EGR 277

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Lecture 9 EGR 261 Signals and Systems
Read Ch. 1, Sect. 1-4, 6-8 in Linear Signals
Systems, 2nd Ed. by Lathi
Systems Recall that a system is any process that
results in the transformation of signals. A
system may be made up of physical components,
such as in electric circuits or mechanical
systems, or a system may be a software algorithm
that modifies a signal. A system is typically
described in terms of inputs and outputs. A
system might have a single input and a single
output or may have multiple inputs/outputs, as
illustrated below.
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Lecture 9 EGR 261 Signals and Systems
System Analysis The analysis of systems is an
important part of this course. Analysis often
involves determining systems for given inputs.
Analysis might involve different methods as well
as analysis in different domains (such as in the
time domain or the frequency domain). Example
Analyze a series RC circuit in the s-domain to
determine the output VC(s) as a function of a
general input, Vin(s).
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Lecture 9 EGR 261 Signals and Systems

• Classification of Systems
• Systems may be classified broadly in the
  following categories
• Linear and non-linear systems
• Time-invariant (or constant parameter) and
  time-varying systems
• Static (instantaneous or memoryless) or dynamic
  (with memory) systems
• Causal and noncausal systems
• Continuous-time and discrete-time systems
• Analog and digital systems
• Invertible and noninvertible systems
• Stable and unstable systems

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Lecture 9 EGR 261 Signals and Systems

• Classification of Systems
• 1. Linear and non-linear systems
• A system is considered to be linear if it
  satisfies two properties
• Homogeneity (or scaling)
• Additivity

Homogeneity (or scaling) Consider a
system with input x(t) and output y(t), where
y(t) f(x(t)). A system satisfies the
homogeneity property if the output y(t) is
proportional to the input x(t) as indicated by
the following relationship. Or, if the input is
increased k-fold, then the output is also
increased k-fold, where k is a real or imaginary
number. Example For the circuit below, if the
input voltage doubles, the output voltage
doubles. Or if the input is multiplied by k, the
output is multiplied by k. So the circuit
satisfies the homogeneity property.
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Lecture 9 EGR 261 Signals and Systems
Additivity A system satisfies the
additivity property if the output y(t) can be
determined by the algebraic sum of the outputs
due to each input acting separately (with all
other inputs set to zero). This relationship can
be expressed by Example The circuit below
satisfies the additivity property since if y1
is the output when the input is x1, and y2 is the
output when the input is x2, then y1 y2 will be
the output when the input is x1 x2.
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Lecture 9 EGR 261 Signals and Systems
Superposition Sometimes the
homogeneity property and the additivity property
are combined to form the superposition property.
Thus a system is linear if it satisfies the
superposition property shown below
Homogeneity
Superposition
Additivity
Example A system is described by the
relationship y mx b. Is the system
linear? (Hint Test for homogeneity and
additivity)
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Lecture 9 EGR 261 Signals and Systems
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Lecture 9 EGR 261 Signals and Systems
Example (Exercise E1.11) A system is described
by the relationship y(t) Rex(t). (Hint
Test for homogeneity (try k j) and additivity.)
Example (Exercise E1.12) Show that a system
described by the following equation is linear
(Hint Test for superposition.)
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Lecture 9 EGR 261 Signals and Systems
Systems described by linear differential
equations The text develops the fact that any
systems that can be described by a linear
differential equation of the form shown below is
a linear system

• Response to a Linear System
• If a system is linear, then the additivity
  property indicates that its output is the sum of
• the components due to its inputs (with the other
  inputs set to zero). Suppose then that
• a systems output is due to
• the input, x(t), to the system
• the initial conditions (state) of the system.
• Therefore, the output of the system can be
  expressed as

Total response zero-input response zero-state
response
where zero-input response output due to
initial conditions only (with input x(t)
0) zero-state response output due to input only
(with zero initial condition)
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Lecture 9 EGR 261 Signals and Systems

• Example
• Recall from EGR 260 that the response to a
  1st-order RC circuit with DC sources has the
  form
• y(t) B Ae-t/RC
• The output could be evaluated using the value of
  y(t) at two points (i.e., y(0) and y(?))
• Solve for B and A in terms of y(0) and y(? )
• Express y(t) in terms of the zero-state
  response and the zero-input response

