AD-A099 829 NAVAL SURFACE WEAPONS CENTER DAHLGREN VA F/B 17/7FORMULATION FOR THE NAVSTAR GEODETIC RECEIVER SYSTEM CNSRS).U

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UNCLASSIFIED* 'E~~%CU,'tITY CLASSIFICATION OF THIS PAGE (Wien Do,. Entered)__________________

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R EP RT! ~-2GVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

,NSWo rr?0.348.I_ _ _ _ _ _ _

4. TITLE (and 51 ' We) S. TYPE OF REPORT A PERIOD COVERED

Formulation for the NAVSTAR ( )Final et.h /61 Geodetic Receiver System (NGRS),. -ItOO *iR NUMBER

7. AUTHOR(0) 11. CONTRACT OR GRANT NUMBER(@)

Bruce R./ermann

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*Naval Surface Weapons Center (K13) 65* Dahlgren, VA 22448 -

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Headquarters // Ma8Defense Mapping Agency NUMS u BfR-O 1

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Approved for public release, distribution unlimited.

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16. SUPPLEMENTARY NOTES

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.4 NAVSTAR Global Positioning System (GPS)NAVSTAR Geodetic Receiver System (NGRS)Geodetic Point PositioningDoppler Frequency (Velocity)Doppler Rate (Acceleration)

20 ASSTRPACT (Continue on reverse aide it naecesry and Identify by block numbet)

The NAVSTAR Global Positioning System (GiPS) can be used for geodetic applications. TheNAVSTAR Geodetic Receiver System (NGRS) has been developed to demonstrate this capability. Abrief description of the NGRS hardware is presented along with the formulation that converts the obser-vations of satellite Doppler frequency to range differences. The sensitivity of the range differences torandom error sources is also discussed-.

SD IO i A,",07 1473 EDITION OFl I NOV 69 IS OBSOLETE UNCLASSIFIEDS'N 01 02-LFP.014.6601 SECURITY CLASSIFICATION OF THIS PASS (hnDate 00,14aO

IN .-

FOREWORD

The development of the NAVSTAR Geodetic Receiver System (NGRS) was funded by theDefense Mapping Agency as a test bed to demonstrate the capability of using satellite Doppler datato achieve high accuracy geodetic point positioning. The receiver was designed by Stanford Tele-communications, Incorporated, under contract to the Naval Surface Weapons Center (NSWC). Thesupporting hardware definition and the system microprocessor controller software were proposedand implemented at NSWC by the Electronics Systems Department, Advanced Projects Division.Technical and administrative assistance was received from the sponsor through the Defense MappingAgency Hydrographic/Topographic Center in Washington, D.C.

This report has been reviewed by R. W. Hill, Head, Space Flight Sciences Branch, R. J. Anderle,Research Associate, and D. R. Brown. Jr., Head, Space and Surface Systems Division.

Released by:

R. T. RYLAND. HeadStrategic Systems Department

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ACKNOWLEDGEMENTS

The author wishes to take this opportunity to acknowledge those who have had a role in thedevelopment of the NGRS. At NSWC, Mr. Richard Anderle, Research Associate, and Mr. RobertHill, Head, Space Flight Sciences Branch, provided both administrative and technical support.Technical assistance was also provided by Mr. William Birmingham, Ms. Dee Batayte and Ms.Louise Gordon provided the programming support, and the author was assisted in the data reduc-tion by Mr. Patrick Hoffman, Mr. Stanly Meyerhoff, and by Ms. D. Jeannie Haynie.

In the Electronics Systems Department, Mr. Theodore Saffos was responsible for the systemhardware design and maintenance while supervising the development of the microprocessor systemcontroller software. This software development was accomplished by Mr. Ralph Dickerson, Mr.Glen Bowen, and Mr. Eric Morgan.

Project direction, planning, and advice were offered by the Defence Mapping Agency throughMr. Benjamin Roth at SAMSO and Mr. Henry Heuerman at the Hydrographic/TopographicCenter. Final authority rested with Dr. Charles Martin at Headquarters.

