17.3 Convolution and correlation

Convolution and correlation are powerful tools in signal processing. They help us understand how systems respond to inputs and how signals relate to each other. These concepts are crucial for analyzing and designing filters, communication systems, and more.

By mastering convolution and correlation, you'll be able to tackle complex signal processing problems. These techniques form the foundation for advanced topics like Fourier analysis and digital filter design, which we'll explore later in the course.

Convolution

Convolution Integral and Sum

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• Convolution integral represents the output of a linear time-invariant (LTI) system as a weighted sum of the input signal
  • Calculated by integrating the product of the input signal and the system's impulse response
  • Impulse response characterizes the system's behavior to a brief input signal (Dirac delta function)
• Convolution sum is the discrete-time equivalent of the convolution integral
  • Computes the output of an LTI system by summing the products of the input signal and the system's impulse response
  • Impulse response in discrete-time is the system's response to a unit impulse (Kronecker delta function)

System Response and Impulse Response

• System response refers to the output of an LTI system when given an input signal
  • Determined by convolving the input signal with the system's impulse response
  • Impulse response fully characterizes the behavior of an LTI system
• Impulse response represents the system's reaction to a brief input signal
  • In continuous-time, the impulse is modeled using the Dirac delta function
  • In discrete-time, the impulse is represented by the Kronecker delta function
  • Impulse response is crucial for analyzing and designing LTI systems (filters, control systems)

Correlation

Cross-Correlation and Auto-Correlation

• Cross-correlation measures the similarity between two signals as a function of the lag or time-shift applied to one of them
  • Helps determine the time delay between two related signals (radar, sonar)
  • Can be used for pattern recognition and template matching (image processing, speech recognition)
• Auto-correlation is the cross-correlation of a signal with itself
  • Measures the similarity between a signal and its time-shifted version
  • Useful for detecting repeating patterns or periodicities within a signal (pitch detection, rhythm analysis)

Properties of Correlation

• Time-shifting property states that the cross-correlation of two signals is a function of the relative time-shift between them
  • Shifting one signal in time affects the cross-correlation result
  • Maximum correlation occurs when the signals are aligned with their most similar features
• Duality property relates correlation and convolution
  • Cross-correlation of two signals is equivalent to the convolution of one signal with the time-reversed version of the other
  • Allows for efficient computation of correlation using convolution algorithms (Fast Fourier Transform)