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Linear Time Invariant Systems (LTI)


Continuous convolution


Discrete convolution


Convolution Property of Continuous Impulse


Convolution Property of Discrete Impulse


Properties of LTI Systems

  1. The commutative property

y(t) = x(t) 688.png h(t) = 693.png

or y(t) = h(t698.png x(t) =703.png

y(n) = x(n708.png h(n) =713.png

or y(n) = h(n719.png x(n) =724.png

  1. The distributive property

The output y(n) = x(n729.png {h1(n) + h2(n)}

y(n) = x(n734.png h1(n) + x(n739.png h2(n)


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

  1. Associative Property of LTI Systems
    The output y(t) = x(t749.png [h1(t754.png h2(t)]
    y(t) = [x(t759.png h1(t)] 764.png h2(t)
    y(t) = x(t770.png h1(t775.png h2(t)
  2. 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  0.
    y(t) = kx(t)
    where K is constant and its impulse response
    h(t) = Kδ(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  0 then the system is called a dynamic system or system with memory.
  3. 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.
    The overall impulse response of a system with impulse response h(t) cascaded with its inverse system with impulse response h1(t) is given as
    h(t780.png h1(t) = δ(t)


Fig. : An inverse system for

continuous-time LTI systems.

  1. 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.
  2. 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

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