# Linear Time Invariant Systems (LTI)

**Continuous convolution**

Discrete convolution

Convolution Property of Continuous Impulse

Convolution Property of Discrete Impulse

Properties of LTI Systems

**The commutative property**

*y*(*t*) = *x*(*t*)* * *h*(*t*) =* *

or *y*(*t*) = *h*(*t*) *x*(*t*) =

*y*(*n*) = *x*(*n*) *h*(*n*) =

or *y*(*n*) = *h*(*n*) *x*(*n*) =

**The distributive property**

The output *y*(*n*) = *x*(*n*) {*h*_{1}(*n*) + *h*_{2}(*n*)}

*y*(*n*) = *x*(*n*) *h*_{1}(*n*) + *x*(*n*) *h*_{2}(*n*)

**Fig. : **The distributive property of convolution sum for a parallel interconnection of discrete-time LTI systems.

**Associative Property of LTI Systems***y*(*t*) =*x*(*t*) [*h*_{1}(*t*)*h*_{2}(*t*)]

*y*(*t*) = [*x*(*t*)*h*_{1}(*t*)]*h*_{2}(*t*)

*y*(*t*) =*x*(*t*)*h*_{1}(*t*)*h*_{2}(*t*)**Static and Dynamic LTI Systems**

Static systems are also known as memoryless systems. A system is known as static if its output at any time depends only on the value of the input at the same time.

In particular, a continuous-time LTI system is memoryless (static) if its unit-impulse response*h*(*t*) is zero for*t*â‰ 0.

*y*(*t*) =*kx*(*t*)

where*K*is constant and its impulse response

*h*(*t*) =*K*Î´(*t*)

If*K*= 1, then these systems are called identity systems.

If the impulse response of a discrete-time LTI system is not identically zero for*n*â‰ 0 then the system is called**a dynamic system or system with memory.****Invertibility of LTI Systems**

A system is known as invertible only if an inverse system exists which, when cascaded (connected in series) with the original system, produces an output equal to the input at first system.*h*(*t*) cascaded with its inverse system with impulse response*h*_{1}(*t*) is given as*h*(*t*)*h*_{1}(*t*) = Î´(*t*)

** **

**Fig. : **An inverse system for

continuous-time LTI systems.

**Causality for LTI System**

This property says that the output of a causal system depends only on the present and past values of the input to the system.

A continuous-time LTI system is called causal system if its impulse response*h*(*t*) is zero for*t*< 0.**Stability for LTI Systems**

A stable system is a system which produces bounded output from every bounded input. Now let us determine conditions under which LTI systems are called stable.

Condition of Stability for Continuous-time LTI System