1.1 1-D Spectra via Method of Stationary Phase

Transcription

1 1.1 1-D SPECTRA VIA METHOD OF STATIONARY PHASE D Spectra via Method of Statioary Phase Now we cosider a differet approximatio of the Fourier trasform that is valid for certai 1-D fuctios at large values of ξ. The process is a applicatio of the method of statioary phase, which was developed by Lord Kelvi i the 1800s to solve itegrals ecoutered i the study of hydrodyamics. I tur, this is a variatio of the method of steepest descets for evaluatig path itegrals of complex fuctios. The method of statioary phase provides useful estimates of itegrals of oscillatig fuctios, ad thus of itegrads with imagiary-valued expoets. This method is particularly applicable to superchirp fuctios e ±iπx, for which o closed form of the spectrum has bee derived. The results obtaied for these fuctios will be applied i several cotexts later i the book. More detailed descriptios are available i Erdelyi (1956) ad Copso (1965). The goverig priciple behid the method of statioary phase will be itroduced by example. Cosider a itegral of the geeral form: I [k] r [x] e ik µ[x] dx (1) where r [x] ad µ [x] are real-valued fuctios ad k is a selectable real-valued parameter. The expoetial fuctio e ik µ[x] oscillates at a rate that depeds o both k ad the fuctioal form of µ [x]. Ifk is large, the rate of oscillatio of the expoetial term must also be large i all regios of the domai where µ [x] 6 0. I such cases, the cotributio to the oscillatig fuctio to the area will be small i ay regio where the expoetial term oscillates more rapidly tha the variatio of r [x], because the areas of the adjacet positive ad egative lobes will approximately cacel. Coversely, i those regios of the domai where k µ [x] is small, the amplitude of the expoetial term will approximate the uit costat. The area of those regios will be determied by the width of this regio of the domai ad the amplitude of the real-valued fuctio r [x]. The real ad imagiary parts of a sample itegrad i these two regios are show i Figure 1c,d. This shows that the itegral i Eq.(1) may be estimated by evaluatig the itegrad oly i those regios where the expoetial term oscillates slowly, i.e., wherever the derivative of the phase fuctio is approximately 0, dµ dx 0; these are the statioary poits of µ [x]. The semi-ifiite itegrals of the real ad imagiary parts over the domai <x a are show i Figure 1e,f. These illustrate that the primary cotributio to the area i both cases arise from the itegrad i the regio of the statioary poit. The example i Figure 1 also shows that the requiremet that r [x] be real valued i Eq.(1) creates o problem whe evaluatig the itegral of a complex-valued fuctio, because the liearity of itegratio allows the itegrals of the idividual parts to be

2 Figure 1.1: Priciple of the Method of Statioary Phase: (a) real-valued modulatio r [x]; (b) phase fuctio m [x], which is approximately statioary i viciity of x ; (c), (d) Real ad imagiary parts of r [x] e iµ[x], showig Z rapid oscillatios away from x statioary poit; (e), (f) Real ad imagiary parts of r [α] e iµ[α] dα, showigthat the primary cotributio to the area is from r [x] i the viciity of the statioary poit.

3 1.1 1-D SPECTRA VIA METHOD OF STATIONARY PHASE 3 performed separately ad summed. I [k] µ (<{r[x]} + i {r [x]}) e ik µ[x] dx µ <{r [x]} e ik µ[x] dx + i {r [x]} e ik µ[x] dx () Uder some circumstaces, other ad more subtle aspects of the method of statioary phase require special treatmet. Sice these occur rarely i the cases of greatest iterest i imagig, they will be metioed oly i passig ad ot cosidered i detail. Iterested readers should cosult sources that cocetrate o this subject, especially the work of Erdelyi ad of Friedma. To illustrate use of the method of statioary phase i Fourier aalysis, cosider the Fourier trasform of a 1-D complex-valued fuctio f [x] expressed i terms of its magitude f [x] ad phase Φ{f [x]}: F 1 {f [x]} f [x] e πiξx dx f [x] e iφ{f[x]} e πiξx dx f [x] e i(φ{f[x]} πξx) dx (3) Note that the form of this itegral is somewhat less geeral tha that i Eq.(1) because the modulatio fuctio f [x] is ot oly real valued, but also oegative, whereas r [x] i Eq.(1) may be egative. TheFourieritegraliEq.(3)mayberewritteitheformofEq.(1)bydefiig: µ [x] 1 Φ {f [x]} πx () ξ ad substitutig the spatial frequecy ξ for the parameter k ad f [x] for r [x]: F [ξ] f [x] e iξ µ[x] dx (5) Note that the phase fuctio µ [x] icludes a factor of ξ 1, but this is reasoable sice ξ is a parameter (rather tha a variable) i the itegrad. Thus the result will be a fuctio of this parameter, as it must be. If the phase fuctio µ [x] has o sigularities (i.e., if all of its derivatives are fiite), the µ [x] may be expaded ito a Taylor series about ay arbitrary locatio x 0 :

