Time-domain analysis of LTIC systems - Continuous and Discrete Time Signals and Systems - page 135

Fig. 3.13. Solution y ( t ) for
Example 3.13 computed using
M ATLAB .

2

1.5

1

0.5

0

−0.5

−1

0 2 4 6 8 10 12 14 16 18 20

time

The solution y ( t ) is plotted in Fig. 3.13. It can be easily verified that the
plot is same as the analytical solution given by Eq. (C.38), which is reproduced
below

y ( t ) = 1 + 21 + 9

34 cos t + 15

− t − 4 t

2 e 17 e 34 sin t

for t ≥ 0 .

3.9 Summary

In Chapter 3, we developed analytical techniques for LTIC systems. We saw
that the output signal y ( t ) of an LTIC system can be evaluated analytically in
the time domain using two different methods. In Section 3.1, we determined the
output of an LTIC by solving a linear, constant-coefficient differential equation.
The solution of such a differential equation can be expressed as a sum of
two components: zero-input response and zero-state response. The zero-input
response is the output produced by the LTIC system because of the initial
conditions. For stable LTIC systems, the zero-input response decays to zero
with increasing time. The zero-state response is due to the input signal. The
overall output of the LTIC system is the sum of the zero-input response and
zero-state response.
An alternative representation for determining the output of an LTIC system
is based on the impulse response of the system. In Section 3.3, we defined the
impulse response h ( t ) as the output of an LTIC system when a unit impulse δ ( t )
is applied at the input of the system. In Section 3.4, we proved that the output y ( t )
of an LTIC system can be obtained by convolving the input signal x ( t ) with its
impulse response h ( t ). The resulting convolution integral can either be solved
analytically or by using a graphical approach. The graphical approach was
illustrated through several examples in Section 3.5. The convolution integral