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# ELECTRIC CIRCUIT ANALYS II ECE 222

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This 68 page Class Notes was uploaded by Miss Chadrick Doyle on Tuesday September 1, 2015. The Class Notes belongs to ECE 222 at Portland State University taught by James McNames in Fall. Since its upload, it has received 56 views. For similar materials see /class/168231/ece-222-portland-state-university in ELECTRICAL AND COMPUTER ENGINEERING at Portland State University.

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Date Created: 09/01/15

Transfer Functions Transfer functions defined Linearity and time invariance defined Examples Sinusoidal steadystate analysis Transfer function synthesis J McNames Portland State University ECE 222 Transfer Functions Ver 169 1 Prerequisite and New Knowledge Prerequisite knowledge 0 Ability to find Laplace transforms of signals 0 Ability to find inverse Laplace transforms 0 Ability to perform Laplace transform circuit analysis New knowledge 0 Ability to characterize a circuit regardless of what voltage or current is driving the circuit 0 Ability to analyze all LTI systems that are defined by constantcoefficient ODEs 0 Understanding of tradeoffs between Laplace transform circuit analysis and transfer function analysis 0 Understanding of how steadystate sinusoidal analysis relates to transfer functions and Laplace transform circuit analysis J McNames Portland State University ECE 222 Transfer Functions Ver 169 2 Transfer Functions Assume zero initial conditions i dkya bdkxa ak dtk k dtk k0 k0 N M Zaksk Y8 Zbksts 160 k0 N M Y8 Zaksk X8 Z bksk k0 k0 m ijobkskgtxs Elioaksk Ys H8Xs J McNames Portland State University ECE 222 Transfer Functions Ver 169 3 Initial Conditions Assume zero initial conditions N 01km M 01km Zak dtk gm dtk k0 N M Zakski s Z bksts 760 k0 All voltages and currents are due to independent sources superposition 0 Energy stored in capacitors and inductors also act like independent sources We will now focus a specific class of circuits Only one independent source input No energy stored in capacitors or inductors Greatly simplifies analysis J McNames Portland State University ECE 222 Transfer Functions Ver 169 4 Time Invariance Time Invariant A system is time invariant if and only if ZEt gt yt implies ZEt to gt yt to 0 Transfer function analysis can be applied to any type of system that is linear and time invariant LTI o A system is time invariant if a time shift in the input signal results in an identical time shift in the output signal 0 Circuits that have nonzero energy stored on capacitors or in inductors at t 0 are generally not timeinvariant o Circuits that have no energy stored are timeinvariant J McNames Portland State University ECE 222 Transfer Functions Ver 169 5 Linearity Linear A system is linear if and only if a1x1t l CLQZBQ gt a1y1t l a2y2t for any constant complex coefficients a1 and a2 and any bounded input signals 96115 and 96215 such that 96105 gt 9105 96205 H 1205 J McNames Portland State University ECE 222 Transfer Functions Ver 169 6 LTI Circuits x0 W W W o Circuits that Are driven by a single independent voltage source the input Have no energy stored initially in the capacitors or inductors are LTI systems 0 More generally any system where the relationship between input and output can be defined as a constantcoefficient ordinary differential equation ODE are LTI J McNames Portland State University ECE 222 Transfer Functions Ver 169 7 Transfer Functions Continued M 8k Ys X8 H8Xs N Zk0 aksk o In the time domain the relationship can be complicated o In the 3 domain the relationship of Y8 to X8 of LTl systems simplifies to a rational function