4 Ã µ [x] µ[x 0 ]+ (x x 0 ) dµ dx xx0! + Ã (x x 0 ) d µ dx xx0! + µ[x 0 ]+(x x 0 ) µ 0 [x 0 ]+ (x x 0) µ 00 [x 0 ] + + (x x 0) µ () [x 0 ]+ (6)! where the commo multiple-prime shorthad otatio for derivatives has bee substituted i the secod expressio for simplicity. We ow select x 0 to be a statioary poit of the phase fuctio, so that µ 0 [x 0 ] 0. For ow, assume that µ [x] has oly oe such statioary poit; extesio to cases with multiple statioary poits is straightforward ad will be cosidered later. The first-order term i the Taylor series vaishes: µ [x] µ [x 0 ]+0+ (x x 0) µ 00 [x 0 ]+ (x x 0) 3 µ 000 [x 0 ]+ (7)! 3! ad the Fourier itegral i Eq.(3) may be rewritte i terms of this series: Z " Ã!# + F [ξ] f [x] exp +iξ µ [x 0 ]+µ 00 (x x 0 ) [x 0 ] + dx Z Ã " Ã!#! + f [x] exp [+iξ µ [x 0 ]] exp +iξ µ 00 [x 0 ] (x x 0) dx Z Ã " Ã!#! + + Y f [x] exp [+iξ µ [x 0 ]] exp +iξ µ () [x 0 ] (x x 0) dx (8) Of course, the zero-order term i the Taylor series is a costat with respect to x ad maybeextractedfromtheitegral: Z Ã " Ã!#! + + Y F [ξ] exp[+iξ µ [x 0 ]] f [x] exp +iξ µ () [x 0 ] (x x 0) dx (9) If the spatial frequecy ξ is assumed to be sufficietly large that the expoetial term oscillates may times over the scale of variatio of f [x], the the itegral may be approximated to evaluate the spectrum. Uder this coditio, the magitude f [x] cotributes sigificat area to the Fourier itegral oly i the viciity of the statioary poit x 0, thus allowig the varyig magitude f [x] to be approximated by the costat f [x 0 ]. I additio, the ifiite limits of the itegral may be chaged to fiite limits i the viciity of the sigle statioary poit. Fially, oly the first ozero term i the Taylor series of order two or larger is sigificat because (x x 0 ) << (x x 0 ) for 3 whe x is i the eighborhood of x 0. The resultig asymptotic form of the

5 1.1 1-D SPECTRA VIA METHOD OF STATIONARY PHASE 5 Fourier itegral is: ˆF [ ξ >> 0] f [x 0 ] exp [+iξ µ [x 0 ]] Z x0 + x 0 " # exp +iξ µ 00 [x 0 ] (x x 0) dx (10) where is a small positive umber. Note that this expressio assumes that µ 00 [x 0 ] 6 0. If the derivatives of secod or larger order of µ [x] also are zero at x 0, the the derivative with the smallest order (other tha 1) that does ot vaish is used i the approximatio. The remaider of the derivatio must be appropriately modified; details are preseted by Friedma. I words, Eq.(10) demostrates that the Fourier itegral of a oscillatig fuctio that icludes a sigle statioary poit i the ifiite domai may be evaluated as the product of some easily evaluated costats ad the fiite itegral of a quadratic-phase expoetial. Sice the area of the quadratic-phase factor also is cocetrated i the viciity of the statioary poit, little additioal error is icurred by substitutig its area over the ifiite domai i Eq.(10). Z " # x0 + exp +iξ µ 00 [x 0 ] (x x 0) Z " Ã!# dx x 0 + exp +iξ µ 00 [x 0 ] (x x 0) dx (11) I short, the fiite itegral is approximated by the total area of the quadratic-phase expoetial, which is easy to evaluate by chagig the itegratio variable to u 1 µ00 [x 0 ](x x 0 ) ad applyig the cetral ordiate theorem from Eq.(9.110). The area of the quadratic-phase term is: Z " Ã!# Ãs! x+ Z exp +iξ µ 00 [x 0 ] (x x 0) π u+ dx exp +iπu du ξµ 00 [x 0 ] x Ãs π ξ µ 00 [x 0 ]! u h exp +i π i This result is substituted ito Eq.(10) to obtai the approximatio for the spectrum that is valid i those cases where the phase of the itegrad of the Fourier trasform is statioary at a sigle coordiate: Ãs! ˆF [ ξ >> 0] π f [x 0 ] ξ µ 00 [x 0 ] h exp +i π i exp [+iξ µ [x 0 ]] (1) (13a)

