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# Convolution and Linear Time-Invariant Systems

## Convolution and Its Numerical Approximation

The output y(t)y(t) size 12{y $$t$$ } {} of a continuous-time linear time-invariant (LTI) system is related to its input x(t)x(t) size 12{x $$t$$ } {} and the system impulse response h(t)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 (Reference) - (Reference) ):

y(t)=h(tτ)x(τ)y(t)=h(tτ)x(τ) size 12{y $$t$$ = Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {h $$t - τ$$ x $$τ$$ dτ} } {}
(1)

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 δΔ(t)δΔ(t) size 12{δ rSub { size 8{Δ} } $$t$$ } {} to be a rectangular pulse of width ΔΔ size 12{Δ} {} and height 1, centered at t=0t=0 size 12{t=0} {}:

δΔ(t)={1Δ/2tΔ/20otherwiseδΔ(t)={1Δ/2tΔ/20otherwise size 12{δ rSub { size 8{Δ} } $$t$$ = left lbrace matrix { 1 {} # - Δ/2 <= t <= Δ/2 {} ## 0 {} # ital "otherwise"{} } right none } {}
(2)

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

xΔ(t)=k=x()δΔ(t)xΔ(t)=k=x()δΔ(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} } $$t - kΔ$$ } } {}
(3)

It can be shown in the limit as Δ0,xΔ(t)x(t)Δ0,xΔ(t)x(t) size 12{Δ rightarrow 0,x rSub { size 8{Δ} } $$t$$ rightarrow x $$t$$ } {}. As an example, Figure 1 shows the approximation of a decaying exponential x(t)=exp(t2)x(t)=exp(t2) size 12{x $$t$$ ="exp" $$- { {t} over {2} }$$ } {} starting from 0 using Δ=1Δ=1 size 12{Δ=1} {}. Similarly, h(t)h(t) size 12{h $$t$$ } {} can be approximated by

hΔ(t)=k=h()δΔ(t)hΔ(t)=k=h()δΔ(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} } $$t - kΔ$$ } } {}
(4)

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

yΔ(t)=hΔ(tτ)xΔ(τ)yΔ(t)=hΔ(tτ)xΔ(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} } {}
(5)

Notice that yΔ(t)yΔ(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:

yΔ()=Δk=x()h((nk)Δ)yΔ()=Δk=x()h((nk)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 \) } } {}
(6)

If one represents the signals hΔ(t)hΔ(t) size 12{h rSub { size 8{Δ} } $$t$$ } {} and xΔ(t)xΔ(t) size 12{x rSub { size 8{Δ} } $$t$$ } {} in a .m file by vectors containing the values of the signals at t=t= size 12{t=nΔ} {}, then Equation (5) can be used to compute an approximation to the convolution of x(t)x(t) size 12{x $$t$$ } {} and h(t)h(t) size 12{h $$t$$ } {}. Compute the discrete convolution sum k=x()h((nk)Δ)k=x()h((nk)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 ΔΔ size 12{Δ} {} to get an estimate of y(t)y(t) size 12{y $$t$$ } {} at t=t= size 12{t=nΔ} {} Note that as ΔΔ size 12{Δ} {} is made smaller, one gets a closer approximation to y(t)y(t) size 12{y $$t$$ } {}.

## Convolution Properties

Convolution satisfies the following three properties (see Figure 2):

• Commutative property
x(t)h(t)=h(t)x(t)x(t)h(t)=h(t)x(t) size 12{x $$t$$ * h $$t$$ =h $$t$$ * x $$t$$ } {}
(7)
• Associative property
x(t)h1(t)h2(t)=x(t){h1(t)h2(t)}x(t)h1(t)h2(t)=x(t){h1(t)h2(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 } {}
(8)
• Distributive property
x(t){h1(t)+h2(t)}=x(t)h1(t)+x(t)h2(t)x(t){h1(t)+h2(t)}=x(t)h1(t)+x(t)h2(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} } $$t$$ } {}
(9)

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