of 8 o Hs is usually a rational ratio of two polynomials o Hs is called the transfer function 0 Specifically the transfer function of an LTl system can be defined as the ratio of Y8 to X8 0 Usually denoted by sometimes G8 0 Without loss of generality usually aN 1 J McNames Portland State University ECE 222 Transfer Functions Ver 169 8 Transfer Function vs Impulse Response 0 Suppose We have a circuit with a transfer function Hs We apply a unit impulse to the circuit ZEt 6t X8 1 Ys H8X8 H8 W Mt Thus the transfer function is the Laplace transform of the system 395 impulse response J McNames Portland State University ECE 222 Transfer Functions Ver 169 9 Transfer Functions and the Impulse Response 0 Because of their relationship both H8 and ht completely characterize the LTI system c If the LTI system is a circuit once you know either H8 or ht you have sufficient information to calculate the output 0 You now have two different approaches to solve for the output of an LTI circuit Solve for X8 and then yt 1X8 Use Laplace transform circuit analysis to solve for the outputs of interest 0 Each has limitations advantages and disadvantages J McNames Portland State University ECE 222 Transfer Functions Ver 169 10 3 Domain Circuit Element Summary Resistor V8 RI8 V RI VIEN Inductor V8 8LI8 V 8L A 1 1 L Capacntor V8 EH8 V E o All of these are in the form Vs ZI8 0 Note similarity to phasor transform Identical if 8 jw Will discuss further later 0 Equations only hold for zero initial conditions J McNames Portland State University ECE 222 Transfer Functions Ver 169 11 Example 2 Transfer Functions vst C v0t Find the transfer function for the circuit above The input is the voltage source 21815 and the output is labeled 21015 J McNames Portland State University ECE 222 Transfer Functions Ver 169 12 Example 3 Transfer Functions R W v r i Find the transfer function for the circuit above Do you recognize this function J McNames Portland State University ECE 222 Transfer Functions Ver 169 13 Example 4 Transfer Functions C W O v00 Find the transfer function for the circuit above Do you recognize this function J McNames Portland State University ECE 222 Transfer Functions Ver 169 14 Example 5 Transfer Functions CB I H RB VVV CA I H RA w Vs v00 Find the transfer function for the circuit above J McNames Portland State University ECE 222 Transfer Functions Ver 169 15 Example 5 Workspace J McNames Portland State University ECE 222 Transfer Functions Ver 169 16 Transfer Function Analysis Tradeoffs Transfer Function yt 1 X8 0 Advantages Reduces differential equation to an algebra problem Usually the easiest approach Easy to find the output for different input signals Provides insights about what the circuit does 0 Disadvantages Can only solve for yt fort gt 0 Requires zero initial conditions Can only be used for SISO systems one independent source J McNames Portland State University ECE 222 Transfer Functions Ver 169 17 Laplace Transform Circuit Analysis Tradeoffs Laplace Transform Circuit Analysis 0 Advantages Elegant method of handling nonzero initial conditions Can handle multiple sources multiple inputs 84 can solve for multiple outputs any voltage or current MIMO systems 0 Disadvantages Can only solve for yt fort gt 0 Cannot account for full history 9615 fort lt 0 Can be tedious Specific to application circuits we did not discuss generalization to other types of systems Doesn39t provide insight about what the circuit does J McNames Portland State University ECE 222 Transfer Functions Ver 169 18 SteadyState Sinusoidal Analysis Assume a system H8 is BIBO stable Consider a sinusoidal input 9515 A coswt A cos COSwt A Sin SinWt coswt gt sinwt 827 X8 Acos ASin 82 w2 A 8 cos w sin 82 W2 