ContinuousTime System Properties

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Continuous-Time System Properties
EE 313 Linear Systems and Signals
Spring 2009
Prof. Brian L. Evans Dept. of Electrical and
Computer Engineering The University of Texas at
Austin
Initial conversion of content to PowerPointby
Dr. Wade C. Schwartzkopf
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Linearity

• A system is linear if it is both
• Homogeneous If we scale the input, then the
  output is scaled by the same amount
• Additive If we add two input signals, then the
  output will be the sum of their respective
  outputs
• Response of a linear system to zero input?

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Examples

• Identity system. Linear?
• Ideal delay by T seconds. Linear?
• Scale by a constant (a.k.a. gain block). Linear?

Role of initial conditions?
Two different but equivalent graphical syntaxes
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Examples

• Tapped delay line
• Linear?

Each T represents a delay of T time units

There are N-1 delays

Continuous Time System
S
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Examples

• Transcendental system
• Answer Nonlinear (in fact, fails both tests)
• Squarer
• Answer Nonlinear (in fact, fails both tests)
• Differentiationis linear
• Homogeneity test
• Additivity test

y(t)
x(t)
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Examples

• Integration
• Homogeneity test
• Additivity test
• Answer Linear
• Human hearing
• Responds to intensity on a logarithmic scale
• Answer Nonlinear (in fact, fails both tests)

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Examples

• Human vision
• Similar to hearing in that we respond to the
  intensity of light in visual scenes on a
  logarithmic scale.
• Answer Nonlinear (in fact, fails both tests)
• Modulation by time
• Answer Linear

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Examples

• Amplitude Modulation (AM) but not AM radio
• y(t) A x(t) cos(2 p fc t)
• fc is the carrier frequency (frequency of radio
  station)
• A is a constant
• Answer Linear

y(t)
A
x(t)
cos(2 p fc t)
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Examples

• Frequency Modulation (FM)
• FM radio
• fc is the carrier frequency (frequency of radio
  station)
• A and kf are constants
• Answer Nonlinear (fails both tests)

Linear
Linear
Linear
Nonlinear
Linear
kf
A

x(t)
y(t)
2pfct
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Time-Invariance

• A system is time-invariant if
• When the input is shifted in time, then its
  output is shifted by the same amount
• This must hold for all possible shifts.
• If a shift in input x(t) by t0 causes a shift in
  output y(t) by t0 for all real-valued t0, then
  system is time-invariant

Does yshifted(t) y(t t0) ?
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Examples

• Identity system
• Step 1 compute yshifted(t) x(t t0)
• Step 2 does yshifted(t) y(t t0) ? YES.
• Answer Time-invariant
• Tapped Delay Line
• Answer Time-invariant

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Examples

• Transcendental system
• Answer Time-invariant
• Squarer
• Answer Time-invariant
• Differentiator
• Answer Time-invariant

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Examples

• Integration
• Answer Time-invariant
• Human hearing
• Answer Time-invariant
• Human vision
• Answer Spatially-varying

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Examples

• Amplitudemodulation(not AM radio)
• FMradio

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Memoryless

• A mathematical description of a system may be
  memoryless, but an implementation of a system may
  use memory.

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Example 1

• Differentiation
• A derivative computes an instantaneous rate of
  change. Ideally, it does not seem to depend on
  what x(t) does at other instances of t than the
  instant being evaluated.
• However, recalldefinition of aderivative
• What happens at a pointof discontinuity? We
  couldaverage left and right limits.
• As a system, differentiation is not memoryless.
  Any implementation of a differentiator would need
  memory.

x(t)
t
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Example 2

• Analog-to-digital conversion
• Lecture 1 mentioned that A/D conversion would
  perform the following operations
• Lowpass filter requires memory
• Quantizer is ideally memoryless, but an
  implementation may not be

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Causality

• System is causal if output depends on current and
  previous inputs and previous outputs
• When a system works in a time domain, causality
  is generally required
• For images, causality is not an issue when the
  entire image is available because we could
  process pixels from upper left-hand corner to
  lower right-hand corner, or vice-versa

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Summary

• If several causes are acting on a linear system,
  then the total effect is the sum of the responses
  from each cause
• In time-invariant systems, system parameters do
  not change with time
• For memoryless systems, the system response at
  any instant t depends only on the present value
  of the input (value at t)

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Summary

• If a system response at t depends on future input
  values (beyond t), then the system is noncausal
• A signal defined for a continuum of values of the
  independent variable (such as time) is a
  continuous-time signal
• A signal whose amplitude can take on any value in
  a continuous range is an analog signal