Linear Time Invariant (LTI) systems have properties that arise from the properties of convolution.

Property 1: Convolution is Commutative

$ x_1(t)*x_2(t) = x_2(t)*x_1(t)\ $

System Example: Convolving the input to a system with its impulse response is the same as convolving the impulse response with the input.

Property 2: Convolution is Distributive

$ \displaystyle x_1(t)* \left ( x_2(t)+x_3(t) \right ) = x_1(t)*x_2(t)+x_1(t)*x_3(t) $

System Example: Convolving a single input with two impulse responses then adding the output is the same as convolving the input with the sum of the impulse responses.

Property 2: Convolution is Associative

$ x_1(t)*x_2(t)*x_3(t) = x_2(t)*x_3(t)*x_1(t)\ $

System Example: Convolving an input with an impulse response and convolving that with the impulse response of another system is the same as convolving the two impulse responses and then the input to the system.

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett