A Linear Time-Invariant (LTI) system is a fundamental concept in signal processing, control systems, and communication engineering. Such systems are characterized by two important properties: linearity and time invariance. Because of these properties, LTI systems are mathematically convenient and widely used for analyzing signals and systems.
Table of Contents
Linearity Property
A system is said to be linear if it satisfies the principle of superposition, which consists of two conditions.
Additivity
If
\[
x_1(t) \rightarrow y_1(t)
\]
and
\[
x_2(t) \rightarrow y_2(t)
\]
then
\[
x_1(t) + x_2(t) \rightarrow y_1(t) + y_2(t)
\]
Homogeneity (Scaling)
If
\[
x(t) \rightarrow y(t)
\]
then
\[
a\,x(t) \rightarrow a\,y(t)
\]
for any constant \(a\).
Combining both conditions gives the superposition property:
\[
a_1 x_1(t) + a_2 x_2(t) \rightarrow a_1 y_1(t) + a_2 y_2(t)
\]
Time-Invariance Property
A system is said to be time invariant if its characteristics do not change with time.
If an input \(x(t)\) produces an output \(y(t)\), then a delayed input
\(x(t – t_0)\) should produce a delayed output
\[
y(t – t_0)
\]
If this condition holds, the system is time invariant.
Impulse Response of LTI Systems
The behavior of an LTI system is completely determined by its
impulse response \(h(t)\).
If the input to the system is an impulse function \(\delta(t)\), the output is
\[
h(t)
\]
Thus,
\[
\delta(t) \rightarrow h(t)
\]
Convolution Representation
For an LTI system, the output is obtained by convolution of the input
signal with the impulse response.
\[
y(t) = x(t) * h(t)
\]
In expanded form,
\[
y(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)\, d\tau
\]
This equation shows that the output of an LTI system depends on the
input signal and the system impulse response.
Frequency Response of LTI Systems
The frequency response of an LTI system is obtained by taking the
Fourier transform of the impulse response.
\[
H(\omega) = \mathcal{F}\{h(t)\}
\]
where
\(H(\omega)\) = system frequency response
\(h(t)\) = impulse response
The output in the frequency domain becomes
\[
Y(\omega) = X(\omega) H(\omega)
\]
Thus, the system modifies the amplitude and phase of frequency components
of the input signal.