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Lecture 9 EGR 261 Signals and Systems
Example The circuit shown below is Figure 1.26
from Linear Systems and Signals, 2nd Edition, by
Lathi. Express the output y(t) in terms of the
zero-state response and the zero-input response.
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Lecture 9 EGR 261 Signals and Systems

• Classification of Systems
• 2. Time-invariant (or constant parameter) and
  time-varying systems
• Systems whose parameters do not change with time
  are time-invariant systems.
• In a time-invariant system, delaying the input
  results in a delay in the output, or
• if y(t) f(x(t)) then y(t T)
  f(x(t-T))
• The relationship above indicates that the delayed
  output will look exactly like the
• original output, other than being delayed by T
  seconds.
• Notes
• Networks that are composed of RLC elements and
  other commonly used active elements such as
  transistors are time-invariant systems.
• Networks that can be described by a linear
  differential equation of the form below are
  time-invariant systems.

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Lecture 9 EGR 261 Signals and Systems
Example If a system output is represented by
y(t) 10e-tu(t), show that the system is
time-invariant. Sketch the original output and
the delayed output. Does the delayed output have
the same form as the original output?
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Lecture 9 EGR 261 Signals and Systems
Classification of Systems 3. Static
(instantaneous or memoryless) or dynamic (with
memory) systems
Static system a static system (or instantaneous
or memoryless) is a system where the output at
any instant t depends only on its input at that
instant. Example Circuits composed only of
resistors are static systems. Find
y(t). Dynamic system a dynamic system is
one where the output at any instant depends on
both present and past inputs. Example Circuits
with inductors and capacitors are dynamic
systems. Find y(t).
Note We will generally be concerned with
dynamic systems. Static systems can be viewed as
a special case of dynamic systems.
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Lecture 9 EGR 261 Signals and Systems
Classification of Systems 4. Causal and
Non-Causal Systems
A causal system is one for which the output at to
depends only on the input for t lt to. A
non-causal system is one for which the output
could also depend on future inputs. Any
practical system that operates in real time must
be causal. A causal system is also called a
realizable system. A non-causal system is also
called a non-realizable system.
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Lecture 9 EGR 261 Signals and Systems
Classification of Systems 5. Continuous-Time and
Discrete-Time Systems
Continuous-Time Systems As mentioned earlier,
signals defined over a continuous range of time
are continuous-time signals (denoted x(t), y(t),
etc). Systems whose inputs and outputs are
continuous-time signals are continuous-time
systems. Discrete-Time Systems Signals only
defined at discrete instants of time (t0, t1, t2,
) are discrete-time signals (x0, x1, x2,
etc.) Systems whose inputs and outputs are
discrete-time signals are discrete-time systems.
A digital computer is an example of a discrete
time system. Continuous-time signals are
sometimes sampled at regular intervals in order
to produce discrete-time signals which can then
be processed using a computer, as illustrated
below.
Reference Linear Signals and Systems, 2nd
Edition, by Lathi.
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Lecture 9 EGR 261 Signals and Systems
Classification of Systems 6. Analog and Digital
Systems
Analog and digital signals were discussed
earlier. Systems whose inputs and outputs are
analog signals are analog systems. Systems whose
inputs and outputs are digital signals are
digital systems. Note that a digital computer is
a digital system as well as a discrete-time
system.
7. Invertible and Noninvertible Systems
A system is invertible if we can obtain the x(t)
back from the corresponding output y(t) by some
operation. This implies that there must be a 11
mapping between inputs and outputs (i.e., every
input must have a unique output) as illustrated
below.
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Lecture 9 EGR 261 Signals and Systems
A system S is invertible if there exists some
system Si such that x(t) can be regenerated from
y(t).
Example A system is represented by y(t) 2x(t)
(an ideal amplifier with a gain of 2). Is the
system invertible?
Example A system is represented by the
relationship y(t) x(t)2. Is the system
invertible?
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Lecture 9 EGR 261 Signals and Systems
Application Invertible systems Inverse systems
are important in signal processing as we may
distort a signal during processing and then need
to restore the signal (undo the distortion) after
processing. Example Dolby recording system
(discuss).
Classification of Systems 8. Stable and Unstable
Systems
A system is stable (externally) if for every
bounded input there is a bounded output. This
type of stability is also known as Bounded-Input
Bounded-Output (BIBO) stability. Example Is a
system represented by y(t) x(t)2 BIBO
stable? Suppose x(t) is limited to the range
-10 lt x(t) lt 10?