An
ideal linear time-invariant system is shown in Figure 1.
Figure 1. Linear Time Invariant System
A system is linear if the principle of superposition applies. In other words, if x1 results in the output signal y1 and x2 results in the output signal y2, and the system is linear, the system must satisfy ax1 + bx2 à ay1 + by2, for any constants a and b. Hence a system is time invariant if time shifts in the input produce output with equal time-shifts. For example, a continuous-time system is time invariant if x(t) à y(t) implies x(t-t0) à y(t-t0) for any time shift t0. Both linearity and time invariance apply to both continuous time and discrete-time systems.
In the system shown in Figure 1, y(t) is the output, h(k) the impulse response at time k and x(t) the input data. Since this is an LTI system, the output y(t) is the convolution sum of the input x(t) and the system response h(k) and the system equation is,
.
(1)
In terms of system identification, if
the system h(k) is not known, the
input x(t), and output y(t) can be used to identify the system h(k) with a deconvolution process.
In reality, noise is present in most of the data acquisition
process. A system with the presence
of noise is shown in Figure 2.
Figure 2. LTI System with the Presence of Noise
The
noise components
and
are added to the clean input x(t)
and filtered output y(t) accordingly,
and 
and
are the data which will be used for the system identification.
The discrete model of this linear system can be redrawn as in Figure 3.
Figure 3. Discrete-Time LTI System with the presence of noise
In
this case, x[n] is the input of length N,
h[m]
is the causal filter with length M,
and
is the final output of the system
with length M+N-1.
The discrete model of the system will be used throughout this paper.