Y3 ENS H8 A 8 COSifL wc 81n i J McNames Portland State University ECE 222 Transfer Functions Ver 169 19 SteadyState Sinusoidal Analysis Continued A 8 COS w sin Y8 H8 82 LUZ k k N kg S jw 8jw lm N 3915 2W coswt 4107415 2 kg e pgtut 51 2 yss ytr yss t yt 2W coswt 4k J McNames Portland State University ECE 222 Transfer Functions Ver 169 20 SteadyState Sinusoidal Analysis Comments If xt Acoswt yssa t1irgoyt 2w cosw 4k o If the input to an LTI system is sinusoidal The steadystate output is sinusoidal at the same frequency The amplitude and phase of yt differ from that of 9615 0 We applied this idea when we did phasor analysis 0 But how is k related to A and p J McNames Portland State University ECE 222 Transfer Functions Ver 169 21 Solving for the Complex Residue Y8 H8 A 8 Cs2 wc SlIl 1 k N kg s jw8jwm A 8 COS w sin 8 jw 8jw A jw COS w sin 2jw A COSW j sin 2 k H8 HOW HOW HjwAej J McNames Portland State University ECE 222 Transfer Functions Ver 169 22 Sinusoidal SteadyState Output Since Hjw is complex we can write it in polar form as HUwiHUwM ZHW Then using the results of the previous slide we have k HUleequ iHjwiAej Hjw MMMwNA zk 2Huw ysst 2W coswt l 4k iHjwiAcos wt ZHjw J McNames Portland State University ECE 222 Transfer Functions Ver 169 23 Sinusoidal SteadyState Output 96t Acoswt ysst iHjwiAcos wt 4Hjw o The input is sinusoidal o The steadystate signal ysst is also a sinusoid Same frequency as 9615 w Amplitude is scaled by iHjwi The phase is shifted by ZHUw o If we know we can easily find the steadystate solution for any sinusoidal input signal J McNames Portland State University ECE 222 Transfer Functions Ver 169 24 Example 8 SteadyState Sinusoidal Analysis R vst C v0t Find the steadystate sinusoidal response to an input voltage of 21815 coswt J McNames Portland State University ECE 222 Transfer Functions Ver 169 25 Example 8 Workspace J McNames Portland State University ECE 222 Transfer Functions Ver 169 26 SteadyState Sinusoidal Analysis Comments c We will study this in depth shortly c There is analytical significance to how the magnitude and phase of Hs vary with 8 jw J McNames Portland State University ECE 222 Transfer Functions Ver 169 27 LTI Systems o If we know the transfer function we have sufficient information to calculate the output for any input o This enables us to treat the circuit more abstractly as H8 o The transfer function may be for another type of system mechanical chemical hydraulic etc o Mathematically they are treated the same 0 Fieldspecific analysis is used only to find H8 J McNames Portland State University ECE 222 Transfer Functions Ver 169 28 Example 9 Transfer Function Analysis b k m I xt gt 39 n Find the transfer function for the linear system shown above The external force ZEt is the input to the system and the displacement yt is the output Find the transfer function J McNames Portland State University ECE 222 Transfer Functions Ver 169 29 Transfer Function Synthesis 0 Thus far we have talked only about circuit analysis 0 We now know several ways to solve for the output of a given system o If there are zero initial conditions then we can find the transfer function Hs of a given circuit 0 Now we will discuss how to design a circuit that implements a given H8 o This is called transfer function synthesis 0 There are many circuits that have the same transfer function J McNames Portland State University ECE 222 Transfer Functions Ver 169 30 Cascade Transfer Function Synthesis xrgt H1 l gt H2s l gt ya H1s gtlt H2s gtlt gtlt Hp8 0 There are many approaches to transfer function synthesis 0 Will discuss how to specify H8 to meet the requirements for a given application later this term 0 The most common and perhaps easiest approach to