Discrete Time Convolution Properties | Discrete Time Signal

Convolution is such an effective tool that can be utilized to determine a linear time-invariant (LTI) system’s output from an input and the impulse response knowledge.

Given two discrete time signals x[n] and h[n], the convolution is defined by

$x\left[ n \right]*h\left[ n \right]=y\left[ n \right]=\sum\limits_^<\infty >>x\left[ i \right]h\left[ n-i \right]~~~~~~~~~~~~~~~~~~~~~~~\left( 1 \right)$


The summation on the right side is called the convolution sum.

It should be noted that the convolution sum exists when x[n] and h[n] are both zero for all integers n

Since the summation in (2) is over a finite range of integers (i=0 to i=n), the convolution sum exists. Hence any two signals that are zero for all integers n

To compute the convolution (1) or (2)

  1. First, change the discrete-time index n to i in the signals x[n] and h[n].
  2. Flip the signals h[i] to obtain h[-i] (it is called folding).
  3. For each output index n, shift by n to get h[n-i] .(Positive value of n gives right shift.)
  4. The product x[n]*h[n] is formed and y[n] is computed by summing the values of x[i]*h[n-i] as i ranges over the set of integers.

Discrete-Time Convolution Example

Suppose that x[n]=a n u[n] and h[n]= a n u[n] Where u[n] is a discrete-time unit-step function and a and b are fixed non zero real numbers.

Step 1:

Change discrete time signal index n to i in both signals:

$h\left[ i \right]=^>u\left[ i \right]$

Step 2:

Flip h[i] to get h[-i]:

$h\left[ -i \right]=^>u\left[ -i \right]$

Step 3:

Shift by n to get h[n-i]:

$h\left[ n-i \right]=^>u\left[ n-i \right]$

Step 4:

Find y[n] by summing the product x[i]h[n-i] over a finite range of i:

$y\left[ n \right]=\sum\limits_^<\infty >>x\left[ i \right]h\left[ n-i \right]~~~~~~~~~~~\left( 3 \right)$

Thus the summation on i in (3) may be taken from i=0 to i=n and the convolution operation is given by,

$y\left[ n \right]=\sum\limits_^>x\left[ i \right]h\left[ n-i \right]$

If a≠b, then standard math relation gives

Discrete-Time Convolution Properties

The convolution operation satisfies a number of useful properties which are given below:

Commutative Property

If x[n] is a signal and h[n] is an impulse response, then

Associative Property

If x[n] is a signal and h1[n] and h2[n] are impulse responses, then

Distributive Property

If x[n] is a signal and h1[n] and h2[n] are impulse responses, then