Lecture 4
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4. SYSTEMS AND CLASSIFICATION OF SYSTEMS
A. System Representation.
B. Continuous Time and Discrete-Time Systems.
C. Systems with Memory and without Memory.
D. Causal and Noncausal Systems.
E. Linear Systems and Nonlinear Systems
F. Time-Invariant and Time-Varying Systems
G. Linear Time-Invariant Systems
H. Stable Systems
I. Feedback Systems
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A. System Representation
A system is a mathematical model of a physical process that
relates the input signal to the output signal.
Let x and y be the input and output signals, respectively, of a
system. Then the system is viewed as a transformation (or mapping)
of x into y. This transformation is represented by the mathematical
notation
y=Tx
where T is the operator representing some well-defined rule by which x
is transformed into y.
This Relationship is depicted as shown in Fig. (a). Multiple input and/or
output signals are possible as shown in Fig. (b).
a b
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B. Continuous Time and Discrete-Time Systems
If the input and output signals x and y are continuous-time signals,
then the system is called a continuous-time system [Fig. (a)].
If the input and output signals are discrete-time signals or sequences,
then the system is called a discrete-time system [Fig. (b)].
(a) Continuous-time system (b) Discrete -time system
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C. Systems with Memory and without Memory
A system is said to be memoryless if the output at any time depends on
only the input at that same time. Otherwise, the system is said to have
memory.
An example of a memoryless system is a resistor R:
Voltage Current
An example of a system with memory is a capacitor C:
A second example of a system with memory is a discrete-time system
whose input and output sequences are related by
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D. Causal and Noncausal Systems
A system is called causal if its output y (t) at an arbitrary time
t = t0, depends on only the input x ( t ) for t ≤t0.
That is, the output of a causal system at the present time
depends on only the present and /or past values of the input,
not on its future values.
Thus, in a causal system, it is not possible to obtain an output
before an input is applied to the system.
A system is called noncausal if it is not causal.
Note that : all memoryless systems are causal, but not vice
versa.
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Examples:
1) 𝒚 𝒕 = 𝒙(𝒕) it is a Causal system
2) 𝒚 𝒕 = 𝒙 𝒕 + 𝒙 𝒕 − 𝟏 it is a Causal system
3) 𝒚 𝒕 = 𝒙 𝒕 + 𝟐 it is a Non-Causal system
4) 𝒚 𝒕 = 𝒙 𝒕 + 𝒙 𝒕 − 𝟏 + 𝒙 𝒕 + 𝟏 it is a Non-Causal system
5) 𝒚 𝒕 = 𝒙 𝟑𝒕 it is a Non-Causal system
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E. Linear Systems and Nonlinear Systems
Any system is called a linear system if it satisfy the two
conditions:
1) Additivity:
Given that T x1 = y1, and T x2 = y2 , Then:
𝐓 𝒙𝟏 + 𝒙𝟐 = 𝒚𝟏 + 𝒚𝟐
2) Homogeneity (or Scaling):
𝐓 𝜶𝒙 = 𝜶𝒚
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Note:
The previous two conditions can be combined as following;
𝐓 𝜶𝟏 𝒙𝟏 + 𝜶𝟐 𝒙𝟐 = 𝜶𝟏 𝒚𝟏 + 𝜶𝟐 𝒚𝟐
where 𝜶𝟏 and 𝜶𝟐 are arbitrary scalars.
This Equation is known as the superposition property.
Examples of linear systems are the resistor and the capacitor.
Examples of nonlinear systems are
y=𝒙𝟐 y=cosx
Examples of linear systems are the resistor and the capacitor.
For any linear systems is that a zero input yields a zero output.
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F. Time-Invariant and Time-Varying Systems
A system is called time-invariant if a time shift (delay or advance)
in the input signal causes the same time shift in the output signal.
Thus, for a continuous-time system, the system is time-invariant if:
𝑻𝒙 𝒕 − 𝝉 =𝒚 𝒕 − 𝝉 F.1
Thus, for a discrete-time system, the system is time-invariant if:
𝑻 𝒙 𝒏−𝒌 =𝒚 𝒏−𝒌 F.2
for any integer k.
A system which does not satisfy Eq. F.1 (continuous-time system) or
Eq. F.2 (discrete-time system) is called a time-varying system.
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