The Fourier transform of the unit step function B. L. Burrows a ; D. J. Colwell a a

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1 This article was downloaded by: [National Taiwan University (Archive)] On: 10 May 2011 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK International Journal of Mathematical Education in Science and Technology Publication details, including instructions for authors and subscription information: The Fourier transform of the unit step function B. L. Burrows a ; D. J. Colwell a a Department of Mathematics, Staffordshire Polytechnic, Beaconside, Stafford, England To cite this Article Burrows, B. L. and Colwell, D. J.(1990) 'The Fourier transform of the unit step function', International Journal of Mathematical Education in Science and Technology, 21: 4, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 INT. J. MATH. EDUC. SCI. TECHNOL., 1990, VOL. 21, NO. 4, The Fourier transform of the unit step function by B. L. BURROWS and D. J. COLWELL Department of Mathematics, Staffordshire Polytechnic, Beaconside, Stafford, England (Received 31 January 1989) The purpose of this paper is to highlight difficulties which can arise when the Fourier transform of the unit step function is introduced to non-mathematics specialists without reference to generalized function theory. 1. Introduction In many textbooks the Fourier transform of the unit step function u{t) is introduced using methods of proof which are deceptively simple and which do not highlight the difficulties inherent in the functions involved. A typical proof takes the following form. Defining the signum function by f 1 t>0 sgn(t)=< 0 t = 0 (.-1 t<0 it may be deduced that and further that Transforming this equation at or, using the time differentiation property of the Fourier transform jc«if{sgn(*)}=2 giving, if division is permissible Hence, since «(0=i+isgn(0 = n5(co) X/90 $ Taylor & Francis Ltd.

3 630 B. L. Burrows and D. J. Colwell This method of proof suggests a more direct approach, which we have not seen used, probably because it raises questions on the value of Since then dt or Hence, apparently Clearly this conflicts with the previous result. However, this conflict could be avoided by noting that so that, jco&{u(t)} = 1 = 1 +}7icoS(co) Unfortunately it could equally well have been suggested that, for example so that = 1 = 1 + JTtco 2 S(co) = 71(0(5(0))+ and again there is conflict on the answer for &{u(t)}. Apart from this type of conflict, there is in any case an interpretation problem to be faced. The result obtained in the first proof: 7t(5(a>)+ JO) involves a term 1/jco which is not defined at co = 0, while, for many authors, the function S(co) is only non-zero at cu = 0. Hence should the formula really be written as 1 jco

4 Fourier transform of the unit step function 631 This question, and indeed the conflict over the precise formula for ^F{u(t)}, highlights the difficulties which can arise when generalized functions are treated as if they were ordinary functions to simplify the mathematics. To answer it some generalized function theory needs to be introduced, but, as such a theory can appear rather daunting, we have tried to develop it in a form which may be readily understood. 2. A generalized function approach In order to use the theory of generalized functions it is first necessary to introduce the concept of a good function. The standard definition [1] is thatg(i) is a good function of t if it is differentiate any number of times everywhere, and such that it and all its derivatives are Ofl^l"^) as ( -KX>, for all positive integers N. However, since it is often necessary to consider precisely how g(t) behaves as t -*oo, it usually suffices to say that g(t) is a good function if it can be differentiated, any number of times everywhere, andg'" 1 ' (<)»0 as < -»oo for all m. The definition of a generalized function fit) may now be introduced using a sequence of good functions f n {t) («= 1,2,...), which is such that the limit lim n->co n-»co J -oo f H (t)g(t)dt exists for any good function G(t). Such a sequence, which is called a regular sequence, is said to define the generalized function/(i) in the sense that the integral is defined to be f(t)g(t)dt J 00 lim f + f n (t)g(t)dt n-*co J oo The Fourier transform of a generalized function/(i) is defined to be n-»oo J -c f n (t)exp(-jcot)dt Since exp ( jcot) is not a good function, this definition does not guarantee that F(a>) is an ordinary function. Indeed it may be shown [1] that sometimes F(co) is a generalized function. These rather abstract ideas may now be used to make a fresh attempt at finding the Fourier transform of the unit step function. As a first step it is useful to introduce a generalized function version of the ordinary unit step function u(t), and this requires an appropriate regular sequence of functions f n (t) to be constructed. Since lim exp ( tin) = 1 n-»oo

