ORDINARY DIFFERENTIAL EQUATIONS

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1 Page 1 of 15 ORDINARY DIFFERENTIAL EQUATIONS Lecture 17 Delta Functions & The Impulse Response (Revised 30 March, 09:40) Professor Stephen H Saperstone Department of Mathematical Sciences George Mason University Fairfax, VA sap@gmu.edu Copyright 2009 by Stephen H Saperstone All rights reserved 17.1 Dirac Delta Function - a Heuristic Development In order to motivate the definition of the delta function, we digress and discuss the impulse of a force. There are times when the ODE has a forcing function that is impulsive; that is, is identically zero except over a short time interval during which its magnitude is very large. For instance, a hammer blow, the collision of billiard balls or the static discharge when you touch a light switch after having walked across a carpeted floor in the winter, or a lightening strike in a thunderstorm, are all examples of an impulsive force. The exact nature of the force during this short time interval is usually not known. Consider the case of a body of mass constrained to move in a straight line. Let denote the velocity of the body at time. If a force is applied to the body over the time interval, then according to Newton's second law we have that (replacing the variable with the dummy variable

2 Page 2 of 15 which is the change in the body's momentum from to As was suggested above, the function can represent physical quantities other than a force. In a static discharge is a current Analogous to Newton's second law we have that so that which is the change in charge from to From a geometric standpoint the impulse of over the interval is the area under the curve on Thus we are led to the following definition which generalizes what we have been discussing. Definition 17.1 (Impulse) The impulse of a function over the time interval is defined by the definite integral We call the force an impulsive force. Figure 17.1 illustrates the nature of a force that can produce a given impulse on a body. For simplicity, let The graph on the left side of Figure 17.1(a) represents a hypothetical force which is nonzero on the interval The graph on the right side of Figure 17.1(a) illustrates the integral of Then By shortening the interval over which the force acts and increasing the height of, we can maintain the same value of the impulse. Thus we arrive at Figure 17.1(b). The value of the integral of over the shortened interval must be. As the interval shrinks down, must get taller in order to maintain the value of the impulse at We have depicted with the dashed line on the right side of Figures 17.(a) and (b) how the velocity changes from to A real impulse like a hammer blow takes place over such a small interval of time that it is convenient to take the length of the interval to be zero. In terms of the evolution depicted in Figure 17.1, we examime what happens when the velocity must jump instantaneously from to This can happen only when the force becomes infinite magnitude at the instant and zero at all other times. No ordinary function behaves this way!

3 Page 3 of 15 Figure 17.1 There is nothing special about the bell-shaped curves in Figures 17.1(a) and (b). Since the impulse in Figure 17.1(c) occurs over an infinitesimal time interval, the shape of is irrelevant. Data about the nature of a hammer blow are almost impossible to obtain. What is relevant and significant is that the integral of the force equals the change in momentum of the body. Figure 17.2 illustrates an alternative to the bell-shaped force.. Figure 17.2

4 Page 4 of 15 The box-shaped force in Figure 17.2(a) when integrated yields the "ramp" like velocity curve in Figure 17.2 (b). The box-shaped force, when applied over a very short time interval, has to be very large. Figure 17.3 depicts this situation for a force of duration starting at If the impulse has magnitude (an increase of momentum from to we can see the result of varying Figure 17.3 It appears that we arrive at the instantaneous velocity jump from to no matter what shape we take for. The box-curves are the easiest to work with so we use them from now on. But first we need to create some new tools for impulsive forces. The first of these is the unit step function also known as the Heaviside function. Definition 17.2 Unit Step (Heaviside) Function Figure 17.4: Unit step (Heaviside) function Definition Translate of the Unit Step Function

5 Page 5 of 15 Figure 17.5: -Translated unit step Function We piece together unit step functions and their translates to create a pulses as demonstrated in the next example. Example 17.4 (Pulses) 1. The unit pulse (introduced in Example 6.2): Figure 17.6: Unit pulse 2. -translate of the unit pulse: Figure 17.7: -Translated Unit Pulse

Page 6 of 15 3. -translate of the unit pulse of width Figure 17.8: -Translated Pulse of width End of Example 17.

