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Convolution (mathematical analysis)

Lecture



The convolution of functions is an operation in functional analysis.

By definition, a convolution is a mathematical operation applied to two functions f and g , generating a third function, which can sometimes be considered as a modified version of one of the original ones. Essentially, this is a special kind of integral transform.

The concept of convolution is generalized for functions defined on groups, as well as measures.

The convolution operation can be interpreted as the "similarity" of one function with a reflected and shifted copy of another.

  Convolution (mathematical analysis)

Convolution of two rectangular pulses: as a result gives a triangular pulse.

  Convolution (mathematical analysis)

Convolution of a rectangular pulse (input signal) with an impulse response of an RC circuit

Content

  • 1 functions screwdriver
    • 1.1 Properties
  • 2 screwdriver on groups
  • 3 Measures screwdriver
    • 3.1 Properties
    • 3.2 Distribution driver

Convolution of functions

Let be   Convolution (mathematical analysis) - two functions that are integrable with respect to Lebesgue measure on the space   Convolution (mathematical analysis) . Then their convolution is called the function   Convolution (mathematical analysis) defined by the formula

  Convolution (mathematical analysis)   Convolution (mathematical analysis)

In particular, when   Convolution (mathematical analysis) the formula takes the form:

  Convolution (mathematical analysis)   Convolution (mathematical analysis)

Convolution   Convolution (mathematical analysis) determined for almost all   Convolution (mathematical analysis) and integrable.

For the first time, integrals that constitute a convolution of two functions are found in the writings of Leonhard Euler (1760s); later, the convolution appears in Laplace, Lacroix, Fourier, Cauchy, Poisson and other mathematicians. The designation of the convolution of functions using an asterisk was first proposed by Vito Volterra in 1912 in his lectures at the Sorbonne (published a year later) [1] .

Properties

  • Commutativity:

  Convolution (mathematical analysis) .

  • Associativity:

  Convolution (mathematical analysis) .

  • Linearity (distributivity and multiplication by a number):

  Convolution (mathematical analysis)

  Convolution (mathematical analysis)

  Convolution (mathematical analysis) .

  • Rule of differentiation:

  Convolution (mathematical analysis) ,

Where   Convolution (mathematical analysis) denotes the derivative of the function   Convolution (mathematical analysis) on any variable.

  • Fourier image property:

  Convolution (mathematical analysis) ,

Where   Convolution (mathematical analysis) denotes the Fourier transform of the function   Convolution (mathematical analysis) .

Convolution on groups

Let be   Convolution (mathematical analysis) - Lee group, equipped with a Haar measure   Convolution (mathematical analysis) and   Convolution (mathematical analysis) - two functions defined on   Convolution (mathematical analysis) . Then their convolution is called the function

  Convolution (mathematical analysis) .

Convolution of measures

Let there be a Borel space   Convolution (mathematical analysis) and two measures   Convolution (mathematical analysis) . Then their convolution is called measure

  Convolution (mathematical analysis) ,

Where   Convolution (mathematical analysis) denotes the product of measures   Convolution (mathematical analysis) and   Convolution (mathematical analysis) .

Properties

  • Let be   Convolution (mathematical analysis) absolutely continuous with respect to Lebesgue measure   Convolution (mathematical analysis) . We denote their Radon-Nicodemus derivatives:

  Convolution (mathematical analysis) .

Then   Convolution (mathematical analysis) also absolutely continuous regarding   Convolution (mathematical analysis) and its derivative Radon - Nicodemus   Convolution (mathematical analysis) has the appearance

  Convolution (mathematical analysis) .

  • If a   Convolution (mathematical analysis) - probability measures, then   Convolution (mathematical analysis) also is a probabilistic measure.

Convolutions distribution

If a   Convolution (mathematical analysis) - distribution of two independent random variables   Convolution (mathematical analysis) and   Convolution (mathematical analysis) then

  Convolution (mathematical analysis) ,

Where   Convolution (mathematical analysis) - amount allocation   Convolution (mathematical analysis) . In particular, if   Convolution (mathematical analysis) absolutely continuous and have densities   Convolution (mathematical analysis) , then a random variable   Convolution (mathematical analysis) also absolutely continuous and its density is:

  Convolution (mathematical analysis) .

see also

  • Autocorrelation function
  • Fourier transform
  • Sequence folding

Convolution of functions is the most important mathematical concept that is used in almost all areas of science and technology, including that it is widely used for evaluating image systems and for processing digital images. The convolution of two functions is a mathematical operation of two functions h (x) and f (x), generating the third function g (x), which can be considered as a modified version of one of the original ones, for example, after averaging or smoothing operations. The convolution h (x) and f (x) is written as h ∗ f (asterisk symbol). For continuous functions, it is defined as the integral of the product of two functions after one is reversed and shifted. Essentially, this is a special kind of integral transformation:

  Convolution (mathematical analysis) (5.9)

The convolution operation is illustrated in Fig. 5.8 for two functions defined as rectangular pulses of different durations.

  Convolution (mathematical analysis)

Fig. 5.8. An example of the convolution of two continuous functions h (x) and f (x). A darker color shows the area equal to the integral (5.9) for different values ​​of x (adapted from [4])

The one-dimensional discrete convolution of two discrete functions h (i) and f (i) of length N is defined as

  Convolution (mathematical analysis) (5.10)

From the point of view of the computational process, an easier and faster way to calculate the convolution of two functions is to use the convolution theorem. This theorem proves that the convolution of two functions is equivalent to the multiplication of their Fourier transforms in the frequency space. Thus, the convolution equation (5.9) can be expressed as

  Convolution (mathematical analysis) (5.11)

where H (u) and F (u) is the Fourier transform of the functions h (x) and f (x) in the frequency space.


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Digital signal processing

Terms: Digital signal processing