Convolution

In mathematics (in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reversed and shifted. ( f ∗ g ) ( t ) ≜   ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ . {displaystyle (f*g)(t) riangleq int _{-infty }^{infty }f( au )g(t- au ),d au .} ( f ∗ g ) [ n ] = ∑ m = − ∞ ∞ f [ m ] g [ n − m ] {displaystyle (f*g)=sum _{m=-infty }^{infty }fg} ( f ∗ g N ) [ n ] = ∑ m = 0 N − 1 f [ m ] g N [ n − m ] = ∑ m = 0 n f [ m ] g [ n − m ] + ∑ m = n + 1 N − 1 f [ m ] g [ N + n − m ] = ∑ m = 0 N − 1 f [ m ] g [ ( n − m ) mod N ] ≜ ( f ∗ N g ) [ n ] {displaystyle {egin{aligned}(f*g_{N})&=sum _{m=0}^{N-1}fg_{N}\&=sum _{m=0}^{n}fg+sum _{m=n+1}^{N-1}fg\&=sum _{m=0}^{N-1}fg riangleq (f*_{_{N}}g)end{aligned}}}     (Eq.1) In mathematics (in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reversed and shifted. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation only in that either f (x) or g(x) is reflected about the y-axis; thus it is a cross-correlation of f (x) and g(−x), or f (−x) and g(x).  For continuous functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, statistics, computer vision, natural language processing, image and signal processing, engineering, and differential equations. The convolution can be defined for functions on Euclidean space, and other groups. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing. Computing the inverse of the convolution operation is known as deconvolution. The convolution of f and g is written f∗g, using an asterisk. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform: An equivalent definition is (see commutativity): While the symbol t is used above, it need not represent the time domain.  But in that context, the convolution formula can be described as 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.  As t changes, the weighting function emphasizes different parts of the input function.