6 6 This complex amplitude may be expressed as magitude ad phase: s ˆF [ ξ >> 0] π f [x 0 ] ξ 1 µ 00 [x 0 ] o Φ ˆF [ ξ >> 0] +ξ µ[x0 ]+ π (13b) (13c) The approximatio for the more geeral form of the Fourier itegral with a real-valued bipolar modulatio is obtaied by a direct substitutio of the real-valued modulatio r [x] for the oegative magitude f [x] : s f [x] r [x] e iφ{f[x]} ˆF [ ξ >> 0] π r [x 0 ] h+i ξ µ 00 [x 0 ] exp π i exp [+iξ µ [x 0 ]] (1) If there are two (or more) poits of statioary phase i the itegrad, the approximatio is evaluated at each ad the results are summed to obtai the asymptotic solutio. A example of such a case is cosidered i the ext sectio. Obviously, a itegral i the frequecy domai similar to that i Eq.(3) may be costructed for the iverse Fourier trasform, which will allow the asymptotic evaluatio of f [x] from a spectrum with a oscillatig expoetial. Such a developmet will be used i Chapter Examples of Spectra via Statioary Phase Uit-Magitude Liear-Phase Expoetial The formulatio i Eq.(1) will be used to evaluate the asymptotic form of the Fourier itegral for a few phase fuctios. Cosider first the spectrum of the liear-phase expoetial f [x] e +πiξ 0x. The Fourier itegral is: F 1 [ξ] 1[x] e +πiξ 0 x e πiξx dx e πi(ξ ξ 0)x dx (15) which may be recast ito the form of Eq.(5) by chagig the itegratio variable ξ to ζ (ξ ξ 0 ) ad idetifyig the phase fuctio µ [x] to be πx: F 1 [ζ] e +iζ µ[x] dx e +iζ (πx) dx (16)

7 1.1 1-D SPECTRA VIA METHOD OF STATIONARY PHASE 7 Thederivativeofthephasefuctioisthepositivecostatµ 0 [x] π, which cofirms the observatio that the itegrad oscillates at the same rate over the etire domai; i other words, there is o poit of statioary phase. Sice the criterio for statioary phase is ot fulfilled, the asymptotic solutio for F [ξ] does ot exist. This result demostrates that the method of statioary phase may be applied oly if the phase of f [x] icludes terms of order or higher Modulated Quadratic-Phase Expoetial Perhaps the most useful applicatio of the method of statioary phase to Fourier aalysis is the determiatio of the asymptotic form of the spectrum of a fuctio f [x] with a quadratic phase scaled by the factor α ad modulated by f [x] : f [x] f [x] e +iπ ( x α) (17) The scale factor α has uits of legth to esure that the expoet is dimesioless. The Fourier itegral may be recast ito the form of Eq.(5): F [ξ] f [x] e +iπ x α dx ξx The phase fuctio ad its derivatives are easy to evaluate: f [x] e +iξ πx α πx ξ dx (18) µ [x] πx α ξ πx (19a) µ x µ 0 [x] π α ξ 1 (19b) µ µ 0 x0 [x 0 ]0π α ξ 1 (19c) x 0 +α ξ (19d) Note that x 0 +a ξ i Eq.(19d) has the required dimesios of legth for a coordiate i the space domai. The phase fuctio ad its derivatives evaluated at this statioary poit are: µ[x 0 ] πα ξ α ξ πα ξ πα ξ (0a) µ 0 [x 0 ]0 (0b) µ 00 [x] π α ξ µ00 [x 0 ] π α ξ (0c) µ () [x 0 ]0for 3 (0d)

