Orthogoal Gaussia Filters for Sigal Processig Mark Mackezie ad Kiet Tieu Mechaical Egieerig Uiversity of Wollogog.S.W. Australia Abstract A Gaussia filter usig the Hermite orthoormal series of fuctios is developed. The filter is compared with a similar filter usig the Hermite- Rodriguez series o Doppler radar sigals. The results idicate that a more compact filter ca be achieved with the Hermite series compared to the Hermite-Rodriguez series. Itroductio Figure explais the operatio of a classical filter, sometimes referred to as a movig average filter. The filter itegrates oly the oisy fuctio, f ( τ, eclosed withi the widow fuctio, w ( t τ, at the locatio t. A average value, fˆ, of the oisy fuctio at t is obtaied. This average value is the correlatio of the oisy fuctio with the widow fuctio. With a Gaussia widow fuctio this is: fˆ f ( τ w( t τ where = dτ ( t τ = f ( τ exp dτ ( σ π σ t ( w t = exp ( σ π σ This itegratio ca be approximated by a discrete summatio i digital applicatios, or as a Mote Carlo itegratio i statistics, which is referred to as Kerel regressio. I this paper we ivestigate the computatio of this correlatio usig a orthogoal series. A large umber of well-kow orthogoal series occur icludig Fourier, Legedre, ad the Tchebychev series. While these are useful i certai applicatios, i this work we choose the Hermite ad Hermite-Rodriguez orthogoal series, which are based o the Gaussia fuctio because this simplifies the calculatio of the correlatio. The layout of this paper is as follows. Sectio explais how the correlatio is calculated usig orthogoal fuctios. Sectio 3 describes the spatial ad frequecy badwidth of the orthogoal series, which is importat i applyig the series to real life problems. Sectio 4 applies the series to demodulate Doppler radar sigals. Orthogoal filters. Hermite series Cosider the approximatio of the oisy fuctio by a Hermite series. The series approximates the fuctio by a fiite expasio of Hermite fuctios, o a iterval { } f f = ah { } = t (3 where { a } are a set of suitably chose weights ad h are the Hermite activatio fuctios o the iterval {, } which satisfy the orthoormal coditio for m = h hm dt = (4 for m The first few Hermite fuctios i this series are h h ( ( t t = exp (5 4 π t exp = 4 π ( t (6 The remaider may be determied from the recurrece relatio = (7 + + t h h h+ 7
Sice the fudametal, h, is the Gaussia fuctio it ca take the place of the filter fuctio. The correlatio itegratio of equatio ( is the approximately equal to the correlatio of the Hermite series with the fudametal Hermite fuctio. ˆ dτ (8 fˆ = ah ( τ h ( t τ f = I order to evaluate this expressio, the correlatio betwee the Hermite fuctios of differet order, is required. This correlatio is give by 3 h m ( h ( t τ dτ = l ( t τ ( m m m (9 where l m is a ormalized associated Laguerre fuctio. Usig this result i equatio (8 we obtai = al ( t fˆ ( = ote that oly those associated Laguerre fuctios of subscript { m = } cotribute, thus greatly simplifyig the result. The associated Laguerre fuctios of subscript are l t t e = (! The usefuless of this expasio is that by fittig a Hermite series to the iput fuctio, oe also immediately obtais the weights of the Laguerre series, { a }, which is the correlatio of the iput with a Gaussia fuctio.. Hermite Rodriguez Fuctios Hermite-Rodriguez fuctios 4,5 are similar to the Hermite fuctios except that a Gaussia widow modulates their amplitude. They are defied as hr ( ( 4 t t h e = π ( where h is a orthoormal Hermite fuctio. The fudametal Hermite-Rodriguez fuctio is also a Gaussia fuctio but of differet width to the fudametal Hermite fuctio. The fudametal is ( hr = exp t (4 The others may be determied usig the recurrece relatio (equatio (7 for the Hermite fuctios ad multiplyig by the Gaussia fuctio. Like the Hermite series, a simple expressio also occurs for the correlatio of the Hermite-Rodriguez fuctios. The correlatio of two Hermite- Rodriguez fuctios is give by 5 hr hr = ( + m! hr ( t m + m + m! m! (5 ote that the scale is reduced ad the order of the Hermite-Rodriguez fuctio icreased by the correlatio operatio. Usig equatio (5, the derivatio of a Gaussia filter with the Hermite-Rodriguez fuctios is similar to the filter derived usig the Hermite series. 