What is a Linear Time Invariant System?

The systems that are both linear and time-invariant are called LTI Systems. The system must be linear and a Time-invariant system. Linear systems have the trait of having a linear relationship between the input and the output. A linear change in the input will also result in a linear change in the output.

In many significant physical systems, these features hold (exactly or approximately), in which case convolution can be used to find the system’s response, y(t), to any given input, x(t). y(t) = (x ∗ h)(t), where ∗ denotes convolution and h(t) is the system’s impulse response

For Continuous Signal

Y(t)=∫^∞_−∞h(α)X(t−α)dα=∫^∞_−∞X(α)h(t−α)dα.

In case of discrete signal integration changes to Sigma ( Σ ).

then formula will be,

y[n] = Σ h[α] x[t-α]

where α range -infinity to + infinity

Where ,

x(t) -> input signal

y(t) -> output signal

h(t) -> transfer function

Linear System

If input x₁(t) produces y₁(t) as output and input x₂(t) produces y₂(t) output, then if the combination of the x₁(t) + x₂(t) will produce the y₂(t) + y₂(t) as output then the system is called as the Linear system.

if,

x₁(t) -> y₁(t),

x₂(t) -> y₂(t)

Let x₃(t) = x₁(t) + x₂(t)

x₃(t) -> y₃(t) , y₃(t) = y₁(t) + y₂(t)

if satisfied the following condition then system called as linear system.

Time-Invariant System

The output signal are different for the different time shift of the signal called called Time-invariant system. suppose x(t) produce output y(t)

x(t) -> y(t)

and shift in time t -> t + t₀

x(t + t₀) -> y(t + t₀)

same for the t -> t – t₀

x(t – t₀) -> y(t – t₀)

x[n] -> y[n]

x[n – n₀] -> y[n – n₀] for( discrete system )

then if the system satisfied following condition called time-invariant system and if the system follow the time-invariant and linear system property then system is called as the linear time-invariant system (LTI).

Homogeneity Principle

If scaling any input signal X(t) also scales the output signal by the same factor, then the system is said to be homogeneous. If x(t) produce the output y(t), Now if the X(t) scale by the factor of the “a” so the respective output also scaled by factor of “a” . As shown in below figure.

Superposition Principle

Its define only for the linear system, if input given to the system is x1(t) , x2(t) and output y1(t) , y2(t) respectively. now x1(t) + x2(t) and the output is y1(t) + y2(t).

for continuous-time linear system,

ay1(t) + by2(t) = a[x1(t)] + b[x2(t)]

for discrete-time linear system,

ay1[n] + by2[n] = a[x1[n]] + b[x2[n]]

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.