7.3 Cross-correlation and auto-correlation functions

Cross-correlation and auto-correlation are powerful tools for analyzing signals. They help us compare different signals or a signal with itself, revealing hidden patterns and similarities. These techniques are crucial for understanding how signals relate to each other over time.

By measuring signal similarity and identifying time lags, we can extract valuable information from complex data. Whether it's finding repeating patterns or matching specific templates, these methods unlock insights in fields like audio processing, seismology, and even finance.

Cross-correlation and Auto-correlation

Comparing Signals with Cross-correlation and Auto-correlation

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• Cross-correlation measures the similarity between two different signals ($f(t)$ and $g(t)$) as a function of the displacement of one relative to the other
  • Involves shifting one signal in time and computing the integral of the product of the two signals at each time shift
  • Helps identify common features or patterns between the signals
  • Can be used to determine the time delay between two related signals (seismic waves)
• Auto-correlation measures the similarity of a signal with a delayed copy of itself as a function of the time lag
  • Compares a signal to its own time-shifted version
  • Helps identify repeating patterns or periodic components within the signal
  • Can be used to estimate the fundamental frequency of a periodic signal (musical pitch detection)

Quantifying Signal Similarity with Correlation Coefficient

• Correlation coefficient quantifies the strength and direction of the linear relationship between two signals
  • Ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation
  • Normalizes the cross-correlation by dividing it by the product of the signal energies
  • Provides a scale-invariant measure of similarity that is independent of signal amplitudes
• Signal similarity refers to the degree to which two signals resemble each other in terms of their shape, patterns, or features
  • Can be assessed using cross-correlation or correlation coefficient
  • High similarity indicates that the signals have a strong relationship or common underlying structure (similar audio waveforms)
  • Low similarity suggests that the signals are unrelated or have distinct characteristics (random noise)

Time-based Analysis

Analyzing Signal Relationships over Time

• Time lag represents the time difference or delay between two signals or between a signal and its time-shifted version
  • Measured in units of time (seconds, milliseconds)
  • Can be positive or negative, indicating a lead or lag relationship
  • Helps identify the temporal relationship or synchronization between signals (audio-video synchronization)
• Periodicity detection involves identifying repeating patterns or cycles within a signal over time
  • Analyzes the auto-correlation of a signal to find peaks at regular intervals
  • Can determine the fundamental period or frequency of a periodic signal
  • Useful in applications like rhythm analysis in music or detecting seasonal trends in data (weather patterns)

Recognizing Patterns and Structures in Time-based Signals

• Pattern recognition involves identifying specific shapes, sequences, or structures within a signal over time
  • Utilizes cross-correlation or template matching techniques
  • Compares a known pattern or template with the signal at different time positions
  • Can detect the presence and location of specific patterns (keyword spotting in speech)
• Time-based analysis techniques enable the extraction of meaningful information and relationships from signals that evolve over time
  • Allows for the detection of delays, synchronization, periodicity, and specific patterns
  • Provides insights into the temporal structure and behavior of signals
  • Finds applications in various domains (audio processing, biomedical signal analysis, financial time series analysis)