synthesis is to break H8 up into lst real poles or 2nd complex poles order components 0 Thus each component has either a lst or 2nd order polynomial in the numerator and denominator J McNames Portland State University ECE 222 Transfer Functions Ver 169 31 Cascade Transfer Function Synthesis Continued xrgt H1 l gt H2s l gt ya 0 There are robust standard circuits for implementing these loworder components o The output of each transfer function is generated by an operational amplifier 0 This is essential for the cascade synthesis to work will explain later 0 Some of these lst and 2nd order components are discussed in the text Chapter 15 0 Others can be found in more advanced analog circuits texts 0 You will probably use cascade synthesis in your projects this term J McNames Portland State University ECE 222 Transfer Functions Ver 169 32 Summary o Circuits with a single input independent source and zero initial conditions can be represented generically by their transfer functions H8 is the Laplace transform of the system impulse response 0 The output of the system is yt 1H8X8 for any causal input signal 0 fort lt 0 o For sinusoidal inputs the output is also sinusoidal at the same frequency but amplified by and shifted in phase by ZHUw 0 Thus transfer functions make sinusoidal steadystate analysis easy 0 Generalization of phasors Transfer function analysis used for all types of LTl systems not just circuits 0 Can synthesize a transfer function using a cascade of lst and 2nd order active circuits J McNames Portland State University ECE 222 Transfer Functions Ver 169 33 Laplace Transform Analysis Illustration Continued Total Transient 08 Steady State vot V 702 704 706 70398 5 1390 15 2390 25 Time ms 21015 etO 001 sin1000t 450 Utrt l US05 J McNames ECE 222 Laplace Transform Portland State University Ver 178 5 Overview of Convolution Integral Topics o Impulse response defined 0 Several derivations of the convolution integral 0 Relationship to circuits and LTI systems J McNames Portland State University ECE 222 Convolution Integral Ver 170 1 Impulse Response Recall that if ZEt 6t the output of the system is called the impulse response o The impulse response is always denoted ht o For a given input it is possible to use ht to solve for yt One method is the convolution integral 0 This is a important concept J McNames Portland State University ECE 222 Convolution Integral Ver 170 2 RC Circuit Impulse Response R C W ll xt ht RC eut 0 Many of the following examples use the impulse response of a simple RC voltage divider o The impulse response can be obtained using the Laplace transform 0 In many of the following examples RC ls J McNames Portland State University ECE 222 Convolution Integral Ver 170 3 ContinuousTime Time Invariance 0 Recall that time invariance means that if the input signal is shifted in time the output will be shifted in time also 0 Consider three separate inputs 90115 505 90105 gt 9175 hlt 96215 505 2 96205 gty2t hlt Q 1133155t 5 96305 gty3t hlt 5 0 Let ht etut g J McNames Portland State University ECE 222 Convolution Integral Ver 170 4 Example 1 ContinuousTime Time Invariance 1 Input 1 Output 5t0 ht0 05 05 X10 y1t 5042 ht2 05 05 X20 y2t O O 2 0 2 4 6 8 10 2 0 2 4 6 8 10 1 1 v v f 05 f 05 I Equot 0 0 2 0 2 4 6 8 10 2 0 2 4 6 8 10 Time s Time s J McNames Portland State University ECE 222 Convolution Integral Ver 170 5 Example 1 MATLAB Code function E TimeInvariance figure1 FigureSet14528 t 2000110 tc025 for cl 11engthtc subplotlengthtc2cl21 h plottccl1 1O 1 b tccl095 bquot seth MarkerFaceColor b seth MarkerSize S st sprintf xdt Atd cltccl dispst ylabe1st box off Xlimmint maxtl ylimo 1 subplot1engthtc2cl2 y1 expttcclt2tcci Unit impulse response hn h