8 8 These results are substituted ito the statioary-phase solutio i Eq.(13) to estimate the amplitude of the spectrum at spatial frequecies distat from the origi: s π π ˆF [ ξ >> 0] f [x 0 ] ξ µ 00 [x 0 ] e+i e +iξ µ[x 0 ] f α ξ v u t ³ π e +i π e +iξ ( πα ξ) ξ π α ξ f α ξ α e +i π e iπα ξ α f α ξ e +i π e iπα ξ (1) The validity of this expressio may be cofirmed by by settig the chirp rate α to uity ad f [x] 1to produce a umodulated chirp. The result may be compared to the kow spectrum of the quadratic-phase expoetial i Eq.(9.9): F ª h 1 exp +iπx ˆF [ ξ >> 0] 1 [ξ] exp +i π i exp iπξ F [ξ] () I the case of a uit-magitude quadratic-phase fuctio, the asymptotic ad exact forms of the spectrum are idetical. This is because µ () [x] 0for >3, ad thus o error i the phase fuctio is icurred by trucatig the Taylor series at the secod-order term. A more geeral quadratic-phase fuctio is obtaied by replacig the oegative modulatio f [x] with a real-valued bipolar fuctio m [x] after scalig by b ad traslatig by x 1 : x x1 f 3 [x] m e ±iπ ( α) x (3) b Eq.(1) may be applied directly to evaluate the asymptotic form of the spectrum: ˆF 3 [ ξ >> 0] α ξ x 1 α m b " ξ x1 # α α m b α e ±i π e iπα ξ e +i π e iπα ξ (a) The estimate of the phase trasfer fuctio of the modulated quadratic-phase filter is idetical to that of the umodulated sigal: o µ Φ ˆF 3 [ξ] π α ξ 1 (b) Eq.() has some very iterestig (ad perhaps uexpected) features. Note that the magitude of the spectrum is the same real-valued modulatio m exhibited by the space-domai fuctio, after traslatig ad scalig by the respective factors x 1 ad α

9 1.1 1-D SPECTRA VIA METHOD OF STATIONARY PHASE 9 b. Both of these factors have dimesios of reciprocal legth, as appropriate for α fuctios i the frequecy domai. I words, icreasig the width parameter of the modulatio of f 3 [x] icreases the the width parameter of the spectrum estimate ˆF 3 [ξ] by a proportioal factor. Similarly, a traslatio of the modulatio of f 3 [x] produces a proportioal traslatio of the modulatio of ˆF 3 [ξ]. These features of the spectrum may seem to violate the scalig ad shiftig theorems of the Fourier trasform, but i fact are artifacts of the quadratic-phase fuctio that will be discussed i more detail i Chapter 17. Because decreasig the scale factor α of the quadratic phase has the effect of icreasig the oscillatio rate of f 3 [x], this coditio improves the accuracy of statioaryphase solutio at a particular spatial frequecy. The predictio of Eq.() will be tested for the specific caseofascaledad traslated SINC fuctio modulated by a quadratic-phase fuctio with α : x f [x] SINC e +iπ ( x ) (5) which is graphed both as real-ad-imagiary parts ad as magitude-ad-phase i Figure ; ote the bipolar modulatio preset i f [x]. The statioary-phase solutio for the spectrum is obtaied by direct substitutio ito Eq.(): ˆF [ ξ >> 0] ξ 1 SINC e +i π e iπ(ξ) (6) 1 where the scale factor α 1ithe argumet of the SINC fuctio corrects the dimesios of both the traslatio ad scale factor. The spectrum is compared to a discrete calculatio i Figure 3. The differeces betwee the approximatios ad the computed spectra are more apparet i the magified views for 1 ξ cycles per uit legth. Note i the approximate magitude spectrum is zero at ξ 1,whilethe exact computed spectrum is ot Statioary-Phase Approximatio for Symmetric Superchirps We ow briefly cosider asymptotic forms of the spectra of umodulated superchirp fuctios that were itroduced i Eq.(6.13). We defied two flavors of superchirps that differ i behavior for odd values of the order : the itrisically symmetric form cos [π x ]±i si [π x ] ad the Hermitia variety cos [π x ]±isgn[x] si[π x ]. We cosider the symmetric form first. Based upo the symmetry argumets developed i Chapter 9.1, we expect the spectra of these complex-valued ad symmetric fuctios to be complex ad symmetric. The Fourier itegral is: o F 1 e +iπ α x ³e +iπ α x e πiξx dx (7) The phase fuctio ad its first derivative are: µ [x] + π ³ x πx (8) ξ α

10 10 Figure 1.: f [x] SINC h x exp +iπ x i as (a) real part; (b) imagiary part; (c) magitude; (d) phase. This fuctio will be used to demostrate the approximatio of the Fourier trasform via the method of statioary phase.

