Digital Signal Processing Reference
In-Depth Information
As in Simulation 1, the input signal
g
(
t
) representing the number of antigens
entering the human body is assumed to be a pulse of duration 0.5 s. The numbers
of antigens
a
(
t
), plasma cells
p
(
t
), and antibodies
b
(
t
) are monitored with the
simulated scope available in simulink and are plotted in Fig. 8.13(c). We observe
that the number of antigens
a
(
t
) increases at an exponential rate. Although the
number of plasma cells
p
(
t
), and consequently the number of antibodies
b
(
t
),
also increases, it does so at a slower pace due to the small value of
β
and large
delay
τ
. Since the number of antigens exceeds the number of plasma cells, the
antibodies are destroyed by the antigens. This is shown by negative values for
the number of antibodies
b
(
t
). In reality, the minimum number of antibodies is
zero. The negative values are observed because of the unconstrained analytical
model. We can make Simulation 2 more realistic by constraining the number
of antigens, plasma cells, and antibodies to be greater than zero.
In summary, Simulation 1 presents a scenario where the patient will survive,
whereas Simulation 2 presents a scenario where the patient will die. Although
this model presents a very simplistic view of a highly complex system, it is pos-
sible to improve the model by using more accurate model parameters. Similar
mathematical models can be used in several applications, such as population
prediction, ecosystem analysis, and weather forecasting.
8.5 Summary
We have presented applications of signal processing in analog communica-
tions, mechanical systems, electrical machines, and human immune systems.
In particular, the CTFT and Laplace transform were used to analyze these
systems. Section 8.1 introduced amplitude modulation (AM) and used the
CTFT to analyze the frequency characteristics of AM-based communication
systems. Both synchronous and asynchronous detection schemes for recon-
structing the information-bearing signals were developed. Sections 8.2 and 8.3
used the Laplace transform to analyze the spring damping system and armature-
controlled dc motor. For the two applications, the transfer function and impulse
response of the overall systems were derived. Section 8.4 used the Laplace
transform to model the human immune system. An analytical model for the
human immune system was developed and later analyzed using the simulink
toolbox available in M
ATLAB
.
Problems
8.1
The information signal given by
x
(
t
)
=
3 sin(2
π
f
1
t
)
+
2 cos(2
π
f
2
t
)
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