6 Page 6 of translate of the unit pulse of width Figure 17.8: -Translated Pulse of width End of Example 17.4 The Dirac Delta Function The following definition specifies a "function" by the following properties. Definition 17.5 (Dirac Delta) This "function" isn't really a function as we know it. In order that the integral of over have the value 1, would have to be infinite. As is not a number, cannot be a function in the ordinary sense. Nevertheless, we persist in calling a function even though it isn't one. We can represent by the spike as shown in Figure We say that has its only "action" at Figure 17.9: Dirac delta "function"

7 Page 7 of 15 Notation 17.6 (Dirac Delta) Because whenever there is no loss in generality in taking the interval of integration to be finite, e.g., where so that "straddles" 0. Thus we can write the integral condition in Definition 17.5 as or even The symbol means that we evaluate the lower limit at some number and take the limit of the result as Thus where the notation means that the limit is taken as approches from below. End of Notation 17.6 Whenever we compute the integral of a delta function, we must take care to see the relationship of to the limits of integration. The next example illustrates some of the many possibilities. Example 17.7 (Impulses) The computation of the value of the following integrals comes directly from the definition of

Page 8 of 15 End of Example 17.7 Approximation by Pulses can be approximated it with actual functions as follows. Define the -pulse where is any small postive number.

8 Page 8 of 15 End of Example 17.7 Approximation by Pulses can be approximated it with actual functions as follows. Define the -pulse where is any small postive number. We can express in terms of the unit-step function by Figure 17.10: -Pulse An -pulse can be symmetric about Thus is a legitimate -pulse. We will have need for this constructiom in the proof of Theorem 17.8 later. Observe that for any the area under any -pulse is 1 Figure indicates how the -pulse satisfies Eqn. (17.3) for any. In particular, no matter how small

Page 9 of 15 we make Eqn. (17.3) is always true. Figure 17.

9 Page 9 of 15 we make Eqn. (17.3) is always true. Figure also illustrates that no matter how narrow the pulse becomes (by taking to be very small), its height grows accordingly to ensure that the area bounded by the pulse remains constant (with value one). Figure 17.11: -pulses, Except for It follows that Time translates of the delta function and the unit step function are important. Definition 17.8 (Translated Dirac Delta) Figure 17.12: -translate of Although the delta function is an ideal mathematical abstraction, there are numerous forces that can be modeled by an -pulse. A bat hitting a baseball or a spark across some gap are but a few examples of events that can be modeled by a delta function. These events share the following behavior: a very large force is

10 Page 10 of 15 applied to some system over an extremely short interval of time IMPULSE RESPONSE What do Dirac delta functions and approximations by pulses have to do with ODEs and convolution? We saw in Green's kernel solution that if the forcing function is piecewise continuous on, the ODE has zero-state response given by where is the Green's kernel for the corresponding homogeneous ODE. We are interested in kowing the zerostate response when is the Dirac delta function The following theorem gives us the answer. Theorem 17.9 (Impulse Response Function) The zero-state response to the ODE is given by Proof: We seek to calculate by approximating so we can write with -pulses. We start by recognizing that the integral in Eqn. (17.6) is a convolution and Since has its "action" at precisely at we must extend the upper limit of integration to. (Typically we write instead of as the upper limit of integration with the understanding that really stands for As is not piecewise continuous we approximate with a symmetric -pulse about, namely Such a pulse is illustrated in Figure along with

is continuous on hence it is continuous on the interval of length When is sufficiently small, the value of will not vary much over that interval.