11 1.1 1-D SPECTRA VIA METHOD OF STATIONARY PHASE 11 Figure 1.3: The statioary phase approximatio to the Fourier trasform of f [x] SINC h x exp +iπ i x. The approximatio is ˆF [ξ] SINC ξ + 1 exp iπ (ξ) exp +i π (a) real part; (b) imagiary part; (c) magitude; ad (d) phase. The differeces betwee the approximatio ad the computed spectra are more visible i the magified views for 1 ξ cycles per uit legth.

12 1 µ 0 [x] + π αξ ³ x α 1 π (9) It is coveiet to cosider the cases of eve ad odd values of separately. Whe is eve, the expoet 1 i µ 0 [x] is odd. The poit(s) of statioary phase (if ay) are the solutios to µ 0 [x 0 ]0, which are determied by the spatial frequecy ξ i the Fourier itegral: 1 1 µ αξ x 0 α (30) Note that x 0 has the required uits of legth. The statioary poits of a eveorder symmetric superchirp are the real-valued solutios of Eq.(30), ad thus are proportioal to odd-order roots of the selected spatial frequecy ξ. For example, if, the statioary poits are proportioal to the real-valued solutios of ξ 1 3.For ξ>0, a sigle real-valued solutio for ξ 1 3 x 0 exists ad it is positive. Whe ξ<0, the sigle real-valued solutio for x 0 is egative. Whe is odd, the phase fuctio of the symmetric superchirp may be decomposed ito two fuctios, oe each for positive ad egative x: µ [x 0] + π ³ + x πx (31a) ξ α µ[x 0] + π ³ x πx (31b) ξ α The first derivatives of the phase i these two regios are: µ 0 [x 0] + π ³ αξ + α 1 x π (3a) µ 0 [x 0] + π ³ αξ α 1 x π (3b) Because is odd, 1 is eve. For positive values of x, the statioary poit(s) must satisfy: µ 1 αξ 1 (x 0 ) + +α (33a) Because it is proportioal to a eve-order root of ξ, this expressio is real valued oly for ξ>0. I other words, the statioary-phase estimate of the spectrum of a odd-order superchirp is ozero oly for positive frequecies. Similarly, the statioary poit for egative x must satisfy the coditio: (x 0 ) α µ 1 αξ 1 (33b) The real-valued eve-order root of ξ agai exists oly for ξ>0, ad thus (x 0 ) < 0. I short, the itegrad has a sigle statioary poit if is symmetric. A example isshowifigurefor, α 1uit, ad ξ +cycles per uit legth. The statioary poit is located at x

13 1.1 1-D SPECTRA VIA METHOD OF STATIONARY PHASE 13 Figure 1.: Statioary poit of exp [+iπx ] exp [ πix ]: (a) real ad imagiary parts of exp [+iπx ]; (b) real ad imagiary parts of exp [ πi x ]; (c) real part, imagiary part, uit magitude, ad phase of exp [+iπx ] exp [ πi x ], showig the statioary poit at x