plotty1 r st sprintf ydt ht d cltccl ylabe1st box off Xlimmint maxtl J McNames Portland State University ECE 222 Convolution Integral Ver 170 6 end subplot1engthtc21 title Input subplot1engthtc22 title 0utput subplotlengthtc21engthtc21 X1abel Time s subplot1engthtc21engthtc2 X1abel Time s AXisSet6 print depsc Timelnvariance J McNames Portland State University ECE 222 Convolution Integral Ver 170 7 Input Decomposition Let ZEt rt 2rt 1 rt 2 t 0 g t g 1 m 1gtg2 0 otherwise 0 We can approximate any bounded signal ZEt as a series of pulses with width 111 and height proportional to ZEt OO ZEt Z u t k u t k k oo J McNames Portland State University ECE 222 Convolution Integral Ver 170 8 Example 2 Input Decomposition 1 In ut 05 0 I I I I 105 0 05 1 15 2 25 o 39n f 05 l a 0 I I I 105 0 05 I I 5 2 2 5 m N a 0 I I I 105 0 0 5 1 1 5 2 2 5 S a 0 I I I 105 0 05 I I5 2 25 q o f 05 B 0 I 705 0 05 1 15 2 25 Time s J McNames Portland State University ECE 222 Convolution Integral Ver 170 9 Example 2 MATLAB Code function J InputDecomposition figure1 FigureSet14528 fs 1000 to 05 t1 25 h 050 025 010 004 subplot1engthh111 t t01fst1 x tt20amptlt1 t 2t21amptlt2 hp plottx sethp MarkerFaceColor b sethp MarkerSize S title Input ylabel xt box off xlimt0 til for cl izlengthh subplot1engthh11ci1 t t0hcit1 x tt20amptlt1 t 2t21amptlt2 hp bartx1 sethp FaceColor w sethp EdgeColor r st sprintf w 42f hci J McNames Portland State University ECE 222 Convolution Integral Ver 170 10 ylabelst box off xlimt0 til end X1abel Time S AXisSet6 print depsc InputDecomposition J McNames Portland State University ECE 222 Convolution Integral Ver 170 11 Replacing Rectangles with Impulses o If an input consists of a pulse that is sufficiently short in duration the continuoustime system will respond the same as it would to an impulse with the same area 0 The following example shows the response of an RC circuit to three different pulses 0 Each pulse has unit area 0 The impulse response is also shown 0 Since the response is the same we can replace our approximate input signal that consists of rectangles with a train of impulses if h is sufficiently small J McNames Portland State University ECE 222 Convolution Integral Ver 170 12 Example 3 RC Pulse Response Input 1 Output 04 h A A yt V 02 K 0395 0 0 2 0 4 6 10 2 4 8 10 Input Output 1 1 ht A A yt V 0 5 K 05 0 0 2 0 2 4 6 10 2 4 8 10 10 Input 1 Output ha A A W V 5 K 05 0 0 2 0 2 4 6 10 2 4 8 10 Time s Time s J McNames Portland State University ECE 222 Convolution Integral Ver 170 Example 3 MATLAB Code function E PulseResponse figure1 FigureSet14528 sys tfO 11 1 Transfer function of an RC circuit with RC 1 t 2000110 ht exptt20 System impulse response h 2 1 01 for c1 11engthh subplotlengthh2c121 X tgtO amp tlthc1hc1 plottx tit1e Input ylabe1 xt box off X1immint maxt ylimO maxxl subplotlengthh2c12 y 1simsysxt Simulate the output of the system plottht g ty r 1egend ht yt tit1e Dutput ylabe1 yt box off X1immint maxtl J McNames Portland State University ECE 222 Convolution Integral Ver 170 end subplot1engthh21engthh21 X1abel Time s subplot1engthh21engthh2 X1abel Time s AXisSet6 print depsc PulseResponse J McNames Portland State University ECE 222 Convolution Integral Ver 170 15 Impulse Input Decomposition 9615 can approximated as a sum of pulses 3315 mt f Wm M t k aw u t k 210 k oo Each pulse can be approximated as a unit impulse M hm ut k w ut kw 6t kw dt UHO w so for small 111 ZEt can also be approximated as a sum of impulses ZEt 95515 Z 6t kw k oo Notethat foo 6t kwdtOO ulttkwgtulttkwgt dt1 w J McNames Portland State University ECE 222 Convolution Integral Ver 170 16 Example 4 Approximations to 9625 1 Approximation with Rectangles 0 5 Approximatior with Impulses 9 f 