Real-time Doppler/Doppler Rate Derivation for Dynamic Applications

Transcription

1 Joural of Global Positioig Systems (5) Vol. 4, No. -: 95-5 Real-time Doppler/Doppler Rate Derivatio for Dyamic Applicatios Jaso Zhag School of athematical ad Geospatial Scieces, RIT Uiversity Tel: Kefei Zhag, Ro Grefell, Yog Li ad Rod Deai School of athematical ad Geospatial Scieces RIT Uiversity Received: 6 November 4 / Accepted: 5 October 5 Abstract. Precise GPS velocity ad acceleratio determiatio relies o Doppler ad/or Doppler rate observatios. There are o direct Doppler rate measuremets i GPS. Although every GPS receiver measures Doppler shifts, some receivers output oly raw Doppler shift measuremets ad some do t output ay at all. I the absece of raw Doppler ad Doppler rate measuremets, a differetiator is ecessary to derive them from other GPS measuremets such as the carrier phase observatios. For real-time dyamic applicatios, a ideal differetiator should have a widebad frequecy respose to cover all the dyamics. It should also have a group delay as short as possible. I additio, a low-order differetiator is more favourable for easy implemetatio. This paper provides a overview of methods i differetiator desig for applicatios of GPS velocity ad acceleratio determiatio. Low-order Fiite Impulse Respose (FIR) differetiators proposed by Kavaagh are itroduced. A class of first-order Ifiite Impulse Respose (IIR) differetiators are developed o the basis of Al-Alaoui s ovel differetiator. For oise atteuatio, it is proposed to selectively use Kavagagh s FIR differetiators, ad the first-order IIR filters derived for adaptatio to differet dyamics. Key words: GPS velocity determiatio, GPS acceleratio determiatio, differetiator desig, FIR filter, IIR filter, Doppler.. INTRODUCTION Previously proposed methods for GPS velocity ad acceleratio determiatio fall i two categories. Oe is to derive velocity ad acceleratio directly from GPS determied positios, aother is based o the Doppler shift method. The latter has several advatages: it does t rely o the precisio of the positios from GPS, or will the accuracy dramatically degrade with a icrease i samplig rate (say Hz or more). Sice there is o direct Doppler rate observatio i GPS measuremets, as a virtual observable, it must be derived i order for the formulae preseted by Jeeli ad Garcia (997) to be applied directly i the Doppler shift method. Every GPS receiver measures Doppler shifts. However, this is primarily a itermediate process to obtai accurate carrier phase measuremets. Thus the quality of Doppler shift output varies from receiver to receiver depedig o maufacturer. The Trimble 57 geodetic receiver, for istace, has a measuremet precisio of ±mm/s. The observed Doppler is from a tracig loop that is updated at a very high rate. This also eables the receiver to sese phase acceleratios (Harvey 4). Ufortuately the sesed phase acc-eleratios ad the Doppler shift o L are discarded. Some other GPS receivers, for example the Superstar II from NovAtel, have oly code ad L phase outputs (SuperstarII 4) ad the Doppler shifts are mased out of the measuremets. For our purposes to obtai accurate velocity ad acceleratio usig these types of receivers, it is ecessary to derive the Doppler shifts, i.e. the chage rates of the carrier phase from the measured carrier phase measuremets. Differetiators are required to get the Doppler rate observable for ay type of receiver, or to get the Doppler shift from the carrier phase. I real-time ad dyamic applicatios it is also desirable that the desiged

2 96 Joural of Global Positioig Systems differetiator should have a widebad frequecy respose to cover the system dyamics. It should also have a group delay as short as possible so as to get the Doppler shift or Doppler rate istataeously. For those receivers that output oly raw Doppler shifts, the derivatio of precise Doppler from the carrier phase plays a ey role i precise velocity ad acceleratio determiatio. This is because the precisio of carrier phase observables ca be fully exploited. The objective of this paper is to explore the techiques to derive Doppler rate from GPS measuremets, or to derive precise Doppler shift from the carrier phase i real time ad i dyamic situatios. Several ivestigatios have bee coducted for this purpose i the GPS measuremet domai, ad the proposed methods ca be categorised ito: () Curve fittig (Feto ad