Properties of LTI System

The unit impulse response of an LTI system can be used to express it in continuous time. It is represented by an integral convolution. Therefore, the LTI system also adheres to the same properties as the continuous time convolution. The significance of an LTI system’s impulse response lies in its ability to fully define its properties.

The property types of LTI system are as follows:

• Invertible System
• Causality
• Commutative Property
• Distributive Property
• Associative Property

These properties of the LTI System are explain below in detail:

• Invertible Systems : If we can determine h(t) such that the output y(t) can be used to recover the original input x(t), then the system is invertible. This requires a one-to-one system in order to hold true.

• Causality : As far as we are aware, a causal system’s output is only dependent on its past or current inputs—it is not dependent on its future inputs. Similarly, an output in a causal system is not dependent on the future; rather, it reacts to an input only after it happens. Put otherwise, a reaction to an input at t = t0 would only happen for t >= t0 and not earlier.

h(t)=0; for t<0

So the equation of y(t) will be,

Output of the causal LTI for non causal input signal.

y(t) = ∫^∞₀h(τ) x(t−τ) dτ = ∫^t_−∞x(τ) h(t−τ) dτ

x(t) -> non causal input signal.

Output of a causal LTI for causal input signal.

y(t) = ∫^t₀ h(τ) x(t−τ) dτ = ∫^t₀ x(τ).h(t−τ) dτ

h(t) -> Transfer function

X(t) -> causal Input signal

• Commutative Property : The convolution of the continuous-time signal is called commutative property.

x(t) ❇ h(t) = h(t) ❇ x(t)

∫^∞_−∞x(τ) . h(t−τ)dτ = ∫^∞_−∞h(τ) x(t−τ)dτ

The output of an LTI system with input x(t) and unit impulse response h(t) is the same as the output of an LTI system with input h(t) and impulse response x(t), given the commutative property of LTI systems.

x(t) -> Input signal.

• Distributive Property: Regarding system connectivity, the distributive property of the LTI system has a helpful meaning. This means that a single system with the impulse response [h₁(t)+h₂(t)] can take the place of the two LTI systems with the impulse responses h1(t) and h2(t) linked in parallel.

x(t) ❇ {h1(t) + h2(t)} = x(t) ❇ h1(t) + x(t) ❇ h2(t)

This is a complex convolution can be divided into multiple simpler convolutions using the distributive property of continuous-time convolution.

Associative Property : According to the given property on changing the order of the signal their will be no change in the convolution, As shown below.

x(t) ❇ [h1(t) ❇ h2(t)] = [x(t) ❇ h1(t)] ❇ h2(t)

Systems that are both linear and time-invariant are known as linear time-invariant systems, or LTI systems for short. When a system’s outputs for a linear combination of inputs match the outputs of a linear combination of each input response separately, the system is said to be linear. Time-invariant systems are ones whose output is independent of the timing of the input application. Long-term behavior in a system is predicted using LTI systems. The term “linear translation-invariant” can be used to describe these systems, giving it the broadest meaning possible. The analogous term in the case of generic discrete-time (i.e., sampled) systems is linear shift-invariant.