Convolution Vs Correlation

Concept of Linear Convolution

The concept of convolution is central to Fourier theory and the analysis of Linear Systems. In fact the convolution property is what really makes Fourier methods useful. In one dimension the convolution between two functions, f (x) and h(x) is defined as:

Where ‘s’ is a dummy variable of integration. This operation may be considered the area of overlap between the function f(x) and the spatial (or time) reversed version of the function h(x). For discrete signals the integration in above equation is replaced by summation. The result of the convolution of two simple one dimensional functions is shown in figure 1.

 

The Convolution theorem relates the convolution between the two real space (or time) domain signals to a multiplication in the Fourier domain, and can be written as;

                                                          
                                                           
Convolution Properties

The convolution is a linear operation which is distributive, so that for three functions f (x), g(x) and h(x) we have that

                                        

and commutative, so that

If the two functions f (x) and h(x) are of finite extent, (are zero outside a finite range of x), then the extent (or width) of the convolution g(x) is given by the sum of the widths the two functions. For example if figure 1 both f (x) and h(x) non-zero over the finite range x = ±1. Thus the convolution g(x) is non-zero over the range x = ±2. If f[n] and h[n] are two finite length sequences of lengths L₁ and L₂ then g[n] will be of length L₁+L₂-1.

Correlation of two Signals

A closely related operation to Convolution is the operation of Correlation of two functions. In Correlation two functions are shifted and the area of overlap formed by integration, but this time without the spatial (or time) reversal involved in convolution. The Correlation between two function f (x) and h(x) is given by

                                   

This is for real signals, for complex signals h^*(s-x) is used, where h^*(x) is the complex conjugate of h(x). This operation is shown for two simple functions in figure 2. If we compare the convolution in figure 1 and the correlation shown in figure 2, the only difference is that the second function in correlation is not spatially (or time) reversed and the direction of the shift is changed. Thus the correlation of two real functions f(t) and h(t), is equivalent to the convolution of f(-t) and h(t) or the convolution of f^*(-t) (in case of  complex functions) and h(t).

 

Figure 2: Correlation of two simple functions.

The correlation is of more importance, if we consider f (t) to be the signal and h(t) to be the target” then we see that the correlation gives a peak where the signal matches the target. This gives the basis of the simple method of target detection. In the Fourier Domain the Correlation Theorem becomes

                                                        

The correlation is a linear operation, which is distributive, but however is not commutative, since if

                                                           

then we can show that,
                                                                           

Autocorrelation

Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them. It is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal which has been buried under noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.