order
  • Helps analyze signals with time-reversed symmetry (e.g., certain ECG components)
  • Useful for understanding the behavior of signals when played backward
• Time scaling changes the duration of a signal without affecting its shape, denoted as $x(at)$
  • $a > 1$ compresses the signal in time, making it shorter (e.g., $x(3t)$ compresses the signal by a factor of 3)
  • $0 < a < 1$ expands the signal in time, making it longer (e.g., $x(0.25t)$ expands the signal by a factor of 4)
  • Helps normalize signals to a standard duration or adjust their length for analysis
  • Useful for comparing signals of different durations or aligning them in time

Combining signals

• Addition and subtraction combine signals by adding or subtracting their corresponding values at each time point
  • Represented as $x(t) + y(t)$ for addition and $x(t) - y(t)$ for subtraction
  • Used to combine signals from multiple sources or compare signals
  • Can remove noise or artifacts by subtracting a reference signal (e.g., removing baseline drift from an ECG)
• Multiplication combines signals by multiplying their corresponding values at each time point, represented as $x(t) \cdot y(t)$
  • Used for modulation, such as amplitude modulation (AM) in communication systems
  • Can gate or window signals, isolating specific segments (e.g., isolating QRS complex in an ECG)
• Convolution combines signals by integrating their product over time, denoted as $x(t) * y(t)$
  • Mathematically expressed as $\int_{-\infty}^{\infty} x(\tau)y(t-\tau)d\tau$
  • Filters or smooths signals, removing noise or unwanted frequencies (e.g., low-pass filtering an EEG signal)
  • Finds the response of a linear time-invariant (LTI) system to an input signal (e.g., the impulse response of a neural network)

Effects on biomedical signals

• Time-shifting aligns signals for comparison or synchronization
  • Helps analyze the delay between stimulus and response in biomedical systems (e.g., nerve conduction studies)
  • Can synchronize signals from different sources (e.g., aligning ECG and blood pressure waveforms)
• Scaling normalizes or adjusts the intensity of biomedical signals
  • Amplitude scaling can normalize signals for comparison (e.g., comparing EEG signals from different patients)
  • Time scaling can normalize signal duration or adjust it for analysis (e.g., normalizing ECG cycle length)
• Reflection helps analyze symmetry or palindromic properties in biomedical signals
  • Can identify specific waveform components (e.g., identifying the P wave in an ECG)
  • Useful for understanding the behavior of signals when reversed in time
• Time reversal helps analyze signals with time-reversed symmetry
  • Can identify specific waveform components (e.g., identifying the T wave in an ECG)
  • Useful for understanding the behavior of signals when played backward
• Addition and subtraction can remove noise or artifacts from biomedical signals
  • Subtracting a reference signal can remove baseline drift or power line interference (e.g., removing 60 Hz noise from an ECG)
  • Helps compare or combine different biomedical signals (e.g., comparing EEG signals from different brain regions)
• Multiplication can gate or window biomedical signals
  • Isolates specific segments of interest (e.g., isolating the QRS complex in an ECG)
  • Can be used for amplitude modulation in biomedical communication systems
• Convolution filters or smooths biomedical signals
  • Removes noise or unwanted frequencies (e.g., removing high-frequency noise from an EEG signal)
  • Analyzes the response of a biomedical system to an input signal (e.g., the impulse response of a cochlear implant)