Towsed, 994); () Kalma smoother/filterig (Hebert, Keith et al. 997); (3) Taylor series approximatio (Hebert, Keith et al. 997; Cao, Lachapelle et al. 998; Bruto, Gleie et al. 999); (4) Fiite Impulse Filter (FIR) by usig Fourier series with widow techiques (Bruto, Gleie et al. 999); ad (5) FIR optimal desig usig the Remez exchage algorithm (ibid). The FIR filterig techique based o Taylor series approximatios was recetly adopted to derive phase acceleratios by Keedy (3). This paper briefly describes the digital differetiator theory ad states the desig problems i real-time dyamic GPS applicatios. It is followed by a comprehesive literature review o each method referred to i the above sectio. By comparig the various differetiator desigs, a series of first-order Ifiite Impulse Filters (IIR) are preseted which are capable of deliverig the derivatives from iput sigals i real-time dyamic situatios. A adaptive scheme is also proposed for oise atteuatio.. Digital filter ad digital differetiator Desig + d m y c x y () m where y is the output of the filter, c ad d are filter coefficiets which are referred to as the impulse respose of the filter, which is the filter respose for a uit iput sigal pulse (ibid). The coefficiets of a filter completely defie the property of the filter ad selectively suppress or ehace particular parts of sigals. Whe the coefficiets of the secod term o the right had side of Eq. () are ozero, the filter is referred to as a recursive filter sice the output of y - has bee used recursively. The filter coefficiets c ad d are usually time-ivariat i classical filter desigs. Their values are carefully chose to achieve the desired filterig result. However, their values ca be assiged olie to respod to the chage of situatios i the so-called adaptive filter desig. For practical applicatios, the legth of a realisable digital filter is always fiite.... Trasfer fuctio The trasfer fuctio of a discrete filter is defied as the Z-trasform of the filter output sigal over the Z- trasform of the iput sigal, i.e. c Z(x ) c z Y(z) H (z) () X(z) d Z(y ) d z m m m where z is a complex variable, ad the Z-trasform is a liear trasform whereby a discrete-time sigal value of x is defied as Z(x ) X (z) z x (3) ad where z - serves as a uit delay operator. The trasfer fuctio is most importat i filter desig ad aalysis. With the trasfer fuctio havig bee determied, oe ca directly write out the impulse respose of the filter (filter coefficiets), ad further aalyse the performace of the filter either i the time domai or i the frequecy domai... Digital Filterig Suppose there is a discrete sigal sequece of x ( is a iteger) with a samplig period of T. A digital filter ca be regarded as a liear combiatio of the discrete samples x -, together with the previous output y -., which ca be defied by the followig formula (Hammig 977, p):... Frequecy ad amplitude respose The frequecy respose of a filter is defied as the discrete Fourier trasform of output sigals over the discrete Fourier trasform of iput sigals

3 Zhag et al.: Real-Time Doppler/Doppler Rate Derivatio for Dyamic Applicatios 97 jω c e Y( ω) H ( ω) (4) X( ω) jωm d e m m The above frequecy respose fuctio is obtaied by simply replacig the variable z i the trasfer fuctio by the Fourier trasform variable e jw. The frequecy respose fuctio allows us to evaluate the frequecy respose of a filter o the uit cycle. Factorig the magitude of the frequecy respose ito the followig form jθ( ω) H( ω ) G( ω) e (5) gives the amplitude respose G(ω) which is the gai of the filter. The phase respose Θ(ω) shows the radia phase shift experieced by each siusoidal compoet of the iput sigal. The phase ad group delays of a filter give the time delay i secods experieced by each siusoidal compoet of the iput sigals: Θ( ω) d phase delay group delay Θ( ω) (6) ω dω I the case of a filter that has a liear phase respose, the group delay ad the phase delay are idetical, for example whe Θ( ω) π ω...3. Noise amplificatio A digital filter is a liear combiatio of iput sigals that are usually cotamiated by oise. For simplicity we assume that the oise is Gaussia white, ad thus the error propagatio law applies. This allows us to estimate the oise amplificatio of the filter. Assume that the oise of a series of L carrier phase measuremets x x + ε is Gaussia white, where x stads for the true value of x, ad the the outcome of the fiite o-recursive