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Convolution and its numerical approximation

The output y ( t ) size 12{y \( t \) } {} of a continuous-time linear time-invariant (LTI) system is related to its input x ( t ) size 12{x \( t \) } {} and the system impulse response h ( t ) size 12{h \( t \) } {} through the convolution integral expressed as (for details on the theory of convolution and LTI systems, refer to signals and systems textbooks, for example, references [link] - [link] ):

y ( t ) = h ( t τ ) x ( τ ) size 12{y \( t \) = Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {h \( t - τ \) x \( τ \) dτ} } {}

For a computer program to perform the above continuous-time convolution integral, a numerical approximation of the integral is needed noting that computer programs operate in a discrete – not continuous – fashion. One way to approximate the continuous functions in the Equation (1) integral is to use piecewise constant functions. Define δ Δ ( t ) size 12{δ rSub { size 8{Δ} } \( t \) } {} to be a rectangular pulse of width Δ size 12{Δ} {} and height 1, centered at t = 0 size 12{t=0} {} :

δ Δ ( t ) = { 1 Δ / 2 t Δ / 2 0 otherwise size 12{δ rSub { size 8{Δ} } \( t \) = left lbrace matrix { 1 {} # - Δ/2<= t<= Δ/2 {} ## 0 {} # ital "otherwise"{}} right none } {}

Approximate a continuous function x ( t ) size 12{x \( t \) } {} with a piecewise constant function x Δ ( t ) size 12{x rSub { size 8{Δ} } \( t \) } {} as a sequence of pulses spaced every Δ size 12{Δ} {} seconds in time with heights x ( ) size 12{x \( kΔ \) } {} :

x Δ ( t ) = k = x ( ) δ Δ ( t ) size 12{x rSub { size 8{Δ} } \( t \) = Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {x \( kΔ \) δ rSub { size 8{Δ} } \( t - kΔ \) } } {}

It can be shown in the limit as Δ 0, x Δ ( t ) x ( t ) size 12{Δ rightarrow 0,x rSub { size 8{Δ} } \( t \) rightarrow x \( t \) } {} . As an example, [link] shows the approximation of a decaying exponential x ( t ) = exp ( t 2 ) size 12{x \( t \) ="exp" \( - { {t} over {2} } \) } {} starting from 0 using Δ = 1 size 12{Δ=1} {} . Similarly, h ( t ) size 12{h \( t \) } {} can be approximated by

h Δ ( t ) = k = h ( ) δ Δ ( t ) size 12{h rSub { size 8{Δ} } \( t \) = Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {h \( kΔ \) δ rSub { size 8{Δ} } \( t - kΔ \) } } {}

One can thus approximate the convolution integral by convolving the two piecewise constant signals as follows:

y Δ ( t ) = h Δ ( t τ ) x Δ ( τ ) size 12{y rSub { size 8{Δ} } \( t \) = Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {h rSub { size 8{Δ} } \( t - τ \) x rSub { size 8{Δ} } \( τ \) dτ} } {}

Approximation of a Decaying Exponential with Rectangular Strips of Width 1

Notice that y Δ ( t ) size 12{y rSub { size 8{Δ} } \( t \) } {} is not necessarily a piecewise constant. For computer representation purposes, discrete output values are needed, which can be obtained by further approximating the convolution integral as indicated below:

y Δ ( ) = Δ k = x ( ) h ( ( n k ) Δ ) size 12{y rSub { size 8{Δ} } \( nΔ \) =Δ Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {x \( kΔ \) h \( \( n - k \) Δ \) } } {}

If one represents the signals h Δ ( t ) size 12{h rSub { size 8{Δ} } \( t \) } {} and x Δ ( t ) size 12{x rSub { size 8{Δ} } \( t \) } {} in a .m file by vectors containing the values of the signals at t = size 12{t=nΔ} {} , then Equation (5) can be used to compute an approximation to the convolution of x ( t ) size 12{x \( t \) } {} and h ( t ) size 12{h \( t \) } {} . Compute the discrete convolution sum k = x ( ) h ( ( n k ) Δ ) size 12{ Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {x \( kΔ \) h \( \( n - k \) Δ \) } } {} with the built-in LabVIEW MathScript command conv . Then, multiply this sum by Δ size 12{Δ} {} to get an estimate of y ( t ) size 12{y \( t \) } {} at t = size 12{t=nΔ} {} Note that as Δ size 12{Δ} {} is made smaller, one gets a closer approximation to y ( t ) size 12{y \( t \) } {} .

Convolution properties

Convolution satisfies the following three properties (see [link] ):

  • Commutative property
x ( t ) h ( t ) = h ( t ) x ( t ) size 12{x \( t \) * h \( t \) =h \( t \) * x \( t \) } {}
  • Associative property
x ( t ) h 1 ( t ) h 2 ( t ) = x ( t ) { h 1 ( t ) h 2 ( t ) } size 12{x \( t \) * h rSub { size 8{1} } \( t \) * h rSub { size 8{2} } \( t \) =x \( t \) * lbrace h rSub { size 8{1} } \( t \) * h rSub { size 8{2} } \( t \) rbrace } {}
  • Distributive property
x ( t ) { h 1 ( t ) + h 2 ( t ) } = x ( t ) h 1 ( t ) + x ( t ) h 2 ( t ) size 12{x \( t \) * lbrace h rSub { size 8{1} } \( t \) +h rSub { size 8{2} } \( t \) rbrace =x \( t \) * h rSub { size 8{1} } \( t \) +x \( t \) * h rSub { size 8{2} } \( t \) } {}
Convolution Properties

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Source:  OpenStax, An interactive approach to signals and systems laboratory. OpenStax CNX. Sep 06, 2012 Download for free at http://cnx.org/content/col10667/1.14
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