14 1 It may be coveiet to recast the itegral i Eq.(7) ito the form of Eq.(5) by chagig variables: where: ³e +iπ x α e πiξx dx x αu e +iv(u u) ³ v π 1 ³ v π 1 αdu (3a) (3b) ξ 1 ³ v 1 1 (3c) α π α The phase fuctio ad its derivatives are easily evaluated i terms of u ad v: µ [u] u u (35a) µ 0 [u] u 1 1 u 0 ( 1) 1 (35b) µ 00 [u] ( 1) u (35c) µ[u 0 ](1 ) ³ ( 1 ) (35d) µ 00 [u 0 ]( 1) ³ ( 1) (35e) Substitutio of these terms ito the statioary-phase solutio of Eq.(13) yields the approximate solutio for the spectrum that is valid for large ν : Ãs! π ˆF [ v >> 0] vµ 00 [u 0 ] e +i π e +iv µ[u 0 ] α µ ³ v 1 π whichmaybewritteithedesiredformbysubstitutigtheformofν from Eq.(3b). The statioary-phase solutio for the spectrum of the umodulated symmetric superchirp fuctio is the complex-valued symmetric fuctio: µ ˆF [ξ] α ( 1) 1 µ α ξ ( ) h exp +i π i à " exp +iπ (1 ) µ α ξ (36) ( 1) #! (37) To check (if ot cofirm) the validity of this expressio, substitute ad compare to the kow spectrum of the quadratic-phase fuctio: o F 1 e +iπ ( α) x ˆF [ξ] α 1 h 1(α ξ ) 0 exp +i π i exp iπ (α ξ ) h α exp +i π i exp iπ (α ξ ) (38) which we kow to be the correct spectrum for all values of ξ from the applicatio of the kow trasform ad the scalig theorem.

15 1.1 1-D SPECTRA VIA METHOD OF STATIONARY PHASE 15 It is perhaps istructive to examie the fuctioal forms of the magitude ad phase of the statioary-phase solutio to the superchirp spectrum. The magitude spectrum is the eve fuctio: s ˆF µ α ξ ( [ξ] (39a) α ( 1) ) The fact that the magitude spectrum is costat for cofirms the observatio that chirp fuctios are composed of siusoids with idetical magitudes at all spatial frequecies. However, for >, the magitude of the spectrum is ot costat, but rather decreases with icreasig frequecy. The rate of declie i the magitude spectrum is ξ 0 1, ξ 1, ξ 1 3,ad ξ 3 8 for 5. I words, the magitude falls off more quickly with icreasig order. This behavior is cosistet with the observatio that the superchirp amplitude is approximately costat over larger regios ad that the spatial frequecy chages more quickly with x as the order is icreased. The phase spectrum of the symmetric superchirp is: Ã oo µ! Φ F 1 e +iπ ( α) x 1 α ξ 1 π +(1 ) (39b) which varies as ξ, ξ 3, ξ 3,ad ξ 5 for 5, respectively. I words, the variatio i phase of the spectrum over ξ decreases as the order of the superchirp is icreased. Examples are show i Figure 5. The iitial phase of the statioary-phase approximatio of the spectrum of all superchirps of the form e +iπ x is + π radias, though we kow from the momet calculatio i Eq.(39) that the iitial phase actually is + π radias. This agai remids us that the statioary-phase calculatio is valid for ξ >> Spectra of Hermitia Superchirp Fuctios via Statioary Phase The symmetry argumets of Chapter 9.1 ad the expasio i terms of momets i Eq.(8) demostrate that the spectra of all odd-order Hermitia superchirps are real valued. The approximate forms of these spectra obtaied from the method of statioary phase demostrate the variatios required whe the itegrad has more tha oe statioary poit. We cosider the case for odd for simplicity. The phase fuctio of the Fourier itegral of the Hermitia fuctio is: µ µ [x] + π ³ x πx (0a) ξ α

16 16 Figure 1.5: Magitude ad phase of spectra of symmetric superchirps f [x] exp [+iπ x ] for, 3,, 5 by discrete computatio ad the approximatio by the method of statioary phase from Eq.(39). Note that the magitude falls off more rapidly ad the phase less rapidly with ξ as icreases.

17 1.1 1-D SPECTRA VIA METHOD OF STATIONARY PHASE 17 ad its first ad secod derivatives are respectively: µ µ 0 [x] + π ³ x 1 π αξ α µ µ 00 [x] + π ³ x ( 1) α ξ α (0b) (0c) The statioary poit(s) (if ay) are the solutios to µ 0 [x 0 ]0: x 0 α µ 1 αξ 1 (1) Sice is odd for all Hermitia superchirps, the ( 1) is eve. I words, the coordiate of the statioary poit is proportioal to the real-valued eve-order roots of the spatial frequecy ξ where the Fourier itegral is evaluated. Whe ξ is large ad positive (ξ >>0), there are two real-valued solutios for x 0 with idetical magitudes ad opposite sig, i.e., at x 0 ± x 0. The cotributios from the two statioary poits must be added to evaluate the asymptotic form of the Fourier itegral for positive frequecies. The situatio is qualitatively differet for ξ<0. The coordiates of the statioary poits are proportioal to eve-order roots of egative umbers, which have NO realvalued solutios; there are o statioary poits if ξ<0, ad thus the statioary-phase approximatio of the Fourier itegral is zero for all ξ<<0. Examples of the itegrad of the Fourier itegral of the cubic Hermitia superchirp are show i Figure 6. It remais to evaluate the phase fuctio ad its secod derivative at the two statioary poits of the odd-order Hermitia superchirp for ξ>0. These may be iserted separately ito Eq.(13) ad summed to evaluate the approximatio. The positive ad egative statioary poits are labelled x + ad x, respectively: µ αξ 0 <x + +α µ αξ 0 >x α (a) (b) where ξ, α, ad are all real-valued positive quatities. The correspodig phase fuctios are obtaied by substitutio of these derivatives ito Eq.(0a). The phase fuctio evaluated for x + is: Ã π αξ µ [x + ](+1) 1 ξ π µ αξ π µ µ 1 α 1 α 1 1 ξ π ξ 1 1! 1 (3a) Sice ξ is assumed to be large ad is positive, the firsttermmustbelargertha

