Types of LTI System
The types of LTI System are mentioned below:
- Continuous-Time LTI Signal
- Discrete Time LTI Signal
Continuous-Time LTI Signal
The impulse response is always taken into account while evaluating LTI systems. In other words, the impulse signal is the input and the impulse response is the output.
here x(t) = δ(t),
and the impulse response the respective output signal is y(t) = h(t) = T. δ(t)
Any signal can be described as a combination of a weighted and shifted impulse signal, according to the shifting property of signals.
x(t)=∫∞−∞x(τ).δ(t−τ)dτ
the impulse response is,
y(t) = T[x(t)] = ∫∞−∞x(τ) . T[δ(t−τ)]dτ,
y(t) = ∫∞−∞x(τ) h(t−τ) dτ
from here we got an equation,
y(t) = x(t) ✲ h(t)
Discrete-Time LTI System
In case of discrete time signal ‘t‘ going to replace by the ‘n‘ , here
x[n] = δ(n)
the discrete time input response is given by,
y[n] = h(n) = T δ(n)
Any signal can be described as a combination of a weighted and shifted impulse signal, according to the shifting property of signals.
x[n]=∑ x[k] δ[n−k] where k= [−∞, ∞]
the impulse response y[n] = T δ[n]
y[n] = ∑ x[k]. T[δ(n−k)]
y[n] = ∑ x[k] h[n−k]
so the final output will be,
y[n] = x[n] ✲ h[n]
LTI System
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.
Table of Content
- LTI System
- Types
- Properties
- Transfer Function
- Convolution
- Sampling Theorem
- Nyquist Rate
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