5 632 B. L. Burrows and D. J. Colwell one possible model is Jexp( tjn) fjso \h n (t) t<0 provided the as yet undeclared form of h a (t) satisfies such conditions as are necessary to ensure that/ n (f) is a good function. Thus, for example, while it is clearly necessary to require that it must also be ensured that/ n (0 is differentiable at f = 0. Full details of an appropriate construction for h n (t) are given in the Appendix, but some readers might prefer to proceed using the resulting form for / (<) which has been sketched in the Figure. In particular, the form taken by f n (t) at t= \jn and t = 0 should be noted. Although it is not immediately clear in the Figure, at these points f n (t) and all its derivatives may be shown to be continuous. The form which has been constructed for h n (t) ensures that -f-. h n(t)exp(-i(ot)dt tends to zero as n-*co. Thus to obtain the Fourier transform off n (t), and thereby the transform of u(t), it is now only necessary to consider the integral F R exp ( tin) exp ( jcot) dt (co)-rj 0 1/f! 1 CO say.

6 Fourier transform of the unit step function 633 Hence the required Fourier transform is the generalized function F(co) such that, for an arbitrary good function G(a>), /* + co rf*oo J -oo F(co)G(co) dftj=lim n->oo J -oo /*+0O F n (co) G(w) dco f + OO = lim A n {co)g(a))d(o+lim,(co)g(co)dco+lim B n (co) G(co) dco (1) n-»ooj oo n-*oo J oo It is convenient to consider the integrals involving A n (co) and B n (co) separately. The A n (co) integral may be written in the form P + OO f + 00 lim ^n(co)g(0)dco+lim ^n(co) (G(co) - G(O)) dco n-»oo J -oo n-*ooj - oo = lim G(0) tan" 1 (nco) +0 * <*> L J-oo = G(O)TC n5(co) G(co) dco J -00 using the definition of the generalized function <5(co). Since B n {oi) is odd, on introducing the even and odd parts of the good function G(co), the integral in equation (1) involving B n (io) may be written as r r + co p + co ") lim \ B n (co) G even (co) dco + B n (co) G odd (co) dco \ B-00 (.J-00 J-00 J (O = lim 2 n-oo J J -oo(l/w As it is not obvious whether this integral will exist when w-»oo, it is useful to note that, using Taylor's theorem for some co*. Then it is clear that lim I n-»oo J B n (w)g(co)dco -cog' odd (co*)dco exists and is equal to r + n j / + «j J-ao)C>> J-ooJ» Gathering the results achieved so far together, equation (1) may be rewritten as f + c lim F n ((o)g(co)d(o= [TzS(co)+ )G(a))d(o JV W

7 634 B. L. Burrows and D. J. Colwell Hence it may be deduced that the Fourier transform of u(t) is given by the generalized function The question of the interpretation of this generalized function still remains. In particular, can the second term be treated as an ordinary function? The ordinary interpretation depends on whether or not / + «j is zero. Hence, as it may readily be seen that n(co)dco limj f i-g CTM ( Q >)dw+ I ±G cycn the function 1/ja) may be treated as an ordinary function, provided the Cauchy principle value interpretation is applied to the integral (1). Appendix In this Appendix an appropriate definition for h n (t) will be introduced and investigated. The definition will make use of the good function s(y)= and it should be noted that this function is zero at y= +1. Furthermore all its derivatives are also zero at y = +1. The definition adopted for h n (t), after some trial and error, was where This definition ensures that I 0 1 ^t t< 1/n n f -r-l/n s(2nx+\)dx Also hjo) = 1, ensuring that/ B (/) is continuous at t = 0. Similarly there is continuity at

8 Fourier transform of the unit step function 635 In the interval 1 /«< t -1 P 1 Kit)= - exp (-1In) s(2nx +1) d*+ exp (- t/n) s(2nt +1) nj± J -l/n " Hence h' n ( 1 /«) = 0 and AJ,(O) = 1 In, so that/j,(o is continuous at both t= l/n * = 0. Similarly, differentiating >w times (wi= 1,2,...) it may be shown that and /# >(-l/n) = 0 and *C(0)=-^-exp (-*/«) at, =0 so that/h"(i) is continuous and differentiable for all m. Hence it may now be concluded that the sequence of functions f n (t) is an appropriate regular sequence. Reference [1] LIGHTHILL, M. J., 1980, An Introduction to Fourier Analysis and Generalised Functions (Cambridge University Press).