S. Boyd EE102Lecture 8Transfer functions and convolution• convolution & transfer functions• properties• examples• interpretation of convolution• representation of linear time-invariant systems8–1Convolution systemsconvolution system with input u (u(t) = 0, t < 0) and output y:y(t) =Zt0h(τ)u(t − τ) dτ =Zt0h(t − τ)u(τ) dτabbreviated: y = h ∗ uin the frequency domain: Y (s) = H(s)U(s)• H is called the transfer function (TF) of the system• h is called the impulse response of the systemblock diagram notation(s):PSfrag replacementsuyuy∗ hHTransfer functions and convolution 8–2Properties1. convolution systems are linear: for all signals u1, u2and all α, β ∈ R,h ∗ (αu1+ βu2) = α(h ∗ u1) + β(h ∗ u2)2. convolution systems are causal: the output y(t) at time t depends onlyon past inputs u(τ), 0 ≤ τ ≤ t3. convolution systems are time-invariant: if we shift the input signal uover T > 0, i.e., apply the inputeu(t) =½0 t < Tu(t − T ) t ≥ 0to the system, the output isey(t) =½0 t < Ty(t − T ) t ≥ 0in other words: convolution systems commute with delayTransfer functions and convolution 8–34. composition of convolution systems corresponds to• multiplication of transfer functions• convolution of impulse responsesPSfrag replacementsuucompositionyyABBAramifications:• can manipulate block diagrams with transfer functions as if they weresimple gains• convolution systems commute with each otherTransfer functions and convolution 8–4Example: feedback connectionPSfrag replacementsuGyin time domain, we have complicated integral equationy(t) =Zt0g(t − τ)(u(τ) − y(τ)) dτwhich is not easy to understand or solve . . .in frequency domain, we have Y = G(U − Y ); solve for Y to getY (s) = H(s)U(s), H(s) =G(s)1 + G(s)(as if G were a simple scaling system!)Transfer functions and convolution 8–5General examplesfirst order LCCODE: y0+ y = u, y(0) = 0take Laplace transform to getY (s) =1s + 1U(s)transfer function is 1/(s + 1); impulse response is e−tintegrator: y(t) =Zt0u(τ) dτtransfer function is 1/s; impulse response is 1delay: with T ≥ 0,y(t) =½0 t < Tu(t − T ) t ≥ Timpulse response is δ(t − T ); transfer function is e−sTTransfer functions and convolution 8–6Vehicle suspension system(simple model of) vehicle suspension system:PSfrag replacementsuymk b• input u is road height (along vehicle path); output y is vehicle height• vehicle dynamics: my00+ by0+ ky = bu0+ kuassuming y(0) = 0, y0(0) = 0, (and u(0−) = 0),(ms2+ bs + k)Y = (bs + k)UTF from road height to vehicle height is H(s) =bs + kms2+ bs + kTransfer functions and convolution 8–7DC motorPSfrag replacementsiθJθ00+ bθ0= ki(J is rotational inertia of shaft & load; b is mechanical resistance of shaft& load; k is motor constant)assuming θ(0) = θ0(0) = 0,Js2Θ(s) + bsΘ(s) = kI(s), Θ(s) =kJs2+ bsI(s)i.e., transfer function H from i to θ isH(s) =kJs2+ bsTransfer functions and convolution 8–8Circuit examplesconsider a circuit with linear elements, zero initial conditions for inductorsand capacitors,• one independent source with value u• y is a voltage or current somewhere in the circuitthen we have Y (s) = H(s)U(s)example: RC circuitPSfrag replacementsuyRCRCy0(t) + y(t) = u(t), Y (s) =11 + sRCU(s)impulse response is L−1µ11 + sRC¶=1RCe−t/RCTransfer functions and convolution 8–9to find H: write circuit equations in frequency domain:• resistor: v(t) = Ri(t) becomes V (s) = RI(s)• capacitor: i(t) = Cv0(t) becomes I(s) = sCV (s)• inductor: v(t) = Li0(t) becomes V (s) = sLI(s)in frequency domain, circuit equations become algebraic equationsTransfer functions and convolution 8–10example:PSfrag replacementsvinvout1Ω1Ω1Ω1F1Fv−v+let’s find TF from vinto vout(assuming zero initial voltages for capacitors)• by voltage divider rule, V+= Vin11 + 1/s= Vinss + 1• current in lefthand resistor is (using V−= V+):I =Vin− V−1Ω=µ1 −ss + 1¶Vin=1s + 1VinTransfer functions and convolution 8–11• I flows through 1Fk1Ω, yielding voltageVin1s + 1(1)(1/s)1 + 1/s= Vin1(s + 1)2• finally we have Vout= V−− Vin1(s + 1)2= Vins2+ s − 1(s + 1)2so transfer function isH(s) =s2+ s − 1(s + 1)2= 1 −1s + 1−1(s + 1)2impulse response ish(t) = L−1(H) = δ(t) − e−t− te−twe havevout(t) = vin(t) −Zt0(1 + τ)e−τvin(t − τ) dτTransfer functions and convolution 8–12Interpretation of convolutiony(t) =Zt0h(τ)u(t − τ) dτ• y(t) is current output; u(t − τ) is what the input was τ seconds ago• h(τ) shows how much current output depends on what input was τseconds agofor example,• h(21) big means current output depends quite a bit on what input was,21sec ago• if h(τ) is small for τ > 3, then y(t) depends mostly on what the inputhas been over the last 3 seconds• h(τ) → 0 as τ → ∞ means y(t) depends less and less on remote pastinputTransfer functions and convolution 8–13Graphical interpretationy(t) =Zt0h(t − τ)u(τ) dτto find y(t):• flip impulse response h(τ) backwards in time (yields h(−τ))• drag to the right over t (yields h(t − τ))• multiply pointwise by u (yields u(τ)h(t − τ))• integrate over τ to get y(t)Transfer functions and convolution 8–14PSfrag replacementsh(τ )u(τ )u(τ )h(−τ )h(t1− τ)h(t2− τ)h(t3− τ)ττττττy = u ∗ ht1t2t3Transfer functions and convolution 8–15Examplecommunication channel, e.g., twisted pair cablePSfrag replacementsuy∗himpulse response:0 2 4 6 8 1000.511.5PSfrag replacementstha delay ≈ 1, plus smoothingTransfer functions and convolution 8–16simple signalling at 0.5 bit/sec; Boolean signal 0, 1, 0, 1, 1, . . .0 2 4 6 8 1000.510 2 4 6 8 1000.51PSfrag replacementsttuyoutput is delayed, smoothed version of input1’s & 0’s easily distinguished in yTransfer functions and convolution 8–17simple signalling at 4 bit/sec; same Boolean signal0 2 4 6 8 1000.510 2 4 6 8 1000.51PSfrag replacementsttuysmoothing makes 1’s & 0’s very hard to distinguish in yTransfer functions and convolution 8–18Linear time-invariant systemsconsider a system A which is• linear• time-invariant (commutes with delays)• causal (y(t) depends only on u(τ) for 0 ≤ τ ≤ t)called a linear time-invariant (LTI) causal systemwe have seen that any convolution system is LTI and causal; the converseis also true: any LTI causal system can be represented by a convolutionsystemconvolution/transfer function representation gives universal