8 February 2019

Computation of linear convolution

The different method used for the computation of linear convolution is as follows :
  • Graphical method
  • Using the mathematical equation of convolution
  • Tabulation method
  • Multiplication method
This article gives information about the graphical method to how to find computation of linear convolution.

Graphical method :
                   
              ∞
y (n)  =  ∑  x (k) h(n-k)
           k= -∞
    
for n = 0              
              ∞
y (0)  =  ∑  x (k) h(-k)
           k= -∞        
    
for n =1

              ∞
y (1)  =  ∑  x (k) h(1-k)
           k= -∞       

Here the term (1-k) can be written as h(-k+1).

              ∞
y (1)  =  ∑  x (k) h(-k+1)
           k= -∞ 

Here h(-k+1) indicates of folded signal h(-k). It indicates that h(-k) is delayed by 1 sample. Similarly for other values of n output y (n) is calculated.

Thus different operation involved in the calculation of linear convolution are as follow :
  • Folding operation: It indicates folding of sequence h (k)
  • Shifting operation: It indicates time shifting of h (-k) e.g h (-k+1)
  • Multiplication: It indicates multiplication of x (k) and h (n-k)
  • Summation: It indicates the addition of all product terms obtained because of multiplication of x (k) and h (n-k)
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