0 5 0 0 705 0 05 1 15 2 25 705 0 05 1 15 2 25 m 02 N o H 0395 01 I I B 0 0 A n T I A A 705 0 05 1 15 2 25 07105 0 05 1 15 2 25 S f 05 005 X X 0 11 ill 705 0 05 1 15 2 25 004 V v 002 0 s39 39L 705 0 05 1 15 2 25 705 0 05 1 15 2 25 Time s Time s J McNames Portland State University ECE 222 Convolution Integral Ver 170 17 Example 4 MATLAB Code function E InputImpulses figure1 FigureSet14528 fs 1000 to 05 t1 25 t t01fst1 N tt20amptlt1 t 2t21amptlt2 h 05025010004l for cl izlengthh subplotlengthh2ci21 th t0hcit1 xr thth20ampthlt1 th2th21ampthlt2 hb barthxr1 sethb FaceColor w sethb EdgeColor b hold on plottx g hold off st sprintf w 42f hci ylabelst box off xlimt0 til J McNames Portland State University ECE 222 Convolution Integral Ver 170 18 subplotlengthh2cl2 for c2 1zlengthth hp plotthc21 1O xrc2hci r thc2O95xrc2hcl rquot hold on sethp2 MarkerFaceColor r sethp2 MarkerSize 2 sethp2 LineWidth 0001 end hold off box off ylim0 hc1 xlimt0 t1 end subplot1engthh21 title Approximation with Rectangles subplot1engthh22 title Approximation with Impulses subplot1engthh21engthh21 X1abel Time s subplot1engthh21engthh2 X1abel Time s AXisSet6 print depsc InputImpulses J McNames Portland State University ECE 222 Convolution Integral Ver 170 19 Applying Linearity ZEt 9550 Z 6t kw k oo then by linearity and time invariance yt y5t Z w ht kw k oo In the limit as w gt 0 if ZEt and yt are integrable we have m lim map00 memos TNT W iiin0y5tOO memos TNT This is the continuoustime convolution integral J McNames Portland State University ECE 222 Convolution Integral Ver 170 20 Example 5 Impulse Approximation and Output 0 5 Input Output 05 0 0 K 0 52 0 2 4 6 8 10 2 0 2 4 6 8 10 0 0 0 52 0 2 4 6 8 10 2 0 2 4 6 8 10 05 0 52 0 2 4 6 8 10 2 0 2 4 6 8 10 05 True Approximate 0 0 2 0 2 4 6 8 10 2 0 2 4 6 8 10 Time s Time 5 J McNames Portland State University ECE 222 Convolution Integral Ver 170 21 Example 5 MATLAB Code function E ImpulseApproximation figure1 clf FigureSet14528 sys tf0 11 1 Transfer function of an RC circuit with RC 1 t0 2 t1 10 t t00001t1 Xt tt20amptlt1 t 2t21amptlt2 Z True system input yt lsimsysxtt Simulate the output of the system y zerossizet h 05 ti 05 10 15 Impulse times Xi 05 10 05lh Impulse Amplitudes for cl 1lengthti subplotlengthti12cl21 hp plottici1 10 Xici b ticlxici bquot sethp MarkerFaceColor b sethp MarkerSize 12 sethp1 LineWidth 0S sethp2 LineWidth 0001 box off ylim0 maxxil Xlimmint maxtl J McNames Portland State University ECE 222 Convolution Integral Ver 170 22 subplotlengthti12cl2 yp xiclexptticittici20 Z Response to a single impulse plottyp r box off ylim0 0661 xlimmint maxtl y y yp Total Response end subplotlengthti12lengthti21 for cl 11engthti hp plottici1 1O xici b ticixicl bquot sethp MarkerFaceColor b sethp MarkerSize 12 sethp1 LineWidth OS sethp2 LineWidth 0001 hold on end hold off box off ylimO maxxi xlimmint maxt xlabel Time s subplotlengthti12lengthti22 hp plottyt g ty r sethp1 LineWidth O8 sethp2 LineWidth OS box off ylim0 0661 xlimmint maxt xlabel Time s J McNames Portland State University ECE 222 Convolution Integral Jer 170 23 legend True Approximate subplot1engthti121 title Input subplot1engthti122 title 0utput AXisSet6 print depsc ImpulseApproximation J McNames Portland State University ECE 222 Convolution Integral Ver 170 24 Example 6 Approximate Outputs versus h Input Output 06 04 True 0 4 Approximate 02 02 0 0 2 0 2 4 6 8 10 2 0 2 4 6 8 10 06 02 True 0 4 Approximate 01 I 02 0 0 2 0 2 4 6 8 10 2 0 2 4 6 8 10 01 06 True 005 02 0 0 2 0 2 4 6 8 10 2 0 2 4 6 8 10 Time s Time s J McNames Portland State University ECE 222 Convolution Integral Ver 170 Example 6 MATLAB Code function E ImpulseTrainApproximations figure1 clf FigureSet14528 sys tf0 11 