18 18 Figure 1.6: Statioary poits of the product exp +iπ x exp [ πix], ( 3, 3 α, ξ +1). There are two statioary poits located at x 0 ±q 1 3 h ±6.53: (a) real ad imagiary parts of exp +iπ i x 3 ; (b) real ad imagiary parts of exp [ πix]; (c) real part, imagiary part, magitude, ad phase of the product, showig the two statioary poits. The cotributios from the two statioary poits to the Fourier itegral are complex cojugates, ad so the imagiary parts cacel.

19 1.1 1-D SPECTRA VIA METHOD OF STATIONARY PHASE 19 the secod, which meas that µ[x + ] must be positive. The phase fuctio evaluated at the egative statioary poit is: Ã π αξ µ [x ]( 1) 1 ξ Ã π! Ã π µ µ α 1 α ξ π µ! 1 αξ 1! 1 1 ξ 1 1 µ[x + ] (3b) Note that the factors e +iξµ[x 0] evaluated at these two statioary poits are complex cojugates. The correspodig solutios for the secod derivative are: µ 00 [x + ] π α ξ µ 00 [x ]+ π α ξ ( 1) µ αξ ( 1) µ αξ 1 (a) 1 µ 00 [x + ] (b) which also have the same magitude ad opposite sig; the secod derivative evaluated at the egative statioary poit x is positive, while that at x + is egative. Substitutio of these results ito Eq.(13) yield the asymptotic form of the Fourier itegral: Ã s s! ˆF [ξ >>0] e +i π e iξµ[x π +] ξµ 00 [x + ] + π eiξµ[x ] ξµ 00 [x ] r Ã! π π e iξµ[x ] ξ e+i p µ 00 [x ] + e+iξµ[x ] p +µ 00 [x ] s µ π π e iξµ[x ] e +iξµ[x ] ξµ 00 [x ] e+i s µ π π e iξµ[x ] e +iξµ[x ] ξµ 00 [x ] e+i + +i +1 s µ π e +i π e iξµ[x ] e +iξµ[x ] + ξµ 00 [x ] e +i π e +i π e i π s π ³e i (ξµ[x ]+ π ξµ 00 ) + e +i(ξµ[x ]+ π ) [x ] s π ³ h cos ξµ[x ξµ 00 ]+ π i [x ] (5)

20 0 Figure 1.7: Fourier trasforms of Hermitia superchirp fuctios compared to approximatios from the method of statioary phase: (a) F{exp [+iπx 3 ]}; (b) F{exp [+iπx ]}. After substitutig the expressios for µ ad µ 00 from Eq.(3) ad Eq.(), the asymptotic solutios for the spectra of odd-order Hermitia superchirp are obtaied: F 1 e +iπ ( x α) o for odd αq ( 1) αξ h cos π (1 ) αξ i 1 π if ξ>>0 0 if ξ<<0 (6) I words, the asymptotic spectrum of the odd-order Hermitia superchirp fuctio is real valued ad also is zero for egative frequecies. The computed forms for 3 ad 5are show i Figure 7. The graph of Eq.(5) i Figure 6 shows that the momet expasio of the spectrum for 3has ozero amplitude for small egative frequecies. This behavior is ot modelled by Eq.(6) because it is valid oly for large ξ. Note that the rate of oscillatio of the spectrum with ξ decreases with the order.