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System
C-T
Y(t)
X[n]
System
D-T
Y[n]
In an electrical system the laws of operation are the voltage-current
relationships for resistors, inductors and other devices. Thus the system
is characterized by its inputs, outputs and its mathematical model.
X(t)
Example:
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Physical systems are interconnection of components, devices or
subsystems.
A system can be an entity that processes a set of inputs(signals) to
yield another set of signals(output). A system is characterized by its
input and outputs (or responses), and the rules of operation that
describe its behavior.
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Continuous-Time and Discrete-Time Systems
Mathematical modeling
Analysis
Design
1.
2.
3.
Study of Systems consists of three major areas:
Continuous-Time and Discrete-Time Systems
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v (t )
i
−
+
− v (t )
v
i
0
(
)
t
i(t ) =
R
i(t )
+
R
v(t )
−
From Ohms law, the current through the resistor
V0 (t ) → Output Signal
Vi (t ) → Input Signal
i
v(t ) = Ri (t )
R
−
+
v (t )
0
To identify a system of linear differential equations that govern the
behavior of a simple R-C circuit with source voltage Vi and capacitor
c\voltage Vo
System Equations
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dv (t )
0
dt
i(t )
+0 −
i(t ) = C
dv (t )
dt
dv (t ) v (t ) v (t )
0 + 0 = i
dt
RC
RC
Equating the right-hand sides of the two equations above
i (t ) = C
Relate i(t) to the rate of change with time of the voltage across the
capacitor
v (t )
System Equations
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so
RC
1
t
d
=D
dt
D = − 1 RC
This is the system response to external input.
−
1 
v (t ) = 0
 0
RC 
v (t ) = Ce
0

D +


dv (t ) v (t )
0 + 0 =0
dt
RC
Total response = zero-input response + zero-state response
System Equations
v (t )
0
−
+
L
∑ v(t ) = 0
C
R
1 t
di(t )
+ i(t ) R
v0 (t ) −
∫ i(t )dt + L
C −∞
dt
i(t )
Write the differential equation governing the behavior of the circuit
given that any voltages around any closed path equals zero.
d 2i(t )
di(t ) i(t ) dv0 (t )
+R
+
=
L
2
dt
C
dt
dt
If the equation is differentiated, a pure differential equation results.
so
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Example
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− 3 − 2 −1
0
y[n] =
2
X [n]
2
3
4
4
2
5
6
1
(X [n] + X [n − 1] + X [n − 2])
3
1
4
6
A simple but useful transformation of a Discrete-time signal is to
compare a “moving average”or “running average”of two or more
consecutive numbers of the sequence, thereby forming a new sequence
of the average values. Averaging is commonly used whenever data
fluctuate and must be smoothed prior to interpretation. Consider a
three-point average method:
Example
2.
1.
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The systems in this class have properties and structures that we can
exploit to gain insight into their behavior.
Many systems of practical importance can be accurately modeled using
systems in this class.
A particular class of systems is referred to as the Linear Time
invariant systems.
Essential component of engineering practice in using the method
developed in this course consist of identifying the range of validity of
the assumptions and ensuring that any analysis or design based on
that model does not violate the assumptions.
2.
1.
Mathematical descriptions of systems from a wide applications have
a great deal in common. This provides motivation for the
development of broadly applicable tools for signals and systems
analysis.
Two important characteristics of the systems are :