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Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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The convolution of two signals in the time domain is equivalent to the multiplication of their representation in frequency domain. Mathematically, we can write the convolution of two signals as

Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Steps for convolution

  • Take signal x1(t) and put t = p there so that it will be x1(p).
  • Take the signal x2(t) and do the step 1 and make it x2(p).
  • Make the folding of the signal i.e. x2(-p).
  • Do the time shifting of the above signal x2[-(p-t)]
  • Then do the multiplication of both the signals. i.e.  Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Example

Let us do the convolution of a step signal u(t) with its own kind.

Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Now this t can be greater than or less than zero, which are shown in below figures


Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

So, with the above case, the result arises with following possibilities

Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of Convolution

Commutative

It states that order of convolution does not matter, which can be shown mathematically as

Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Associative

It states that order of convolution involving three signals, can be anything. Mathematically, it can be shown as;

Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Distributive

Two signals can be added first, and then their convolution can be made to the third signal. This is equivalent to convolution of two signals individually with the third signal and added finally. Mathematically, this can be written as;

Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Area

If a signal is the result of convolution of two signals then the area of the signal is the multiplication of those individual signals. Mathematically this can be written

If  Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Then, Area of y(t) = Area of x1(t) X Area of x2(t) 

Scaling

If two signals are scaled to some unknown constant “a” and convolution is done then resultant signal will also be convoluted to same constant “a” and will be divided by that quantity as shown below.

If,   Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)
Then,  Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Delay

Suppose a signal y(t) is a result from the convolution of two signals x1(t) and x2(t). If the two signals are delayed by time t1 and t2 respectively, then the resultant signal y(t) will be delayed by (t1+t2). Mathematically, it can be written as −

If,  Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)
Then,   Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Solved Examples

Example 1 − Find the convolution of the signals u(t-1) and u(t-2).

Solution − Given signals are u(t-1) and u(t-2). Their convolution can be done as shown below −

Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Example 2 − Find the convolution of two signals given by

Convolution - Operations on Signals Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Solution

x2(n) can be decoded as x2(n) = {2,2,2,2,2} Original first

x1(n) is previously given = {3,−2,3} = 3−2Z−1 + 2Z−2

Similarly, x2(z) = 2 + 2Z−1 + 2Z−2 + 2Z−3 + 2Z−4

Resultant signal,

X(Z) = X1(Z)X2(z)

= {3−2Z−1 + 2Z−2} × {2+2Z−1 + 2Z−2 + 2Z−3 + 2Z−4}

= 6+2Z−1 + 6Z−2 + 6Z−3 + 6Z−4 + 6Z−5

Taking inverse Z-transformation of the above, we will get the resultant signal as

x(n) = {6,2,6,6,6,0,4} Origin at the first

Example 3 − Determine the convolution of following 2 signals -

x(n) = {2,1,0,1}

h(n) = {1,2,3,1}

Solution

Taking the Z-transformation of the signals, we get,

x(z) = 2 + 2Z−1 + 2Z−3

And h(n) = 1 + 2Z−1 + 3Z−2 + Z−3

Now convolution of two signal means multiplication of their Z-transformations

That is 

Y(Z) = X(Z) × h(Z)

= {2 + 2Z−1 + 2Z−3} × {1 + 2Z−1 + 3Z−2 + Z−3}

= {2 + 5Z−1 + 8Z−2 + 6Z−3 + 3Z−4 + 3Z−5 + Z−6}

Taking the inverse Z-transformation, the resultant signal can be written as;

y(n) = {2,5,8,6,6,1} Original first

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