Discrete time convolution

As the name suggests the two basic properties of a LTI system are:

1) Linearity

A linear system (continuous or discrete time ) is a system that possesses the property of SUPERPOSITION. The principle of superposition states that the response of sum of two or more weighted inputs is the sum of the weighted responses of each of the signals. Mathematically

y[n] = S a_k y_k[n] = a₁y₁[n] + a₂y₂ [n] + ......

Superposition combines in itself the properties of ADDITIVITY and HOMOGENEITY. This is a powerful property and allows us to evaluate the response for an arbitrary input, if it can be expressed as a sum of functions whose responses are known.

2) Time Invariance

It allows us to find the response to a function which is delayed or advanced in time; but similar in shape to a function whose response is known.

Given the response of a system to a particular input, these two properties enable us to find the response to all its delays or advances and their linear combination.