7.1 Definition and properties of convolutions

Convolution is a powerful mathematical tool that blends two functions to create a third. It's like mixing ingredients in a recipe, where the result depends on how you combine them. This concept is crucial in signal processing and other fields.

Understanding convolution helps us grasp how signals interact and transform. It's closely tied to Fourier transforms, making it a key player in analyzing and manipulating complex data in various domains.

Definition and Properties of Convolution

Convolution as an Integral Transform

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• Convolution is a mathematical operation that combines two functions to produce a third function
• Defined as an integral that expresses the amount of overlap of one function $g$ as it is shifted over another function $f$
• The resulting function [f * g](https://www.fiveableKeyTerm:f_*_g) is a weighted average of the function $f(τ)$ at the moment $t$ where the weighting is given by $g(−τ)$ simply shifted by amount $t$ ($τ$ is the dummy variable of integration)
• Mathematically, convolution is defined as: $(f * g)(t) = \int_{-\infty}^{\infty} f(τ)g(t - τ) dτ$
• Can be thought of as a blending of two functions (signal processing, image processing)

Properties of Convolution

• Convolution is commutative: $f * g = g * f$
  • The order of the functions does not affect the result
  • Follows from the commutativity of multiplication and the symmetry of the integration limits
• Convolution is associative: $(f * g) * h = f * (g * h)$
  • Grouping of the functions does not affect the result
  • Allows for convolution of more than two functions
• Convolution is distributive over addition: $f * (g + h) = (f * g) + (f * h)$
  • Follows from the linearity of integration
• The identity element under convolution is the Dirac delta function $δ(t)$: $f * δ = f$
  • The Dirac delta function is a generalized function that is zero everywhere except at zero, with an integral of one over the entire real line

Convolution and Fourier Transform

Fourier Transform of Convolution

• The Fourier transform of a convolution is the pointwise product of Fourier transforms
• Mathematically: $\mathcal{F}\{f * g\} = \mathcal{F}\{f\} · \mathcal{F}\{g\}$
  • $\mathcal{F}\{f\}$ denotes the Fourier transform of $f$
  • The dot $·$ denotes pointwise multiplication
• This property is often used in signal processing and other applications to simplify calculations
  • Convolution in the time domain corresponds to multiplication in the frequency domain (audio signal processing)
  • Convolution in the spatial domain corresponds to multiplication in the Fourier domain (image processing)

Convolution Theorem

• The convolution theorem states that under suitable conditions, the Fourier transform of a convolution is the point-wise product of Fourier transforms
• In other words, convolution in one domain equals point-wise multiplication in the other domain
• Mathematically: $\mathcal{F}\{f * g\} = \mathcal{F}\{f\} · \mathcal{F}\{g\}$ and $\mathcal{F}^{-1}\{\mathcal{F}\{f\} · \mathcal{F}\{g\}\} = f * g$
  • $\mathcal{F}^{-1}$ denotes the inverse Fourier transform
• The convolution theorem is used in many applications such as:
  • Filtering signals by convolution with a filter function (low-pass filter, high-pass filter)
  • Multiplying large integers by using the Fast Fourier Transform (FFT) to compute the convolution of their digits