3 Method 3. Sigal duratio ad badwidth Applicatio requires the duratio ad badwidth of the orthogoal series to be matched to the sigal beig modeled. The Hermite series behaves as a widow i the time domai. Outside this widow the Hermite fuctios decay expoetially, limitig the effective rage over which a fuctio may be approximated to withi the widow. The width of the Hermite series widow is equal to the duratio of the largest order Hermite fuctio, h, occurrig i the series. The useful rage of applicatio of the Hermite series iterpolatio is the t + (6 Where the right had side, equal to the duratio of the Hermite fuctio of order { }, may be determied via the Quatum mechaic solutio of the Harmoic oscillator as the locatio where the oscillator eergy becomes egative 6. The Fourier trasform 3 of a Hermite fuctio is { h } j h ( ω F (7 I view of this isomorphic Fourier trasform, a similar widowig effect occurs i the complex frequecy domai. The useful badwidth is 73
ω + (8 Together, the badwidth ad widow width of equatio ( ad ( defie the size,, of the eural series required to approximate a fuctio. Ulike the Hermite series, which icreases i duratio with the order { } of the fuctio, the Hermite-Rodriguez series is idepedet of. Istead it is limited i duratio by the Gaussia amplitude modulatio fuctio to the rage 5 t 3 (9 The Fourier trasform of the Hermite-Rodriguez fuctio is a associated Laguerre fuctio F { hr } ( j l ( ω = ( From this Fourier trasform it ca be show that that the badwidth i the frequecy domai icreases with the order { } of the fuctio accordig to ω + ( which is the same as for the Hermite fuctio (equatio (. 3. Scalig Applicatio to practical problems requires scalig of the orthogoal series. The procedure is illustrated for the Hermite-Rodrigues series. The scaled Hermite-Rodriguez series is obtaied by troducig the variable t t α ( Scalig chages the duratio ad badwidth of the Hermite-Rodriguez series to, respectively, t 3α (3 ad + ω (4 α Usig scaled Hermite-Rodriguez fuctios 5, the correlatio with the Gaussia fuctio is hr α γ (5 ( t α hr ( t β = hr ( t γ where α ad β are the scalig factors ad γ + = α β. 3.3 Optimisig the weights of the orthogoal series The weights of the Hermite ad Hermite-Rodriguez series were both obtaied usig the same method, which is described i this sectio for the Hermite series. For a cotiuous fuctio defied o {, }, the weights of the Hermite series are optimum 7 with respect to the mea square error (equatio (3 whe A = a h (dt t (6 We use a simple summatio, similar to Euler itegratio, give by i I ( ti h ( ti = = A a (7 i= where t is the itegratio step size which was fixed to the samplig rate. A feature of this type of itegratio is that it is also suitable for radomly sampled data. For radom data, this type of umerical itegratio geeralizes to Mote-Carlo itegratio. umerical itegratio is oly a approximatio to the aalytical cotiuous itegratio. I additio, the data most ofte ecoutered i practice is discrete, ofte corrupted with oise. To cope with these situatios the gradiet descet algorithm 8 was applied after the weights had bee estimated with the itegratio. Gradiet descet reduces the mea square error betwee the Hermite series approximatio ad the discrete data by successive iteratios of the followig algorithm A, = A a A h h k ew k + µ k (8 = where µ is the feedback costat chose i the rage. to.. 4 Applicatio to the demodulatio of Doppler radar sigals 74
The objective of this sectio is to compare the performace of the Hermite with the Hermite- Rodriguez series o a practical sigal processig problem. Figure (a shows the sigal received from the detector of a Doppler radar system. The carrier frequecy, which is proportioal to the velocity of the target, is removed by filterig to obtai the target rage. The frequecy spectrum of the Doppler sigal, from which the velocity may be determied, is give by the ew series formed by replacig the Hermite fuctios i the Hermite series iterpolatio of the Doppler sigal by their Fourier trasform (equatio (9. Removal of the carrier frequecy to obtai the rage of the target is achieved by takig the correlatio of the Hermite series with the Gaussia fuctio. The followig mathematical model of a Doppler sigal was ivestigated a ( + ( πft e t σ = cos (6 where the frequecy, f = Hz, ad the Gaussia width, σ = 3.. Simulated radom oise, with a uiform probability desity, of varyig stregth was added. A Hermite series of 5 elemets has a width of. secods ad a badwidth