1 Transfer function of an RC circuit with RC 1 t0 2 t1 10 t t00001t1 x tt20amptlt1 t 2t21amptlt2 Z True system input yt lsimsysxt Simulate the output of the system h 050 025 010 for cl 1lengthh subplotlengthh2ci21 th t0hcit1 Impulse times xr thth20ampthlt1 th2th21ampthlt2 Z Impulse amplitudes for c2 1lengthth hp plotthc21 10 xrc2hcl b thc2Xrc2hci bquot hold on sethp MarkerFaceColor b sethp MarkerSize 12 sethp1 LineWidth 0S sethp2 LineWidth 0001 end hold off J McNames Portland State University ECE 222 Convolution Integral Ver 170 26 box off Xlimmint maxt ylimO h61 subplotlengthh2cl2 y zerossizet for c2 1zlengthth y y hclxrc2exptthc2tthc220 end hp plottyt g ty r sethp1 LineWidth O8 sethp2 LineWidth OS legend True Approximate box off Xlimmint maxt ylim0 0661 end subplot1engthh21 title Input subplot1engthh22 title 0utput subplot1engthh21engthh21 X1abel Time s subplot1engthh21engthh2 X1abel Time s AXisSet6 print depsc ImpulseTrainApproximations J McNames Portland State University ECE 222 Convolution Integral Ver 170 27 ContinuousTime Convolution Derivation Summary mos ya De nition of ht 60 he Time Invanance 6t T ht T Linearit rm 6t T m ht T Linearit ff rm 6t 7 d7 ff rm ht 7 d7 De nition of 6t 9615 ff magma TNT OO J McNames Portland State University ECE 222 Convolution Integral Ver 170 28 Convolution Integral Alternative Form yt Ox7 ht TNT Letu t r then Tt uand dTZ du ya x0 u Mu dugt xt uhudu 00 ZEt Th7 d7 OO 0 Both forms are called the convolution integral 0 This is often written as yt ZEt gtllt ht J McNames Portland State University ECE 222 Convolution Integral Ver 170 29 Relationship of ht to Transfer Functions Suppose ht ZEt 0 fort lt 0 00 yt ZE7 ht 739 d7 00 Y8 yt ytestdt m ht 7 d7 e stdt 0 w ht Te8t dt d7 J McNames Portland State University ECE 222 Convolution Integral Ver 170 30 Relationship of ht to Transfer Functions Continued Recall the effect of a translation in time on the Laplace transform of a signal ht T We T e STH8 and since ht 0 fort lt 0 we have ht htut and ht 739 ht 739ut 739 Thus YsOx7 ht Te8tdt d7 J McNames Portland State University ECE 222 Convolution Integral Ver 170 31 Transfer Functions and Impulse Responses W foo rm ht TNT 00 00 ZEt 739h739d739 OO Y8 178 X8 0 Thus we see once again that the transfer function and the impulse response form a Laplace transform pair 0 Either is sufficient to calculate the output of the system for any causal ie 9615 0 fort lt 0 input signal 9615 0 Unlike the transfer function analysis the convolution integral does not require 9615 or ht to be causal J McNames Portland State University ECE 222 Convolution Integral Ver 170 32 Impulse Response Intuition 00 yt xtht memos TNT OO 0 Key point The system output at t is in response to all past and present values of the input signal not just at time t o Conceptually you can think of convolution as weighted averaging o The signal xt is being averaged o The impulse response ht determines how it is weighted o Lowpass filters tend to have smooth slowly varying weights 0 Highpass and bandpass filters tend to be oscillatory J McNames Portland State University ECE 222 Convolution Integral Ver 170 33 Summary xt W ya xtht OO 9525 7 hT d7 00 o The convolution integral describes how the output yt is related to the input signal ZEt and the unit impulse ht 0 Only two assumptions were made about the system Linear Time lnvariant 0 Key points The impulse response ht completely defines the behavior of continuoustime LTl systems If ht is known then the output of a continuoustime LTl system can be calculated for any input ZEt using the convolution integral J McNames Portland State University ECE 222 Convolution Integral Ver 170 34

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