11 Page 11 of 15 Figure 17.13: Symmetric -pulse at time Since the lower limit of integration needs to include we will use as the lower limit. When done, we take the limit of the resulting integral as Now proceed to calculate the approximate value of In view of the construction of the -pulse, we have that consequently Now the Green's kernel is continuous on hence it is continuous on the interval of length When is sufficiently small, the value of will not vary much over that interval. Hence we can approximate with the value This yields Thus we have shown that the zero-state response is precisely the Green's kernel. But the possible forms of from Theorem 14.16, namely are defined at all This presents a dilemma: The solution precedes the input How can the solution be defined for when the input isn't initiated until The response of a real system to an input cannot occur before the input is applied. This is the result of the commonsense idea of cause

12 Page 12 of 15 and effect. To ensure that the solution to is zero for we can modify the solution by multiplying it by the unit step function Finally we obtain Q.E.D. The solution is called the impulse response. That the impulse response must be zero prior to the onset of the impulse function reflects a physical requirement called causality CAUSALITY Definition (Impulse Response & System Causality) The zero-state solution to the ODE with impulsive input is called the impulse response function and is denoted by Any system for which the response occurs only during or after the time in which the input is applied is a causal system. Any practical system that operates in real time must necessarily be causal. We do not know how to build a system that responds to future inputs (inputs not yet applied). (A system in which the output at present depends on the future values of an input is called noncausal or anticipative.) Note that the ODE itself does not embody causality. Indeed, the general solution to this ODE is defined for all It is a failing of the ODE to model a physical system that forces us to "tack on" Suitable modified by we revisit the input-output schematic of Figure 16.7 to get Figure 17.14: Input-Output Schematic for a Causal System Thus the impulse response is the zero-state response of a causal system to the impulsive input, Because of the special nature of the impulse response function we call a causal function as well. When the input is a translated Dirac delta function, the impulse response solution is also delayed. This follows directly from Eqn. (17.9). Theorem (Impulse Response - Delayed Input)

13 Page 13 of 15 For any the solution to the IVP is It follows that Figure 17.15: Response of a Causal System to a Delayed Impulsive Input A causal system can have any piecewise continuous input - not necessarily the impulsive input So let's examine the zero-state response of a causal system to an arbitrary input. Eqn. (17.5) still applies: To ensure that Since most inputs start at there is no loss in generality in assuming that each such is zero prior to Such inputs are called causal, too. Definition (Causal Functions) Any piecewise continuous function on is called causal if for all Since causality of a system means that the response at any time depends only on the values of an input up to the time i.e., the values of for each In particular if the input is zero for all the output must also be zero for all In view of the nature of a causal system, Eqn. (17.12) becomes Eqn. (17.13) is a convolution product of with Since is causal, we must have whenever i.e.,. Thus we may extend the upper limit of integration to as for a so can write Since is also causal, we must have whenever Thus we may extend the lower limit of integration to to obtain We conclude this lecture with an important theorem on how to resolve a continuous function into a continuum of delta functions. This reasonable in view of the approximation of a continuous function by a "staircase."

14 Page 14 of : SIFTING Theorem (Sifting) Suppose is any continuous function on Then for any Click here for Proof. We can interpret the sifting theorem to say that the translated delta function value of at time See Figure below "sifts out" or selects the Figure 17.16: Sifting pulse at time Note that in view of commutativity of convolution, Eqn. (17.15) can be rewritten as The limits of integration need not range from to they can be any values to If the delta spike falls at either or, we extend or to or to include the value where the spike occurs. Example (Sifting) The computation of the value of the following integrals comes directly from the Sifting Theorem

15 Page 15 of 15 End of Example : LINEAR TIME-INVARIANT (LTI) SYSTEMS UNDER CONSTRUCTION A generalization of the zero-state response leads to the abstract representation of a linear system, namely where represents an input (forcing function in the terminology of ODEs) and represents the output or response to the input Note that both and are functions, typically functions of time. For this reason somes Eqn. (17.10) is written as where the symbols and indicate that and are functions - not numbers. When the system is ODE, the symbol represents the integral operation that produces the zero-state response. For instance the zero-state response that corresponds to the Green's kernel is given by Here is represented by the integral operator of Eqn. (17.18):

ORDINARY DIFFERENTIAL EQUATIONS

Page 1 of 20 ORDINARY DIFFERENTIAL EQUATIONS Lecture 11 Fundamental Solution Sets for Linear Homogeneous Second-Order ODEs (Revised 22 March 2009 @ 09:16) Professor Stephen H Saperstone Department of Mathematical