Time domain analysis of LTI systems

Convolution (folding together) is a complicated operation involving integrating, multiplying, adding, and time-shifting two signals together. Convolution is a key component to the rest of the material in this book.

The convolution a * b of two functions a and b is defined as the function:
 * .jpg

This is the format of convolution.

The greek letter τ (tau) is used as the integration variable, because the letter t is already in use. τ is used as a "dummy variable" because we use it merely to calculate the integral.In the convolution integral, all references to t are replaced with τ, except for the -t in the argument to the function b. Function b is time inverted by changing τ to -τ. Graphically, this process moves everything from the right-side of the y axis to the left side and vice-versa. Time inversion turns the function into a mirror image of itself.Next, function b is time-shifted by the variable t. Remember, once we replace everything with tau, we are now computing in the tau domain, and not in the time domain like we were previously. Because of this,we can use t as shift parameter.

We multiply the two functions together, time shifting along the way, and we take the area under the resulting curve at each point. Two functions overlap in increasing amounts until some "watershed" after which the two functions overlap less and less. Where the two functions overlap in the t domain, there is a value for the convolution. If one (or both) of the functions do not exist over any given range, the value of the convolution operation at that range will be zero.

Last step is the definite integral plugs the variable t back in for remaining references of the variable τ, and we have a function of t again. It is important to remember that the resulting function will be a combination of the two input functions, and will share some properties of both.

Youtube video

Properties of Convolution
The convolution function satisfies certain conditions:
 * Commutativity
 * Community.png


 * Associativity
 * Associativity.png
 * Associativity.png


 * Distributivity
 * Distribuvity.png
 * Distribuvity.png


 * Associativity With Scalar Multiplication
 * Associativity with scalar.png
 * Associativity with scalar.png

Differentiated rule  MIT video explain about Convolution
 * Differentiation rule.png

Practice Problems
Practice problem from MIT : Problem Solution

Practice problem for discrete time convolution: Problem Solution

Practice problem for continuous convolution: Problem Solution

Extra Resources
MIT course