Convolution and Linear Time-Invariant Systems

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 - ): y(t)=∫−∞∞h(t−τ)x(τ)dτ 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 δΔ(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} {}: δΔ(t)={1−Δ/2≤t≤Δ/20otherwise 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 xΔ(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(kΔ) size 12{x \( kΔ \) } {}: xΔ(t)=∑k=−∞∞x(kΔ)δΔ(t−kΔ) 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,xΔ(t)→x(t) size 12{Δ rightarrow 0,x rSub { size 8{Δ} } \( t \) rightarrow x \( t \) } {}. As an example, shows the approximation of a decaying exponential x(t)=exp(−t2) 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 hΔ(t)=∑k=−∞∞h(kΔ)δΔ(t−kΔ) 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: yΔ(t)=∫−∞∞hΔ(t−τ)xΔ(τ)dτ 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} } {}

Notice that 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(nΔ)=Δ∑k=−∞∞x(kΔ)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 hΔ(t) size 12{h rSub { size 8{Δ} } \( t \) } {} and xΔ(t) size 12{x rSub { size 8{Δ} } \( t \) } {} in a .m file by vectors containing the values of the signals at t=nΔ 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(kΔ)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=nΔ 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 \( t \) } {}.