Linear Convolution

The relation between input to the shift invariant system, x[n] or x(t) and output y[n] or y(t) is given by the convolution of input x[n] or x(t) and h[n] or h(t). Where h[n] and h(t) are the impulse responses of discrete time and continuous time LTI systems respectively.
Mathematically the convolution is defined as,                                               

                                                 
                                                 

Reader should remember that all the relations further are described with respect to the discrete convolution but as far as the concepts are concern they are also valid for the continuous time convolution.

Properties of Convolution
(a) Commutative Property
                                                           

(b) Associative Property
                                      

(c) Distributive Property
                                     

How to perform Linear Convolution?
Having considered some of the properties of the convolution operator, we now look at the mechanics of performing convolutions. There are several different approaches that may be used, and the one that is the easiest will depend upon the form and type of sequences that are to be convolved.


(a) Direct Approach
When the sequences that are being convolved may be described by simple closed-form mathematical expressions, the convolution is often most easily performed by directly evaluating the sum given. In performing convolutions directly, it is usually necessary to evaluate finite or infinite sums. Table given below shows the closed-form expressions for some of the more commonly encountered series.

Example:
                                 
With the direct evaluation sum we find,                     
                                 

As u(k) is equal to zero for k < 0 and u(n - k) is equal to zero for k > n, so when n < 0, there are no non-zero terms in the sum and y(n) = 0. On the other hand, if n is greater then 0, we have

                   
                            

(b) Graphical Approach

In addition to the direct method, convolutions may also be performed graphically. The steps involved in using the graphical approach are as follows

• Plot both sequences, x(k) and h(k), as functions of k.
• Choose one of the sequences, say h(k), and time-reverse it to form the sequence h(-k).
• Shift the time-reversed sequence by n. [Note:If n > 0, this corresponds to a shift to the right (delay), whereas if n < 0, this corresponds to a shift to the left (advance)].
• Multiply the two sequences x(k) and h(n-k) and sum the product for all values of k. The resulting value will be equal to y(n). This process is repeated for all possible shifts, n.

Example:

To illustrate the graphical approach to convolution, let us evaluate y(n) = x(n)*h(n) where x(n) and h(n) are the sequences shown in Fig (a) and (b), respectively. To perform this convolution, we follow the steps listed above:

• Because x(k) and h(k) are both plotted as a function of k in (a) and (b), we next choose one of the sequences to reverse in time. In this example, we time-reverse h(k), which is shown in Fig (c).
• Forming the product, x(k)h(-k), and summing over k, we find that y(0) = 1.
• Shifting h(k) to the right by one results in the sequence h(l-k) shown in Fig (d). Forming the product, x(k)h(l-k), and summing over k, we find that y(1) = 3.
• Shifting h(l-k) to the right again gives the sequence h(2-k) shown in Fig (e). Forming the product,x(k)h(2-k), and summing over k, we find that y(2) = 6.
• Continuing in this manner, we find that y(3) = 5. y(4) = 3, and y(n) = 0 for n > 4. 6. We next take h(-k)and shift it to the left by one as shown in Fig (f ). Because the product, x(k)h(-1-k), is equal to zero for all k, we find that y(-1) = 0. In fact. y(n) = 0 for all n < 0. 

  

Some facts:

• If x(n) is of length M and h(n) is of length N, then y(n)= x(n)*h(n) will be of length L=M+N-1. 
• If the nonzero values of x(n) are contained in the interval [M, N] and the nonzero values of h(n) are contained in the interval [J,K], the nonzero values of y(n) will be confined to the interval [M+J, N+K].