Linear Time-Invariant (

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Linear Time-Invariant(LTI) Systems

• Montek Singh
• Thurs., Feb. 7, 2002330-445 pm, SN115

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What we will learn

• How to represent a circuit as an input-output
  system (black box)
• What are LTI systems?
• How is their behavior described?

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Why treat circuits as I/O systems?

• A system representation
• is not bound to a particular input
• allows us to distill the essence of an
  arbitrarily complex circuit into a concise
  description
• e.g., Thevenin and Norton equivalents

• can incorporate other (non-electrical)
  technologies
• e.g., acoustic, optical, magnetic etc.

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What are LTI systems?

• LTI systems are linear and time-invariant
• Linearity
• output for a sum of inputs sum of individual
  outputs
• i.e.,

• Time-Invariance
• inherent system properties do not change with
  time
• delaying the input by time ? simply delays the
  output by ?
• i.e.,

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Examples

• LTI systems
• Most physical systems when operated at small
  amplitudes
• an LCR electrical network
• a mechanical spring, a glass prism, a loudspeaker
  
• Non-linear systems
• Most physical systems when stretched to the
  limit
• a blaring loudspeaker
• Some systems that are intentionally operated in
  that mode
• diodes, transistors, logic gates, digital systems
  
• Time-variant systems
• Systems whose properties change with time
• a resistor getting hotter
• the human eye

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An LTI systems behavior

• Systems behavior mapping from input to output
• How to represent?
• Describe the underlying physical phenomena
• goes back to circuit theory
• Enumerate all (interesting) input-output pairs
• unwieldy description
• Describe output for a select set of inputs
• choose some special input
• compute output behavior for that input
• infer behavior for arbitrary inputs

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Choosing that special input

• Unit impulse function ?(t)

• Unit impulse a pulse of
• infinitesimal duration
• infinite amplitude
• unit area
• Also known as Dirac delta function

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Unit impulse properties

• Examples

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Unit impulse used as a sampler
Sampling Theorem

• Multiplying a signal by ?(t-a) and integrating
  has the effect of sampling it at t a.

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Reconstituting a signal from samples (1)
Sampling Theorem

• Swap the roles of t and a

x(t) can be regarded as an infinite sum of
infinitesimal samples, i.e., sample x(a) summed
over all a.
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Reconstituting a signal from samples (2)
da
x(t)
x(t)
x(a)?(t-a)da
1/da
a
a
a
t
t
x(t)
a
t
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Unit impulse systems response

• Output of a system when input ?(t) is called
  theunit impulse response
• Denoted by h(t)
• Example human eye

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Generalization Arbitrary input

• Given unit impulse response h(t), i.e.,
• Find system response y(t) to an arbitrary input
  x(t)
• Method
• express input x(t) as an infinite sum of weighted
  impulses
• compute response to each individual impulse
• weight and add up all the individual responses

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Convolution

• Definition y(t) is the convolution of x(t)
  and h(t) if

• Notation
• Properties
• commutativity
• associativity
• distributivity
• scalability
• derivatives

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Convolution example
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Check it out!
http//www.jhu.edu/signals/convolve/index.html
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Homework Due 2/19

• The output of a particular system S is the time
  derivative of its input.
• Prove that system S is linear time-invariant
  (LTI).
• What is the unit impulse response of this system?
• Prove Property 5. That is, prove that, for an
  arbitrary LTI system, for a given input waveform
  x(t), the time derivative of its output is
  identical to the output of that system when
  subjected to the time derivative of its input.
  In other words, differentiation on the input and
  output sides are equivalent.