The Dirac Delta Function and Convolution 1 The Dirac Delta ...

1 Dirac Delta Function and Convolution1 The Dirac Delta (Impulse) FunctionThe Dirac Delta Function is a non-physical, singularity Function with the following definition (x)= 0forx =0undefinedatx=0(1)but with the requirement that (x)dx=1,(2)that is, the Function has unit ) Unit pulses of different extentsb) The impulse function1/T11/T21/T31/T4td(t)0t d (x)XFigure 1: Unit pulses and the Dirac Delta 1 shows aunit pulsefunction T(t), that is a brief rectangular pulse Function of durationT, defined to have a constant amplitude 1/Tover its extent, so that the areaT 1/Tunder thepulse is unity: T(t)= 0fort 01/T0<t T0fort>0.

2 (3)The Dirac Delta Function (also known as the impulse Function ) can be defined as the limiting formof the unit pulse T(t) as the durationTapproaches zero. As the durationTof T(t) decreases,the amplitude of the pulse increases to maintain the requirement of unit area under the Function ,and (t) = limT 0 T(t).(4)The impulse is therefore defined to exist only at timet= 0, and although its value is strictlyundefined at that time, it must tend toward infinity so as to maintain the property of unit area inthe limit. Thestrengthof a scaled impulseK (t) is defined by its limiting form of many other functions may be used to approximate the impulse.

3 Commonfunctions include triangular, gaussian, and sinc (sin(x)/x) impulse Function is used extensively in the study of linear systems, both spatial and tem-poral. Although true impulse functions are not found in nature, they are approximated by shortduration, high amplitude phenomena such as a hammer impact on a structure, or a lightning strikeon a radio antenna. As we will see below, the response of a causal linear system to an impulsedefines its response to all impulse occurring att=ais (t a). The Sifting Property of the ImpulseWhen an impulse appears in a product within an integrand, it has the property of sifting outthe value of the integrand at the point of its occurrence: f(t) (t a)dt=f(a)(5)This is easily seen by noting that (t a) is zero except att=a, and for its infinitesimal durationf(t) may be considered a constant and taken outside the integral, so that f(t) (t a)dt=f(a) (t a)dt=f(a)(6)from the unit area ConvolutionConsider a linear continuous-time system with inputu(t), and responsey(t), as shown in Fig.

4 Assume that the system is initially at rest, that is all initial conditions are zero at timet=0,and examine the time-domain forced responsey(t) to a continuous input waveformu(t).Linear Systemu(t)y(t)Figure 2: A linear Fig. 3 an arbitrary continuous input functionu(t) has been approximated by astaircasefunction uT(t) u(t), consisting of a series ofpiecewise constantsections each of an arbitraryfixed duration,T, where uT(t)=u(nT)fornT t<(n+1)T(7)for alln. It can be seen from Fig. 3 that as the intervalTis reduced, the approximation becomesmore exact, and in the limitu(t) = limT 0 uT(t).

5 The staircase approximation uT(t) may be considered to be a sum of non-overlapping delayed pulsespn(t), each with durationTbut with a different amplitudeu(nT): uT(t)= n= pn(t)(8)2u(t) (3T)00 System InputSystem Inputu (t)T~u (t)T~u(t)u(t)Figure 3: Staircase approximation to a continuous input functionu(t).systemnT(n+1)Td (t-nT)Td (t-nT)Tt1/T00y (t)Ty (t)T00nT(n+1)TtFigure 4: System response to a unit pulse of (t)iResponse (t)T~Total ResponsetTime00 Figure 5: System response to individual pulses in the staircase approximation tou(t).3wherepn(t)= u(nT)nT t<(n+1)T0otherwise(9)Each component pulsepn(t) may be written in terms of a delayed unit pulse T(t) defined in , that is:pn(t)=u(nT) T(t nT)T(10)so that Eq.

6 (8) may be written: uT(t)= n= u(nT) T(t nT)T.(11)We now assume that the system response to T(t) is a known Function and is designatedhT(t)as shown in Fig. 4. Then if the system is linear and time-invariant, the response to a delayed unitpulse, occurring at timenT, is simply a delayed version of the pulse response:yn(t)=hT(t nT).(12)The principle of superposition allows the total system response to uT(t) to be written as the sumof the responses to all of the component weighted pulses in Eq. (11): yT(t)= n= u(nT)hT(t nT)T(13)as shown in Fig. 5. For physical systems the pulse responsehT(t) is zero for timet<0, and futurecomponents of the input do not contribute to the sum, so that the upper limit of the summationmay be rewritten: yT(t)=N n= u(nT)hT(t nT)TforNT t<(N+1)T.

7 (14)Equation (14) expresses the system response to the staircase approximation of the input in termsof the system pulse responsehT(t). If we now let the pulse widthTbecome very small, and writenT= ,T=d , and note that limT 0 T(t)= (t), the summation becomes an integral:y(t) = limT 0N n= u(nT)hT(t nT)T(15)= t u( )h(t )d (16)whereh(t) is defined to be the systemimpulse response,h(t) = limT 0hT(t).(17)Equation (16) is an important integral in the study of linear systems and is known as theconvolutionorsuperpositionintegral. It states that the system is entirelycharacterizedby its response to animpulse Function (t), in the sense that the forced response to any arbitrary inputu(t)maybecomputed from knowledge of the impulse response alone.

8 The Convolution operation is often writtenusing the symbol :y(t)=u(t) h(t)= t u( )h(t )d .(18)4multiplicationtttth(t)0time reversaltimeshiftingh(t -t)0th(-t)System impulse responseSystem inputu(t)0u(t)h(t-t)0tintegrationSystem responset0y(t)t1t111response at time t is defined by thearea under the 6: Graphical demonstration of the Convolution (18) is in the form of a linear operator, in that it transforms, or maps, an input functionto an output Function through a linear operation. It is a direct computational form of the systemtransfer operatorH{u(t)}, that is:y(t)=H{u(t)} u(t) h(t).

9 The form of the integral in Eq. (16) is difficult to interpret because it contains the termh(t )inwhich the variable of integration has been negated. The steps implicitly involved in computing theconvolution integral may be demonstrated graphically as in Fig. 6, in which the impulse responseh( ) is reflected about the origin to createh( ), and then shifted to the right bytto formh(t ). The productu(t)h(t ) is then evaluated and integrated to find the response. Thisgraphical representation is useful for defining the limits necessary in the integration.

10 For example,since for a physical system the impulse responseh(t) is zero for allt<0, the reflected and shiftedimpulse responseh(t ) will be zero for all time >t. The upper limit in the integral is thenat mostt. If in addition the inputu(t) is time limited, that isu(t) 0fort<t1andt>t2, thelimits are:yf(t)= tt1u( )h(t )d fort<t2 t2t1u( )h(t )d fort t2(19)ExampleA mass element, shown in Fig. 7 at rest on a viscous plane, is subjected to a very shortunit impulsive force of duration seconds and magnitude 1000 newtons, and isobserved to respond with a velocityvm(t)=e 3t.