of. 5Hz, which is sufficiet to accurately iterpolate the Doppler sigal. Figure (a, (b, ad (c show the sigal, Gaussia filtered output ad frequecy spectrum respectively from the Hermite series iterpolatio cotaiig 5 elemets. The root mea square error (RMSE of the Hermite series iterpolatio of the Doppler sigal versus the umber of Hermite fuctios i the series is show i figure 3(a. For the purpose of compariso, the scalig parameters were chose so that the Hermite- Rodriguez series matches the Hermite series. With β =, the order { = }, Hermite-Rodriguez fuctio is matched exactly to the order { = }, Hermite fuctio, that is ( t h ( t hr =. Similarly, with α =.85, the Hermite- Rodriguez series is matched i duratio to the Hermite series. Figure 3(b, shows the root mea square error of the Hermite-Rodriguez iterpolatio as a fuctio of the umber of elemets of the series. More tha 4 Hermite-Rodriguez fuctios are eeded before the error drops to a sufficietly small value. The reaso for this is because the frequecy respose of the scaled Hermite- Rodriguez series is cosiderably smaller tha a Hermite series of the same duratio. Figure 4(a ad 4(b shows the frequecy respose of the Hermite series ad Hermite-Rodriguez series o a uit amplitude cosie with 5 ad 4 Hermite- Rodriguez fuctios respectively. The poor frequecy respose is uavoidable because a sufficietly large α must be chose to esure the series has sufficiet duratio to iterpolate the Doppler sigal. Figure 5 shows the output sigal to oise ratio of the Hermite series, = 5, ad the Hermite- Rodriguez series, = 4, versus the iput sigal to oise ratio. early idetical sigal to oise ratios are achieved, which is to be expected sice a idetical Gaussia correlatio fuctio was used, but the Hermite series is cosiderably more efficiet sice it achieves the same result with much fewer terms i the series. I view of their Gaussia widow, the Hermite-Rodriguez series may prove more suitable for correctly aalyzig the frequecy spectrum of sigals subject to glitches. Coclusio A Gaussia filter usig Hermite fuctios was developed. A compariso with a similar filter usig the Hermite-Rodriguez series favours the Hermite series because it has a greater badwidth for a series with the same umber of elemets. A compariso of the Hermite Gaussia filter with a Gaussia filter usig Fourier fuctios is uder ivestiagtio. Refereces ( G. Arfke, Mathematical methods for physicists, d editio, Academic Press, ew York, 97. ( G. G. Walter, Wavelets ad other orthogoal systems with applicatios, CRC Press, USA, 994. (3 M. R. Mackezie ad A. K. Tieu, Hermite Correlatio ad Applicatio, submitted to IEEE Sigal Processig, Sept. (4 C. Kostatopoulos, L. Mittag, ad G. Sadri, Decovolutio of Gaussia filters ad atidiffusio, J. Appl. Phys., vol. 68, o. 4, pp 45-4, 99. (5 L. R. Lo Cote, R. Merletti, ad G. V. Sadri, Hermite expasios of compact support waveforms: applicatios to myoelectric sigals, IEEE Tras. Biomed. Eg., vol. 4, o., pp. 47-59, 995. (6 P. A. Lidsay, Itroductio to quatum mechaics for electrical egieers, McGraw-Hill, ew York, 967. (7 E. Kreyszig, Advaced egieerig mathematics, 7 th editio, Joh Wiley ad Sos, Uited States, 993. 75
(8 B. Widrow, 99, 3 Years of adaptive eural etworks: perceptro, madalie, ad backpropagatio, Proc. IEEE, vol. 78, o. 9, pp 45-44, 99..5 t amplitude.5 oisy fuctio f(t Gaussia widow w(τ 4 6 8 t Figure Operatio of a movig average filter f(t 3 amplitude.5.5 - - t (s - - t (s Figure (b Hermite filtered Doppler radar.sigal Figure (a Doppler radar sigal 76
3.5 Power Amplitude.5 Hermite (5 Gaussia Hermite-Rodriguez (5.5.5 Frequecy (Hz -.5 3 Frequecy Figure (c Power spectrum of Doppler radar sigal by Hermite series. Figure 4(a Frequecy respose of Hermite ad Hermite-Rodriguez series with 5 elemets. root mea square error 7 6 5 4 3 3 4 5 6 7 umber of Hermite fuctios Amplitude.5.5 -.5 Hermite (5 Gaussia Hermite-Rodriguez (4 3 Frequecy Figure 3(a Root mea square error versus umber of Hermite Fuctios. Figure 4(b Frequecy respose of Hermite (5 elemets ad Hermite-Rodriguez series (4 elemets. 7 root mea square error 6 5 4 3 3 4 5 umber of Hermite-Rodriguez fuctios Output SR.. Hermite Hermite-Rodriguez 3 4 5 6 Iput SR Figure 3(b Root mea square error versus umber of Hermite-Rodriguez fuctios. Figure 5 Comparative sigal to oise ratios of Hermite ad Hermite-Rodriguez Doppler radar demodulator. 77