filter is K K K K y c x c x + c ε (7) K K ad the variace of the filter ca be evaluated by (Hammig 977, p4) K K K E cε ce[ ε ] σx c (8) K K K This shows that the sum of the squares of each coefficiet of a filter determies the oise amplificatio of the filterig process. Supposig that the variace of a recursive filter is σ y, ad applyig the precedig procedures, we have σ y σ x c + d σ y m m Let us further assume that σy σy σ y σ y, ad the the variace of the filter ca be estimated by σ c y d m m x (9) σ () This idicates that we ca either roughly estimate the variace of the recursive filter or precisely calculate the filter variace by computig the iitial variace of the recursive filter usig Eq. (), ad the estimatig the variace of the filtered sigals usig Eq. (9)... Statemet of Problem of Differetiator Desig Differetiator desig has bee the subject of extesive ivestigatio i digital sigal processig. A mai issue is that a differetiator amplifies oise at high-frequecies (Carlsso, Ahle et al. 99). As GPS sigals are of low

4 98 Joural of Global Positioig Systems frequecy character, (see Fig. ), it is suggested that a low pass filter would be suitable for the desig of differetiators. However, the chage of dyamics i a system is ormally of high frequecy. Hece we have to deal with the complicated high frequecies with a broad/full bad differetiator. Aother complicatio arises from the sigal correlatio. It is show that the GPS carrier sigals ca be regarded as Gaussia white oly whe the samplig rate is lower tha Hz; whe the samplig rate goes higher, time correlatios must be cosidered (Boa ; Borre ad Tiberius ). Thirdly, the differetiatio may be affected due to lac of iformatio o future sigals sice the applicatio is real time orieted. Fially there might be aliasig problems due to samplig. So the problem is to get the derivative from GPS observatios where both the sigals ad oise have radom characteristics. I the case of corruptig oise beig widebad white ad the sigal beig a Gauss- arov process (mostliely for GPS applicatios), it is apparet that o differetiator is goig to be perfect i passig the desired derivative whilst suppressig the oise (Brow ad Hwag 99, p7 ). This is a typical Wieer filter problem (ibid). The solutio is a compromise betwee good differetiatio ad low oise sesitivity to achieve a small total error. The Kalma filter is a space-state solutio of the Wieer filter problem (ibid), which is formulated by usig the miimum mea-square-error estimatio criterio i a two-step recursive procedure. By assumig that both the process drivig oise ad the measuremet oise are Gaussia white ad there is o correlatio betwee them, it first predicts the sigal state usig the system dyamic equatio, ad the updates the predictio with measuremets to get estimates. A successful Kalma filter is subject to proper modellig of system dyamics ad the associated stochastic radom process. It is suggested that the less tha satisfactory performace of Fig. : Power spectral desities for the Hz ad Hz carrier phase sigals the Kalma filter i the case of Heber et al. (997) is ot due to the Kalma filter approach itself, but due to the improper modellig of the system state whe it is highly dyamic. Whe the samplig rate is high, the theoretical difficulties i Kalma filterig are maily i the determiatio of the radom process of system drivig oise, ad the hadlig of correlatios of measuremet oise ad the crosscorrelatio betwee the measuremet ad sigal oises. Aother associated practical problem is the heavy computatioal load i real-time data processig. Fially the outcome of a Kalma filter is a smooth, bad-limited solutio (Bruto, Gleie et al. 999). Therefore, it is reasoable to fid solutios i the frequecy domai rather tha i the state space usig Kalma filters. The digital differetiator desig orieted i the frequecy domai should still cosider the variace of the output. Thus the criteria of the differetiator may be summarised as follows: the magitude of frequecy respose is accurate i low frequecies ad is as close to the ideal differetiator H ( ω) jω (Stears 3, p7) as possible i a broad bad sese depedig o the system dyamics; the phase respose is liear or approximately liear; the group delay is acceptably small; the sum of the squares of filter coefficiets ca be miimized; ad easy to be implemeted i real time, i.e. to be causal ad low order sice there are cycle slips ad loss-ofloc of sigals.

5 Zhag et al.: Real-Time Doppler/Doppler Rate Derivatio for Dyamic Applicatios Taylor series approximatios Taylor series approximatios have bee widely used to derive differetiators. The differetiators used by (Cao, Lachapelle et al. 998), Hebert (997) ad Keedy (; 3) are of low order Taylor series. They are all i the form of cetral differece approximatios such as N y c x () N where N is the order of Taylor series approximatio. Fig. depicts the frequecy resposes of some low order cetral differece Taylor series approximatios Normalized agitude Respose: Low order Taylor series approximatios ideal Fig. : Frequecy respose of low order cetral differece Taylor series approximatio It is apparet that the higher the order, the closer that a Taylor series approximatio is to the ideal differetiator. This suggests that broad bad differetiators ca be desiged based o Taylor series, ad this ca be observed i Kha ad Ohba (999), who gave the explicit coefficiets c by c c c ( ) N! (N )!(N + )! Normalized agitude Resposes th 3rd st th 5th th Fig.3: Frequecy resposes for arbitrary order Taylor series approximatios 5 () As ca be see from Fig. 3, this type of differetiator is characterised as havig zero amplitude respose i both ω ad ω (Nyquist frequecy). Actually this is the property of type III FIR filters (Che,p99 ) which will be discussed later. 4. Curve fittig with widow Jeeli ad Garcia (997) used fifth-order B-splies to derive phase acceleratios, ad Feto ad Towsed (994) adopted parabolic fuctios to obtai the precise Doppler. The refereced curve fittig techiques use slidig widows wherei the data are fitted ito polyomials usig the least squares approach. The derivative of the cetral poit of a widow is obtaied by differetiatig the polyomials with respect to time accordigly. Bruto (999) gave a i-depth review of the curve fittig differetiators. It is cocluded that whether a curve fittig uses a polyomial, a parabola, or a cubic splie, the resultat differetiator approaches the ideal oly at lower frequecies. Sice it is bad-limited ad lowpass, it is suitable oly for low dyamic or static applicatios. Furthermore, performig the least squares estimatio ivolves itesive computatio. oreover, to obtai the curret derivative at t, the curve fittig with widow requires the iput at t, which is a sigal i the future. Therefore we may coclude that the widowed curve fittig approach is iappropriate for real-time dyamic applicatios. 5. FIR filters A Fiite Impulse Respose (FIR) filter cosists of a series of multiplicatios followed by a summatio. The FIR filter operatio ca be represeted by the followig equatio (Hammig 977) K y c x (3) i i K i A filter i this form is amed FIR because the respose to a impulse dies away i a fiite umber of samples. Note that this form is o-causal ad urealisable. I order to preset a causal FIR differetiator, chagig the form is required. This leads to N y c x (4) i i N The Fourier series with widow are classical i the desig of FIR filters where the impulse respose is calculated by the iverse discrete Fourier trasform of the trasfer fuctio, i.e. (Che )

6 Joural of Global Positioig Systems ω cd[ ] ω π π π jω e j( ) ω dω cos[( ) π ] si[( ) π ] ( ) π ( ) (5) where N/ ad the ifiite legth of Fourier terms is trucated ito fiite terms. The trucatio may cause a discotiuity at the edges of the widow ad leads to residual oscillatios amed Gibbs oscillatios (ripples i the amplitude respose agaist frequecy). Differet widow methods ca be used to smooth the glitches, trucate the filter coefficiets, ad sharpe the frequecy respose. Fig.4 gives the compariso betwee direct trucatio ad applyig the Kaiser widow techique. 5.. Type III FIR Differetiator Desig A FIR filter of type III has a odd legth ad atisymmetric impulse respose. I this case, the differetiator s coefficiets are cos[( ) π] for c d () ( ) (6) for where the sie term i Eq. (5) vaishes. To elimiate the Gibbs pheomeo due to the fiite trucatio, a widow fuctio is required. Amog may widows that are available, the Kaiser widow is most popular. It ca be evaluated to ay desired degree of accuracy usig the rapidly covergig series of the zero-order Bessel fuctio of the first id (Farlex 4). The ripple of the stopbad ca also be cotrolled by a adjustable variable α to meet the optimal criteria give by Kumar ad Roy (988) ad Selesic (). With the above procedures, oe ca also desig FIR differetiators with differet cutoff frequecies.. Normalized agitude Resposes w/wt Kaiser widow ideal direct trucatio Kaiser widow respose to the Nyquist frequecy is zero, it is impossible to desig a full bad type III differetiator. Although such filters are causal ad are liear i phase, the actual derivative obtaied is with respect to time t-(n/)t. This meas that the more taps i a FIR filter, the loger the group delay will be. This property of the FIR filter is detrimetal to the real-time requiremets. However it ca be alleviated if the samplig period T is small. The difficulty is that icreasig the samplig frequecy will result i more oisy derivatives. Therefore trade-off ad compromise must be made to itroduce this type of FIR i real-time applicatios. 5.. Type IV FIR Differetiators Sice a FIR filter of type III has the limitatio that the amplitude respose must go to zero at the Nyquist frequecy, it is impossible to get a full bad differetiator usig a fiite umber of coefficiets. This ca be show i Fig. 3 where trasitio frequecy rage of.85~. is associated with the 5 th order (legth of 3) cetral differece Taylor series differetiator. A FIR filter of type IV has a eve legth ad atisymmetric impulse respose. The type IV FIR is preferable to a type III as a differetiator i terms of the frequecy respose. This ca be evideced by the simplest FIR differetiator of y x -x -, which has a frequecy respose of H(z) Y(z) X(z) H( ω) e z jω ω j si e ω j (7) The correspodig amplitude respose agaist low order Taylor series approximatios is show i Fig Normalized agitude Respose 5th order ideal -pt differetiator rd order st order Fig. 4: agitude resposes of FIR differetiators based o the widow techique Theoretically, FIR filters of type III ca be desiged to meet requiremets at early all frequecies, as log as we icrease the filter order. However, sice the frequecy Fig. 5: Frequecy respose of the simplest IV differetiator agaist low order Taylor Series FIR filters It ca be see that eve though the differetiator is the simplest form, it is closest to the ideal at low frequecies (<.). It has a better amplitude respose for the rest of frequecy bad tha its type III couterpart of first-order. It also has a liear frequecy respose ad therefore has a costat group delay at half the samplig period. The

7 Zhag et al.: Real-Time Doppler/Doppler Rate Derivatio for Dyamic Applicatios type IV FIR differetiators are superior to the type III FIR differetiators i terms of the frequecy respose, sice they have o disadvatageous characteristic of beig zero at ω. Details of type IV differetiator desig are referred to Che (,p33 ). A example differetiator of legth 8 (7 th order) is give with the trasfer fuctio of H(z) z.73z 4 which miimizes E π π H(e jω +.45z ) jωe jnω /.45z 5 dω.59z +.73z z (8) (9) Therefore it is a optimal differetiator i the sese of least squares with a excellet frequecy respose at high frequecy bad. The oise amplificatio ca be calculated from Eq. () as σ y 3.887, which is acceptable so far. It may be expected that a type IV FIR obtaied from the Remez exchage algorithm (Pars ad cclella 97) would be able to deliver a better performace. This is because the Remez exchage algorithm is a miimax optimal, i.e. miimize {maximum [H ideal (ω)-h disiged (ω)]} for all frequecies, ad is more difficult to mathematically compute, but guaratees that the worst case error has bee reduced to a quatifiable value. To verify this, the frequecy resposes have bee depicted i Fig. 6 for the 7 th ad 5 th -order filters respectively by the Remez algorithm Normalized agitude Respose:7th order Remez agitude Respose of the 5th Order Remez type IV Differetiator Fig. 6: Frequecy respose of type IV FIR filters by the Remez exchage algorithm The FIR filter desig by the Remez algorithm is referred to as the equal ripple desig. This is because the method ca suppress the ripples from the Gibbs pheomeo (Atoiou 993) to a certai level ad tur them ito equal ripples i both the passbad ad stopbad. It seems that type IV FIR differetiators usig the Remez exchage algorithm will give us a closig solutio. However, the resultat filters provide the first derivative with system biases ad higher level of oise. Type IV FIR differetiators based o Taylor series (Kha, Ohba et al. ) have also bee tested i this research. It has bee foud that widebad type IV differetiators are associated with heavy oise amplificatios ad big biases. Our ivestigatio of type IV FIR filters for differetiator desig is still at a early stage ad cotiuig Other FIR Differetiators I a series of publicatios, Kumar ad Dutta (988; 988; 989; 989) preseted optimal ad maximally liear FIR differetiators for low-frequecy, mid-frequecy, ad aroud specific frequecy respectively. They gave the explicit formulae ad efficiet recursive algorithms to calculate the impulse respose of filters. Their cotributios are highly appreciated, for example, as the state of art differetiators by Al-Alaoui (993). I the case of sigals that have low frequecy compoets cotamiated by widebad oise, FIR differetiators of optimum white-oise atteuatio are desired. Kavaagh () ivestigated the impact of quatizatio oise o sigal from systems with low-frequecy rates of chage. It is showed that the differetiator proposed by Vaiio et al. (997) 6(N ) h N () N(N ) has a optimum white-oise atteuatio ad a costat group delay. Kavaagh also proposed a better differetiator for the rate experiecig slow chages N h N () N N This differetiator has the characteristic of miimisig the worst-case error. Clearly whe N (type IV), this becomes the simplest two-poit differetiator ad whe N3 (type III), this turs ito the three-poit first-order differetiator of a Taylor series approximatio.

8 Joural of Global Positioig Systems 6. IIR filters There is aother category of filters ow as the Ifiite Impulse Filter (IIR). A causal IIR filter is represeted by N + dm m y c x y () m where the output sigal at a give istat is obtaied as the weighted sum of the sigal x -, ad the past outputs of y -m. As suggested by its ame, a impulse iput has a respose that lasts forever sice the output will be recursively used. It is the recursive characteristic that allows IIR filters to be implemeted with a lesser order ad better performace whe compared with FIR filters. Thus IIR filters are attractive for real-time applicatios. A IIR filter is ustable if its respose to a trasiet iput icreases without boud. Poles ad zeros are used to aalyse the stability of a IIR filter. The poles are the roots of the deomiator ad the zeros are the roots of the umerator i the trasfer fuctio. A IIR filter is stable if ad oly if, all poles of H(z) are iside the uit circle o the z-plae (Stears, 3, p83). The IIR filter caot be desiged by calculatig the impulse respose from the ow frequecy respose as is the case i FIR desigs. ay IIR filters ca be derived from the aalogue filter desigs ad the trasformed ito the sampled z-plae. Aother popular method is the biliear trasform. The IIR differetiator desig has bee of cosiderable iterest (Rabier ad Steiglitz, 97). Amog various recursive differetiator desigs, Al-Alaoui s secod order IIR family (99; 993; 994) has bee highly acowledged ad widely used, for example (Che ad Lee, 995).The ovel approach of desigig digital differetiators by Al-Alaoui is a extesio of the method i desigig aalogue differetiators by usig itegrators. That is, i the aalogue sigal processig, differetiators are ofte obtaied by ivertig the trasfer fuctios of aalogue itegrators. The geeral procedures to derive the Al-Alaoui family are as follows desig a itegrator that has the same rage ad accuracy as the desired differetiator; ivert the obtaied trasfer fuctio of the itegrator; reflect the poles that lie outside the uit circle to iside, i order to stabilise the resultat trasfer fuctio; ad compesate the magitude usig the reciprocals of the poles that lie outside the circle. 6.. Al-Alaoui s First-Order Differetiator A first-order IIR differetiator was developed by Al- Alaoui (993) with a effective rage.78 of the Nyquist frequecy based o a o-miimum phase digital itegrator. The itegrator is a sythesis of the rectagular itegrator ad the trapezoidal itegrator. By assigig weightig factors of ¾ ad ¼ to the trasfer fuctios of the itegrators respectively, the ideal itegrator, which has the followig trasfer fuctio, is approximated 3 HI(z) HR (z) + HT (z) T T(z + ) T z z 4 (z ) 8 z (3) Reflectig the zero z-7 with its reciprocal -/7, ad compesatig the magitude by multiplyig r7, results i a miimum phase digital itegrator with the trasfer fuctio (z) H I 7 T z + 7 (4) 8 z Ivertig the above trasfer fuctio yields the Al- Alaoui s stabilized IIR differetiator of the first order 8 7 z H D (z) (5) T z + 7 The characteristics of this differetiator is show i Fig. 7. This differetiator is able to approximate the ideal differetiator up to.78 of the full bad, ad has a outstadig liear phase respose. Al-Alaoui reported that withi the effective frequecy rage, it has a less tha.% magitude error. Sice it is of first-order, the delay of the filter is just half of the sample thus it meets every requiremet to be used i real-time.

9 Zhag et al.: Real-Time Doppler/Doppler Rate Derivatio for Dyamic Applicatios 3.5 Ideal Al-Alaoui Normalized agitude Respose Phase respose of Al-Alaoui's Differetiator Dif(Al-Alaoui) Ideal 5 Imagiary Part Zero/Pole respose of Al-Alaoui's differetiator Zeros at Poles at -/ Real Part Fig. 7: Characteristics of the first order IIR differetiator 6.. First-order IIR Differetiator Family Al-Alaoui cotributes the above differetiator as a idividual. However, a family of such first order differetiators ca be derived followig his methodology. That is, while Al-Alaoui desigates the weightig factors of ¾ ad ¼ empirically, we may get the optimal weights experimetally. To achieve this, a variable α is itroduced to adjust the weightig factor i the way of H (z) αh I R (z) + ( α)h T (z) + α T( α) z T T(z ) + + ( ) α α + α z (z ) (z ) (6) where <α< serves as a tuer to adjust the itegrator so that it better closes to the ideal. α ¾ ca be used as a good referece to refie the itegrator i the desired rage of frequecies. Obviously it has a zero outside the uit circle. Applyig Al-Alaoui s procedure to reflect the zero with its reciprocal ad to compesate the magitude, a variable itegrator is obtaied as (z) H I α T( + α) z + + α (7) (z ) Ivertig the trasfer fuctio gives a ew set of differetiators with trasfer fuctios as (z ) H D (z) (8) α T( + α) z + + α Sice -α<+α, the pole is well located iside the uit circle ad the resultat differetiators are, therefore, stable. Settig α¾ gives the trasfer fuctio proposed by Al-Alaoui, ad slightly chagig α aroud ¾ results i differetiators which outperform i target badwidth. The oise amplificatio of this id of differetiator ca be evaluated usig Eq. (), which is oly slightly oisier tha the simplest two-poit differetiator. 7. Coclusios The geeral theory o digital filter desig has bee itroduced. The aim of this research is to fid appropriate differetiators that ca be used to derive Doppler shifts/doppler rates from GPS observables i real-time, dyamic applicatios. It is cocluded that the differetiators obtaied from both curve fittig ad Kalma filterig require itesive computatio ad are lowpass. Thus they are ot suitable for real-time dyamic applicatios. Type III FIR differetiators have the iheret ature i frequecy respose of approachig zero at Nyquist frequecy. To exted the performace of type III FIR filters i the higher frequecy bads, oe has to icrease the filter taps. This causes difficulties i maagig the data sice there are cycle-slips ad loss-of-loc sigals. It also results i a loger group delay that is detrimetal for real-time applicatios where istat respose is desired. Type IV FIR differetiators usig Fourier series have bee foud to have outstadig frequecy respose, however, they are oisy ad biased. It is foud that oly the Kavaagh s differetiators of type IV deliver good first derivatives. However, they approximate the ideal differetiator oly at low frequecies (lower tha. of Nyquist frequecy). Type III FIR filters ca be used to derive Doppler/Doppler rate observables i post processig mode. Higher order cetral differece approximatios usig Taylor series might outperform widowed Fourier series sice there is o trucatio ad the associated Gibbs pheomeo. It is demostrated that IIR filters are more favourable for real-time applicatio. Sice the outputs of the filter are recursively used, they have much lower orders tha the FIR filters. The first-order IIR differetiator from Al- Alaoui is ideal i terms of the frequecy respose, phase liearity ad half sample group delay. The proposed class of first-order IIR differetiators allows us to choose a optimum i the desired frequecy rage.

10 4 Joural of Global Positioig Systems It is suggested that Kavaagh s differetiators ca be used i static or i costat velocity modes. The proposed first-order IIR differetiators ca be adaptively used whe systems experiece high dyamics. ACKNOWLEDGEENTS The research described i this paper was partially supported by the Cooperative Research Cetre (CRC) for microtechology, Australia edorsed to A/Prof. Kefei Zhag. Partial fiacial support from the Australia Research Coucil Liage Project (LP4557) edorsed to a research cosortium led by A/Prof Kefei Zhag is highly appreciated. r. Ravidra Babu at the SNAP, UNSW is highly acowledged for cosultatios ad discussios. REFERENCES Al-Alaoui.A. (99): Novel Approach to Desigig Digital Differetiators. Electroics Letters 8(5): Al-Alaoui.A. (993): Novel Digital Itegrator ad Differetiator. Electroics Letters 9(4): Al-Alaoui.A. (994): Novel IIR Differetiator from the Simpso Itegratio Rule. IEEE Trasactios o Circuits ad System-I Fudametal, Theory ad Applicatios 4(). Atoiou A. (993): Digital Filters: Aalysis, Desig ad Appicatios. New Yor, cgraw-hill -7-X. Boa P. (): Precisio, Cross Correlatio, ad Time Correlatio of GPS Phase ad Code Observatios. GPS solutios 4(): 3-3. Borre K. ad Tiberius C. (): Time Series Aalysis of GPS Observables. proceedigs of the ION GPS, Salt Lae City, UT, Brow R.G. ad Hwag P.Y.C. (99): Itroductio to Radom Sigals ad Applied Kalma Filterig. New Yor, Joh Wiley &Sos, INC Bruto A.., Gleie C.L. ad Schwarz K.P. (999): Differetiatio for High-Precisio GPS Velocity ad Acceleratio Determiatio. GPS solutios (4): 7-. Cao.E., Lachapelle G., Szarmes.C., Hebert J.., Keith J. ad Joerst S. (998): DGPS Kiematic Carrier Phase Sigal Simulatio Aalysis for Precise Velocity ad Positio Determiatio. Joural of the Istitute of Navigatio 44(): Carlsso B., Ahle A. ad Sterad. (99): Optimal Differetiatio Based o Stochastic Sigal odels. IEEE Trasactios o Sigal Processig 39(): Che C.-K. ad Lee J.-H. (995): Desig of High-Order Digital Differetiators usig L Error Criteria. IEEE Trasactios o Circuits ad Systems II: Aalogy ad Digital Sigal Processig 4(4): Che C.-T. (): Digital Sigal Processig: Spectral Computatio ad Filter Desig. New Yor, Oxford, Oxford Uiversity Press Farlex (4): TheFreeDictioaryCom: Kaiser Widow. ecyclopedia.thefreedictioary.com/kaiser%widow. Feto P. ad Towsed B. (994): NowAtel Commuicatios Ltd,-What's New? Proceedigs of KIS'94, Baff, AB, CA, 5-9. Hammig R.W. (977): Digital Filters. Eglewood Cliffs, Pretice-Hall, Ic Harvey N. (4): Doppler ad Carrier Phase easures i Trimble 57 GPS Receiver. Christchurch, New Zealad (Persoal commuicatio). Hebert C.J. (997): High Accuracy GPS Velocity Usig the Carrier Phase Observable. Proceedigs of KIS'97, Baff, AB, Caada, Hebert C. J., Keith J., Rya S., Szarmes., Lachapelle G. ad Cao. E. (997): DGPS Kiematic Carrier Phase Sigal Simulatio Aalysis for Precise Aircraft Velocity Determiatio. Proceedigs of the ION GPS-97, Albuquerque, N, Jeeli C. ad Garcia R.. (997): GPS Phase Acceleratios for ovig-base Vector Gravimetry. Joural of Geodesy 7(): Kavaagh R.C. (): FIR Differetiators for Quatized Sigals. IEEE Trasactios o Sigal Processig 49(): Keedy S. (): Precise Acceleratio Determiatio from Carrier Phase easuremets. Proceedig of the ION GPS-, Portald Orego, USA, Keedy S.L. (3): Precise Acceleratio Determiatio from Carrier-Phase easuremets. Navigatio, Joural of the Istitute of Navigatio 5(): 9-9. Kha I.R. ad Ohba R. (999): New Desig of Full Bad Differetiators Based o Taylor Series. IEEE Proceedigs-Visio, Image ad Sigal Processig 46(4): Kha I. R., Ohba R. ad Hozumi N. (): athematical Proof of Explicit Formulas for Tap-coefficiets of Fullbad FIR Digital Differetiators. IEE Proceedigs- Visio, Image ad Sigal Processig 47(6): Kumar B. ad Dutta S. C. R. (988): Coefficiets of aximally Liear, FIR Digital Differetiators for Low Frequecies. Electroics Letters 4(9): Kumar B. ad Dutta S.C.R. (988): Desig of Digital Differetiators for Low Frequecies. Proceedigs of the IEEE 76(3): Kumar B. ad Dutta S.C.R. (989): Desig of Efficiet FIR Digital Differetiators ad Hilbert Trasforms for idbad Frequecy Rages. Iteral 36(6). Kumar B. ad Dutta S.C.R. (989): aximally Liear FIR Digital Differetiators for idbad Frequecies. It. J. Circuit Theory Appl. 7:

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