Chapter 3 Convolution Representation

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Chapter 3Convolution Representation
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DT Unit-Impulse Response

• Consider the DT SISO system
• If the input signal is and the
  system has no energy at , the output
  is called the
  impulse response of the system

System
System
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Example

• Consider the DT system described by
• Its impulse response can be found to be

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Representing Signals in Terms ofShifted and
Scaled Impulses

• Let xn be an arbitrary input signal to a DT LTI
  system
• Suppose that for
• This signal can be represented as

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Exploiting Time-Invariance and Linearity
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The Convolution Sum

• This particular summation is called the
  convolution sum
• Equation is called
  the convolution representation of the system
• Remark a DT LTI system is completely described
  by its impulse response hn

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Block Diagram Representation of DT LTI Systems

• Since the impulse response hn provides the
  complete description of a DT LTI system, we write

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The Convolution Sum for Noncausal Signals

• Suppose that we have two signals xn and vn
  that are not zero for negative times (noncausal
  signals)
• Then, their convolution is expressed by the
  two-sided series

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Example Convolution of Two Rectangular Pulses

• Suppose that both xn and vn are equal to the
  rectangular pulse pn (causal signal) depicted
  below

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The Folded Pulse

• The signal is equal to the pulse pi
  folded about the vertical axis

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Sliding over
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Sliding over - Contd
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Plot of
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Properties of the Convolution Sum

• Associativity
• Commutativity
• Distributivity w.r.t. addition

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Properties of the Convolution Sum - Contd

• Shift property define
• Convolution with the unit impulse
• Convolution with the shifted unit impulse

then
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Example Computing Convolution with Matlab

• Consider the DT LTI system
• impulse response
• input signal

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Example Computing Convolution with Matlab
Contd
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Example Computing Convolution with Matlab
Contd

• Suppose we want to compute yn for
• Matlab code

n040 xsin(0.2n) hsin(0.5n) yconv(x,h) s
tem(n,y(1length(n)))
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Example Computing Convolution with Matlab
Contd
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CT Unit-Impulse Response

• Consider the CT SISO system
• If the input signal is and
  the system has no energy at , the output
  is called the
  impulse response of the system

System
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Exploiting Time-Invariance

• Let xn be an arbitrary input signal with
  for
• Using the sifting property of , we may
  write
• Exploiting time-invariance, it is

System
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Exploiting Time-Invariance
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Exploiting Linearity

• Exploiting linearity, it is
• If the integrand does not
  contain an impulse located at , the
  lower limit of the integral can be taken to be
  0,i.e.,

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The Convolution Integral

• This particular integration is called the
  convolution integral
• Equation is called
  the convolution representation of the system
• Remark a CT LTI system is completely described
  by its impulse response h(t)

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Block Diagram Representation of CT LTI Systems

• Since the impulse response h(t) provides the
  complete description of a CT LTI system, we write

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Example Analytical Computation of the
Convolution Integral

• Suppose that where
  p(t) is the rectangular pulse depicted in figure

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Example Contd

• In order to compute the convolution integral
• we have to consider four cases

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Example Contd

• Case 1

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Example Contd

• Case 2

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Example Contd

• Case 3

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Example Contd

• Case 4

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Example Contd
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Properties of the Convolution Integral

• Associativity
• Commutativity
• Distributivity w.r.t. addition

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Properties of the Convolution Integral - Contd

• Shift property define
• Convolution with the unit impulse
• Convolution with the shifted unit impulse

then
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Properties of the Convolution Integral - Contd

• Derivative property if the signal x(t) is
  differentiable, then it is
• If both x(t) and v(t) are differentiable, then it
  is also

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Properties of the Convolution Integral - Contd

• Integration property define

then
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Representation of a CT LTI System in Terms of the
Unit-Step Response

• Let g(t) be the response of a system with impulse
  response h(t) when with no initial energy
  at time , i.e.,
• Therefore, it is

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Representation of a CT LTI System in Terms of the
Unit-Step Response Contd

• Differentiating both sides
• Recalling that
• it is

and
or