Theory:

 

 

 Convolution

 Convolution properties

 Correlation

       (a)Cross-correlation

       (b)Auto-correlation

 Properties of Auto-correlation

 

 

 

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Convolution:

 

Convolution is a mathematical operation which can be perform on two signals 'f' and 'g' to produce a third signal which is typically viewed as the modified version of one of the original signals. A convolution is an integral that express the overlap of one signal 'g' as it is shifted over another signal 'f'.

Convolution of two signals 'f' and 'g' over a finite range [0 → t] can be defined as

                                                               

Here the symbol [f*g](t) denotes the convolution of 'f' and 'g'. Convolution is more often taken over an infinite range like,

                                                 

 The convolution of two discrete time signals f(n) and g(n) over an infinite range can be defined as

                                                      

 The convolution of two continuous time signals are shown in the Figure 1.

                                                      

                                                                               

                                                                        

                                                                                                    

                                                                       

                                                                                                   

                                                                      

                                                                                                    

                                                                       

                                                                                                   

                                                                      

                                                                                                  

                                                                      

                                                                                                  

                                                                      

                                                                             Fig.1 Convolution of two signals

 

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Convolution properties:

 

There are some important properties of convolution that perform on continuous time signal which we have listed below. The commutativity, associativity, distributivity properties are given below.

 

Commutativity	f(t) * g(t) = g(t) * f(t)
Associativity	[f(t) * g(t)] * h(t) = f(t) * [g(t) * h(t)]
Distributivity	f(t) * [g(t) + h(t)] = f(t) * g(t) + f(t) * h(t)

 

 

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Correlation:

 

So far in the above discussion we have discussed about the convolution of two continous time signals which is used to find the output y(t) of a system. Correlation is a mathematical operation that closely resembles convolution. Correlation is basically used to compare two signals. Correlation is the measure of the degree to which two signals are similar. The correlation of two signals is divided into two ways: (i) Cross-correlation, (ii) Auto-correlation.

 

 Cross-correlation: 

 

Cross correlation is a measure of similarity between two waveforms as a function of time gap or delay applied to one of them. The cross correlation between a pair of continuous time signals f(t) and g(t) is given by 

                                     where      --------- (1)

and it can be derived for discrete time signal f(n) and g(n) as

                                                 where   

The index and are the shift parameters for continuous time and discrete time signals respectively. The order of subscripts 'fg'  indicates that f(t) or f(n) are the reference sequence in continuous-time and discrete-time respectively that remains unshifted in time whereas the sequence g(t) or g(n) are shifted '' or 'k' units in time with respect to f(t) or f(n) respectively.

If we want to fix g(t) and to shift f(t), then the correlation of two sequences can be written as

                                              

                                                           -----------------------(2)

and for discrete time signals, the cross-correlation can be written as

                                                   

                                                                              

Now comparing eqⁿ. (1) and eqⁿ. (2) we find that

                                                                

and similarly for discrete time signal 

                                                                

Where for continuous time signals is the folded version of about =0 and for discrete time signal  is the folded version of  about k=0.

Now we can rewrite the eqⁿ. (1) as         

-----------------(3)

                                                    

and similarly for discrete time signal

                                                                

 

From the above equation(3) we find that the correlation process is essentially the convolution of two data sequence in which one of the sequence has been reversed.

 

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Auto-correlation: 

 

Auto correlation of a continuous time signal is the correlation of the signal with itself. The auto correlation of a continuous time signal f(t) is defined as

                                                                   

    

for discrete time signal f(n) it is defined as

                                                                  

or equivalently we can write                      

 
      

and similarly for discrete time signal f(n) we can write

                                                                 

       

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Relation to Signal energy and Signal power:

 

 The auto-correlation function of a periodic signal is itself a periodic signal with a period the same as that of the original signal.

 If f(t) is an energy signal, its auto-correlation is

                                                                   

         and                                                    

for continuous-time and discrete-time signals respectively.

After applying a zero shift it becomes

          or                                                     

 which is the total signal energy of the signal.

If f(t) or f[n] is a power signal, the auto-correlation at zero shift is

                                                                    

for continuous-time and discrete-time respectively which is the average signal power of the signal.

 

Properties of Auto-correlation:

 

The auto-correlation depends on the choice of the amount of shift applied. we can say from the observation that the value of the auto-correlation can never be bigger than it is at zero shift. That is,

                                                                

It will happen because at a zero shift, the correlation with itself is obviously as large as it can get since the shifted and unshifted versions coincide.

 Another property of auto-correlation function is that all auto-correlation functions are even functions (but not all correlation functions).

  or                                                         

 If we make the change of variable

                                                                       and   

We can show that                                    

It can be shown by a similar technique for discrete-time signal also

                                                                

 Another characteristic of auto-correlation function is that a time shift of a signal does not make any change of its auto-correlation.

 Letbe the auto-correlation function of a discrete-time energy signal f[n]. Then 

                                                             

 Now let y[n] = f [n-n₀] . Then

                                                            

                                                                                        

Now we can make a change of variable q = n-n₀ . Then

                                                             

 

Another characteristic of auto-correlation function is that the auto-correlation of a sum of sinusoids of different frequencies is the sum of auto-correlation of the individual sinusoids. To demonstrate this idea let a continuous-time power signal f(t) be a sum of two sinusoids f₁(t) and f₂(t), where

                                                       

The auto-correlation of this signal is

                                                  

                                    

 Therefore,                                  

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