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CONTENTS

Page

INTRODUCTION........................................................... ISYSTEM DEFINITION.......................................................1ITIME CORRECTIONS ....................................................... 4DOPPLER OBSERVATIONS.................................................. 5CLOCK MODEL ............................................................ 6DOPPLER SIGNAL MODEL.................................................. 7ERROR ANALYSIS ......................................................... 11EXPECTED OBSERVATIONAL ERROR........................................ 16CONCLUSIONS............................................................ 28REFERENCES ............................................................. 28APPENDIXES

A-Frequency Spectrum of the P Code Modulation ............................ 31B-Local Clock Correction From Pseudo-Range Measurement .................... 51C-Maximum Doppler and Doppler Rate.................................... 55

DISTRIBUTION

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FIGURES

Figure Page

1 NG RS Block Diagram ....................................................... 22 Relationship Between the Time Systems and Pseudo-Range ........................ 33 Measurements That Comprise the Doppler Observations .......................... 34 20-Min L, Range Difference RMS (cm) Vs Date, GPS SV-4, Yuma Proving Grounds,

D ays 36-80 1979 ............................................................. 195 20-Min L, Range Difference RMS (cm) Vs Date, GPS SV-6, Yuma Proving Grounds,

D ays 36-80 1979 ............................................................. 206 20-Min L, Range Difference RMS (cm) Vs Date, GPS SV-7, Yuma Proving Grounds,

D ays 36-80 1979 .................................... ........................ 217 20-Min L, Range Difference RMS (cm) Vs Date, GPS SV-8, Yuma Proving Grounds,

D ays 36-80 1979 ............................................................. 228 20-Min Ionospheric Correction to L, RMS (cm) Vs Date, GPS SV-4, Yuma Proving

Grounds, Days 36-80 1979 .................................................... 239 20-Min Ionospheric Correction to L, RMS (cm) Vs Date, GPS SV-6, Yuma Proving

Grounds, Days 36-80 1979 .................................................... 2410 20-Min Ionospheric Correction to L, RMS (cm) Vs Date, GPS SV-7, Yuma Proving

Grounds, Days 36-80 1979 .................................................... 25II 20-Min Ionospheric Correction to L, RMS (cm) Vs Date, GPS SV-8, Yuma Proving

Grounds, Days 36-80 1979 .................................................... 26A-1 Random Gate Function ...................................................... 31A-2 Elementary Gate Function Centered On Zero .................................... 31A-3 Elementary Gate Function Centered On Zero Frequency .......................... 34A-4 Frequency Domain Gate Function Transformed to the Time Domain, coo = PI/TAU .. 35A-5 Elementary Gate Function in Time Domain, TAU = 97.752 ns ..................... 36A-6 Convolution of Elementary Gate With Transformed Gate, co = PI/TAU (Pulse Shape

of Received Chips Due to Finite Bandwidth) ..................................... 37A-7 Cross Correlation of Received Chip With Locally Generated Chip, coo = PI/TAU ..... 38A-8 Half Chip Early Minus Half Chip Late Correlation Curve, co = PI/TAU ............ 39A-9 Cross Correlation Function of Two Gate Functions of Width T ........................ 40A-10 Half Chip Early Minus Half Chip Late Correlation Curve ......................... 40A-1 Frequency Domain Gate Function Transformed to the Time Domain, coo = 2 PI/TAU 41A-12 Convolution of Elementary Gate With Transformed Gate, Wo = 2 PI/TAU (Pulse

Shape of Received Chips Due to Finite Bandwidth) ............................... 42A-13 Cross Correlation of Received Chip With Locally Generated Chip, coo = 2 PI/TAU... 43A-14 Half Chip Early Minus Half Chip Late Correlation Curve, coo = 2 PI/TAU .......... 44A-15 Frequency Domain Gate Function Transformed to the Time Domain, Wo = 3 PI/TAU 45A-16 Convolution of Elementary Gate With Transformed Gate, co = 3 PI/TAU (Pulse

Shape of Received Chips Due to Finite Bandwidth) ............................... 46A-17 Cross Correlation of Received Chip With Locally Generated Chip, co = 3 PI/TAU... 47A-18 Half Chip Early Minus Half Chip Late Correlation Curve, coo = PI/TAU ........... 48C-I Retrograde Equitorial Orbit Geometry ......................................... 55

ix

TABLES

Table Page

I N G RS C om ponents ......................................................... 22 Satellite Phase Noise Derived From Doppler Observations ......................... 27

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'11

INTRODUCTION

The Astronautics and Geodesy Division in conjunction with the Advanced Projects Division ofNSWC, with sponsorship by the Defense Mapping Agency, have developed a receiving system thatallows the NAVSTAR Global Positioning System (GPS) to be used for geodetic point positioning.Previous experience with the Navy Navigation Satellite System has shown that observations of Dop-pler frequency from many satellite passes can be used to obtain positions at the receiving site with anerror of only a few meters. It was hoped that similar techniques applied to the GPS constellationwould allow even better results to be realized. It was to demonstrate this potential that the NGRSwas developed.

SYSTEM DEFINITION

The heart of the NGRS is the GPS receiver, which was developed by the Stanford Telecom-munications, Incorporated, under contract to NSWC. The receiver was delivered in September1978. Integration of the receiver and the other equipment that comprise the NGRS progressedthrough the end of 1978. First data were processed in January 1979, and the system was declaredoperational at the end of that month. The truck containing the NGRS was driven to Yuma ProvingGround in early February for the initial tests alongside the Mobile Test Van, which contained aMagnavox "X-set" configured for static real-time positioning. The NGRS retrained at the Yumasite until day 80 of 1979.

Figure I shows a block diagram of the NGRS, and a list of the equipment is given in Table 1. Adescription of the satellites and the signal structure is described in a special issue of the Journal ofthe Institute of Navigation'. Each satellite transmits on two L band frequencies: L, = 1575.42 MHzand L 2 = 1227.6 MHz. The bandwidth of the signals is about 20 MHz and consists of a carrierbiphase modulated by two pseudo-random sequences having frequencies of 1.023 MHz (C/A) and10.23 MHz (P). A data bit stream of frequency 50 Hz is added to each of the pseudo-random se-quences. A replica of the C/A and P code is generated in the receiver and correlated with the incom-ing signal. The replica is shifted in time until a correlation peak is obtained. If this correlation peakis maintained, the local code is then locked to the incoming code from the satellite. The dependenceof this correlation peak on bandwidth is described in Appendix A.

The pseudo-range measurement is obtained by recording the time interval between the localtime epoch and the same epoch received from the satellite. This measurement consists of two parts:the propagation delay and the time offset between the local and satellite clocks. Figure 2 shows thisrelationship.

The analog Doppler signal derived from the reconstructed carrier in the receiver is used to ob-tain Doppler counts on both L, and L 2. The time interval over which the count is made is deter-mined by the local clock and is nominally 60-s duration. A continuous running counter is strobed atthe end of the interval to provide the integer count. The time from the end of the interval until the

...

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T± r.

GPS TIME

GPS LOCAL SATELLITE LOCAL TIMEEPOCH EPOCH EPOCH AT WHICH

SATELLITE EPOCH RECEIVED

Tr =PROPAGATION DELAYTo = EPOCH OFFSET (SATELLITE - LOCAL)rp = OBSERVATION OF PSEUDO-RANGE = ro + rrT = EPOCH OFFSET (LOCAL - GPS)s = EPOCH OFFSET (SATELLITE - GPS) -ro + rf

A VALUE FOR rs IS FOUND IN THE SATELLITE DATA MESSAGE

Figure 2. Relationship Between the TimeSystems and Pseudo Range

- .0. [-- li.1

... :

CL T 1ME t..*" __1 TL2 OHI rL2 i

L2 m i' r I I

~LOCAL TIME tt.

n AND m -= THE ACCUMULATED COUNTS AT EACH LOCAL 60-a EPOCH

rL1 AND rL2 TIMES FROM THE LOCAL 60- EPOCH TO THE NEXT POSITIVE GOING ZERO CROSSINGS

ZERO DOPPLER FREQUENCY = 28750 Hz

Figure 3. Measurements That Comprise the Doppler Observations

3

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next positive zero crossing of the analog Doppler signal is also recorded. This allows a precisemeasurement of the true time interval to be made. A pictorial description of the process is shown inFigure 3.

The system operation is controlled by a microprocessor. To begin observing, the operator isasked to key in the options he wishes to use for the current session. Each response is displayed on thevideo terminal and sets flags in the controlling program. After a cassette tape is loaded into the tapedrive, the system is ready to begin observations. A search is initiated for the first satellite using theDoppler estimate, which has been entered by the operator.

Two methods of operation are available: local and remote. In the local mode, the operatorloads the acquisition information directly into the receiver via the front panel switches. This infor-mation consists of the satellite code identification number, the expected Doppler frequency with atolerance of ± 500 Hz, and the differential ionospheric delay between the L, and L2 channels. Ac-quisition of different satellites is made by entering new information when desired. The remote modeoffers the option of entering a list of acquisition information and start times into the microprocessorvia the the keyboard. The computer will then operate the receiver remotely, sending the new acquisi-tion information at the preset times.

When the receiver has acquired the first satellite, pseudo-range measurements obtained fromthe time-interval counter are sent to the microprocessor each 6s. At the end of a minute, as deter-mined by the local clock, the two Doppler counts and the two fractional counts are sent to themicroprocessor. All the data accumulated during that minute are then recorded on the taperecorder. The system continues tracking in this 1-minute cycle until the satellite goes out of view, orthe time arrives to search for a different satellite. The tape holds about 12 hr of continuous observa-tions, and it can switch to a second drive when the first is full. The data from these cassettes aretransferred to a nine-track tape and are then preprocessed. The preprocessed data are kept on diskpacks for further processing as desired.

TIME CORRECTIONS

Time corrections are necessary so that the Doppler observations and the satellite reference tra-jectory are referred to the same time system. For our purposes, time is kept at the Master ControlStation (MCS) at Vandenburg Air Force Base. This is GPS time and is allowed to follow its owncourse; that is, it is not corrected for the variations in the earth's rotation as is Universal Time.

Each GPS satellite has an independent frequency standard that must be corrected to GPS time.The corrections for each satellite are predicted by the MCS and uploaded. Each satellite, in turn,transmits these predicted corrections to its apparent time in its navigation message. The user must

4

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then apply these corrections when he uses the data. In this way, all the data can be converted to acommon time system.

In the NGRS, the single frequency (L,) pseudo-range measurement is used to relate the localtime to GPS time (Appendix B). The pseudo-range observation gets its name because it is just atime-of-fliht range measurement treated as if the clocks on both ends were synchronized. In realitythey are not synchronized; and so the measurement is not a true range, but includes the clock errorsplus propagation effects. Figure 2 illustrates the measurement. Epoch is an arbitrary instant atwhich each clock registers the same numerical time. The observable is T, and is time tagged by thesatellite epoch. However, the satellite epoch is offset from GPS time by T, (note that T, as well as T,

and ro, are not constant but vary with time). The navigation message from each GPS satellite givesthe predicted values for T, as a function of GPS time. Thus, the true time (GPS time) tag for theobservation is the satellite epoch time minus T_. This is the correction that needs to be made to thepseudo-range time tags to indicate the true time of transmission. With the use of the T,- observations,the local clock offset T,, can be obtained by subtracting out the propagation delay r,. To calculate T,,the satellite position and the receiver position are needed. However, extreme position accuracy is notnecess , because the relative velocity is much less than the propagation velocity.

If a worst-case relative velocity of 1000 m/s is assumed, the slant range changes by only 0.001 min I ps. Light travels 300 m in I Ms. Therefore, a position error of 300 m in the slant range would pro-duce only i-ps error in the value of T,,. Microsecond accuracy in T. is adaquate for this purpose.With Tr, known, the local epoch can also be converted to GPS time. If t, is the local time, the corre-sponding epoch converted to GPS would be l*

where

6 = j + T, - Ti

In this equation, the senses in Figure 2 are used to determine the signs.

DOPPLER OBSERVATIONS

The Doppler observation consists of two measurements: the Doppler count is an integer thatrepresents the accumulated number of Doppler cycles and is recorded at a regular interval T; and thetime interval measurement is equivalent to a fractional cycle count. The interval measured is be-tween each Tepoch and the next positive going zero crossing of the analog Doppler signal. Both ofthese measurvieens are illustrated in Figure 3.

The signal that is counted is the dfference between an offset frequency f, and the true Dopplerf,. Since 1f - 28750 H/1, a count of zcro l)oppler would register 1,725,000 counts. The fractionalcount is combined with the integer count in the following manner. Referring to Figure 3, the time in-terval that represents the integer count is T + T, I - T,. Therefore, the average frequency is in (T+ T,-, - T,) H/. From the average frequency, the approximaie number of counts in tie interval Tcan be determined. This count, m ,, is

4

* . • * , ' " V -r-

mT = m + (Ti,--Ti) mT + T,- -Ti

The error introduced into m, by using the average frequency is discussed in Appendix C.

The Doppler data consist of two simultaneous observations: one from L, and the other fromL. These two frequencies will be refracted differently by the ionosphere because it is a dispersivemedium. Solution of the resulting simultaneous equations allows a correction term to be evaluated,and substitution of this term into either of the original two equations will produce a corrected obser-vation. The mechanics of this process will be developed in the sections that follow.

CLOCK MODEL

The clock model used in this development will be expressed as a phase function 4(t).

0(t) = + 2n(v0 + v(t)) t + YI(t) (I)

where

0(t) total phase accumulated since t = 0+ phase at t = 0; i.e., 0(0) = +Vo . nominal oscillator frequency (constant)v(t) deterministic frequency variationYI(t) random phase noise term

This form of the phase will represent both the local standard reference as well as the satellitereference.

The instantaneous frequency is 4(t)

where

4(t) = 2n(vo+ v()) + 2iv(t)t + n'(1) (2)

and at t =0,

0(0) 2n(v,, + v(O)) + 17(0)

The argument r will represent GPS time as kept by the MCS. All other clocks will have offset and rate dif-ferences when compared with the reference standard. A clock keeping perfect GPS time can be represented bya phase function of the following form:

I, J0 2nv,,t (3)

6

where v. is a known invariant constant. Time is kept by keeping track of the phase 0,(t). A time

interval would be represented by the difference of two phases at different instants.

ti - t, ___I [t ,,, (t2) - 4 ,, (t, )I seconds

With the clock model defined as above, let the satellite clock be represented by

()= * + 2n(v,,, + v()) t + 17(t) (4)

and the local clock by

0, (t) = *, + 2n(vo, + v,(t))t + v1,(t)

DOPPLER SIGNAL MODEL

The signals recovered will be the transmitted signals perturbed by the ionosphere (troposphere andrelativity effects will be ignored in this development). The ionosphere will be introduced as a time delaysimilar to the geometric propagation time. Let the total propagation delay be represented by TP

where

.to=- (r + 1) and r(t) = slant range from the satellite to receiverC

The second term, 1(t), is the ionospheric delay.

Signals transmitted at /r are received at a later time t- + T,, or conversely the signals currentlyreceived were transmitted earlier at tR - TP. Before transmission, Equation 4 was multiplied by aconstant q, for broadcast on L,. Similarly, Equation 4 was multiplied by q2 for broadcast on L2 .The received signals are represented by Equations 6a and 6b.

(0,(t. - -r) q, [€+ 2n(v, + v,)(t. - "P) + t T)] (6a)

4),(1, - T P) = q21[, + 2n(v., + v,)(,, - T) + t- a)] (6b)

In the receiver, two similar signals are derived from the local clocks; these are represented byEquations 7a and 7b.

1O,(t)=I,[I + 2ur(v,, +v, t,+r,(t)t (7a)

€,,tm) = ,[+. + 21(vo,. + v,.) 6 + 1,(0) (7b)

The receiver mixes the locally generated signals with those received and derives an intermediate fre-quency which is then processed. This intermediate frequency phase function will be written as thedifference between Equations 6 and 7:

~ ~7

(0F,(tR) = '.(IR - T,) - 0,(tR)

= qI - QI+R + 2nq,vo, + v,)(tR T) -Q,(vo,. + v,)tR.

+ q, ?(t. - To) -QVL (t.) (8a)

Of,(t.) = 0,(tR - Tp) - D,,UR)

= q2+ - Q2+R + 2n[q2 (vo. + v,,)(t. - Tp) -Q2(Vo, + v, )tR]

+ q2 1,tR - T) -Q21 Y(IRt) (8b)

The Doppler observation is essentially a measurement of phase at two different times. For ex-ample, from Equation 8a, the phase difference T seconds apart would be

PI(t. + T) _0,(t.)

A detailed expansion is written below as AO:

A = I(tR + T)- 0,(t1R)

AO = 2niq,[v,, + v,(tR + T)][t, + T - To(It + T)J - q, [v., + v,(t.)llt - TP(tR)]

-Q,[VO + v1(IR + T)I[t. + fl + Q,[iVO + v,.(tR)] tR}

+ q, [,(tR + T- T,(t. + T)) - r,(tR - T(t.))] - Q,[nj,(t + T) - n,(10 (9)

This expression does not contain q,+. or QI+ because these are constants and disappear in the sub-traction. In like manner, if there were a channel bias term +,, in Equation 8a, it would also not ap-pear in Equation 9. Therefore, this observation of AO is not sensitive to a channel bias as long as thebias remains constant over the interval T.

It will be assumed that the frequency drifts contained in v, and v, are small enough so that thedifferences over a time Tcan be neglected. To demonstrate this, suppose v.(t) is on the order of 4 x10-" Hz/s. This is a representative number obtained from the specification of the HP5065A Rubidium fre-quency standard'. The frequency drift for this unit is given as less than ± I x 10-" parts per month. Multiply-ing this by 10.23 MHz and dividing by the number of seconds per month produces the number stated above.Over the interval T. the difference q,lv.(t. + T)(tR + T)- V.(t) tIR equals 4 x 10-"q. T. Using T = 60 s andq. = 154. the resulting phase drift is 2n(4 x 10-") (154) (60) = 2.3 x 10-" radians, which is small enough to ig-nore. However. the random variations represented by YI(t) must be considered. Frequency instability is usual-ly represented as a fractional frequency variation Af/f over an interval T. This implies a phase instability.which can be illustrated if the following operations are performed.

LetSAf Tr) =k

Then

8

*.M(T) J,

and

n(it + T) -P7(t) =f ftdt f d .IT ( (T)N 'I1

Therefore, the drift due to the random contribution% from the local reference frequency standard[assuming T = 60 s, AI-f.(/T) - I x 10 "'.f - 5 x 106 H/ and p, - 3151 is 2n(il' 2 ) (315) (5 x 10()(60) = 0.47 rad. This magnitude of phase intability is considerable and constitutes a major sourceof error in determining the phase difference A10. This phase noise will be collected and representedby the parameter tp where

tP = IqIVt,N . F- r(/. + F))- 7. U, - T,l.))I - Q,Irh(U + T)- nh(fu)l (10)

Since v.1I) and v, Mt) were shovn to sary little in an interval T. equation 9 can be simplified to

AO/(. 4 T) - 2. +q, ( )+ v,)IT + r ,i.)- T,(I, 4 T)) - ,(v,,, + v,) T) +

Division of A0 by 2?r produces a number N. shich is the Doppler count (th: number of cycles in theinterval T).

MOP )+ T qL,,+ v) l r ( I T,(/,, + T) - ,(11,, + v,,) T + 4'(Il)2n 2n

This is easily rcarrangtd to place t ti i'lere.ice in r 's on the left hand side.

T( ) T,,[, + F) I [%, - (q,(v,,. + v T- Q,(v,, + v, )F) - (12)r,.,,((1) ,,l ) ,(VO. + V.) 2n]

in Equation 12, N, is the obseration. Fhe ret ofthe terns in the brackets represent that portion otthe count that is due to ti e Doppler offset and the instability of the frequency reference.

Let us hypothesi/e thit tle propagation dcla. r.(t) is coms cd of' the geometric slant ranger(it) and the ionospheric delay /1(). which ss erc introduced earlier. I fthis is substittwed into Uquation12, the result is

r~.) ,'t. + i+ /F, ,(. + ) +. " ,,( ,+v )' I v, +v, I, -tw, 2rrl I lit)

In like manner, an e\pression for I , can be de\eloped:

r )- r( , + ) + I,(t ) I, (' + T) c i, ( v,,. + i ) F -- .Lv, + v, ) ] - 4, 2n (0 1 )I +i1(v. + V,) J

Since the geometric range rPl "ill be common to the i%%o frequencies, it can he eliminated b\ sub-Iracting i taltion 1 3a lt from Fqut ioni 13h.

I,(t.) - ,((,. + IF) j/,(it I - l (if + /1)

LiC, + v,,, + , i] I (4,, (14)114. + v, . ( (h (I lit "2n (1 (1

-' :'P .-- , . . + I I ml. ,-V - = - -" 9 - . -% , .. . ..v - . . . . ,, .

This expression equates the part of the range that is frequency dependent, on the left, to the obser-vations N,, N2, and Ton the right. The ionosphere can be modeled by expressing the quantity /(I)by an integral expression'.

lt)-2w f'.'"' (r- kr)',1 f (r ) ___ dri_ (15)

where

r,(t,) = satellite position at INr,(/,) = recL. -er position at t, + T,

w, = 2nqv,., + v,)

A similar expression can be written for I,(I. + T) having the limits r.(t. + T) and rR(t,, + T). Thetransmitted frequency appears only in the constant coefficient w,. The difference I /,It) - I,( + T)can be written as the difference of tvo integrals like Equation (15).

/,(t) - /.(r. + T) -l r (2 rrdr w,(r) rdr (16)

r - A )' 2 1 - k'2 ' 1

The expres'.on in braces sill he identical for L., and so it can be redefined as a constant I such that

r w 2 (r) rwr - w(r) rdrI f kwi,- (17)

Then qnat ion 16 can be simplified to the following form:

I,(r,.) -IIr. * F) I

2w

! ikcw iw 1

Inserting these into Equation 14 and soIking for / produces an empirical result for the value of thets4o integrals defined in inquation 17.

• 8n €'qlq (V-, - V I% ' + 0, + Ti) 1 , 0

I [4: 1 (1,)

This can now be substituted into either Equation 13a or Equation 13b to produce an ionosphere-corrected observation of the range difference r(i,) - r(t + T). These are written as Equations 19aand 19b:

r(+) r(, + r) c x,~ -[q, v),lv,., -, W,., +v,) 1] , 2n.fq,(v,,. + v )

+ (19a)

10

- - - - - -, - I : -"

rI)-r(th + T) N2 - [q 2(v., + vj T- Q2( VOL + VL)7' - tp2/2tr}- q2 ( aq + vI .

+ !(I 9a)8. 'q2(v,, + v.) 2

ERROR ANALYSIS

Now that the conversion from raw Doppler counts to range difference has been developed, itwill be informative to investigate the sensitivity of these expressions to errors in the measurements.The last term in Equation 19 can be combined with Equation 18 to eliminate some of the constants.This new parameter is defined in Equations 20a and 20b:

I cq, N M + (,, +v,8J 8q q(v,,, + v,) (ql' - q2)(vo, + v,) q-- q

T [,- Q]- h W. _t,.] (20a)

I cq 2 fN, _ N 2 + (v,, + v, )8Tn2q (v. + V,)2 (q2 - q)(v,,, + v,) q, q2

_- - [ 1 _ 2 (20b)

The parameters that will be considered to he uncertain are v-. vI, N,. N2 . lp,. w2 and T.

Special consideration must be given to Wp, and t12, since they are highly correlated. The form of'Equation 10 can be used to describe both tp, and tp,

tp, = q,[l,(tu + T- Tj) - qj(t - T11 - Q'l1,(I + T)- P71(to)l

I2 = (I I7,(f + T - re,) - , - - ,1'71 (t + T) - v,

Using these, the quantity tp, ,q, - p, "q,, which is common to both J,, and A-, can be written:

WI' _ P 2 = j + FT,(T - I r)-n, - i)l - Q' [v),(f + T) - ,(t,1)q z. q,

-l[rl.(lR + T- r,,)- Y.(I T,,) Q' [,i,(t + T)-r, t)]'

Here the satellite phase noise contribution combines to add to /ero, which leaves only a reducedcontribution from the local frequency reference.

II

- 9- .

_1_ - _p = (02 -_ (F_,R, + T) -7L(I.))q, q2 q q,]

-1.06 x l0-[,L(t, + 7) - 17.(IR)J

Thus, in addition to partials of J. and J, with respect to W, and P2, the partials Stp,/Sr andatpz/SiL will be defined.

3P, q, (_2 -

Sri q2 q1 /3We2 =_atP, qIaS q

where

= [(6,(t + 7) -T

The partials of J with respect to the seven parameters are written below:

SJ,, = -cq 21 N,__ N_2 + (Vo, +v,)

aV, 2qI q2 (v,), + v,)2 q,__+, 021 - _, _,,__+

T [__ _ Q ,+ Q I+ - I [ ,,, + +,q I q2] 2n q.,

J,, = -cql I FN, - + (Vil +VI,(q - q) (vo, + ) 2

Sq , 2 2nq q q

<J" cq *+ T [ - Q_ 2Si,, (qI - q )(v,,, + v,

_J_ = (q-cq + T]S2,, 2 )(V, q2SVL + V') qqj

22

SN2 2q -)(V, + V,)

IN ql)(V-I- + V')

aJ_ = -cq,aN2 (q11 - ql)(vo, + v,)qz

aJ. = -cq2L, (q,? - ql)(vo. + v.)2nq,

ai = -cq,ap, (q2 - q2)(vo. + v.)2n

aL. = cq2

atpa (qI - ql)(vo. + v.)2n

ail = cq,

aW2 (q? - qz)(v., + v,)2nq2

__J. = cq,,,woL + VL) rQ. - Q]

T (q, - )(v. + v,) [q q

a b= (i (VOL +VL) [Q.afT (q'- .,"v,, + v,) q q

The error in J due to these parameters is obtained from the total derivative:

dJ a J dv,+ aJ dvy+ aJ dN + A_J dw+ aJ dT (21)-v, av, 8N 8 aT

If values are assigned to the constants, the sensitivity of J to each of the parameters can beevaluated. The appropriate constants are

vo., 10.23 x 10' HzV- 0

VOt 5.00 x 10' Hzv/i =---0

C 2.99792458 x 10' m/sq, 154, q, = 120), 315.07825, 0 = 245.51425T= 60 sN, = N2 = 1725000PS = t= 0

Using these, the numerical values for the partials are

J ..= - 0.0992 m/Hz7. = - 0.1634 m/Hzav, 8v.

SaJo = 0.0288 m/H,, 3Jb = 0.0474 m/Hz

13

-7 't-

3J. = 0.2941 m/count, Ji = 0.4844 m/countSN, SN'

0Jo = - 0.3775 m/count, 3ib = _ 0.6217 m/countS N 2 S N 2

SJ = - 0.0468 m/rad, O = - 0.0771 m/rad-1p. a 1P,

"J = 0.0601 m/rad, IJ2 = 0.0989 m/rad

01. = 2396.0128 m/s, 1-4 = 3946.1000 m/sST aT

In a similar fashion, Equation 19 allows the definition of D. and D, as the uncorrected L, andL, range difference observations:

D, = c IN, - [q,(vo, + v )T- Q I(VOL + vL)T] - y,/2T (22a)I q,(vo-, + v,,)J

c IN 2 -[ ° +(v)T- O( °L + v L) T ]

- /2ni1 (22b)* I q2(V0 ., + V,)J

The partials of D. and D, with respect to the same set of parameters are

SD = -cT _- C N. - [q,(v., + v)T - Q,(VOL + VL)T] - ,

av, v0. + v., q,(vo, + v) 2 2

SDh = -cT - C N, - q,(vo, + v.)T - Q,(VoL + -L)] ',Sv, vo, + v, q,(vo., + VJ ' 2J

S D = -cQ, TSiL q,(vo + v.)

a D,, = -CQ, T

aV, q,(vo, + v,)

aD, = aD = 0SN, SN,

aD = c SDC = CSN, q,(vo, + v,) SN2 q2(vO, + V,)

S D, _ -c

S3P, 2nq,(vo. + v,)

14

'1 1 4-" . . i - , : ' , - " . . - , , .. . . .. ' . . . . ." . 7 f i - . . , . . ..

SDO, = -c

051,o, 2nq 2(vo., + v,)

SD,, = D. = 0

D0 = C [- q,(vo, + v,) + Q,(vo, + viL)]a T q,(vo. + v,)

aD = c [- q2(v0. + v,) + Q,(voL + vL)I

ST q2(vo' + V,)

The sensitivity of D to each of the parameters can be evaluated using the same set of constantsas before. The partials with respect to v, are a function of the actual Doppler count and are aminimum when the Doppler frequency is zero. This is because the terms

q,(vo., + v) T- Q,(VOL + VL)T

and

q2(v0 , + v.) T- Q2 (Vo. + v,)T

equal 1,725,000. When N, = N, = 1,725,000, the term in braces is zero. In order to represent theworst case, N will be adjusted to the equivalent of a ±4000 Hz Doppler frequency in these two par-tials only.

SD0 = -1758.3036 m/Hz, aD,, = -1758.3048 m/HzSir, a r,

SD. = -3597.4143 m/Hz, SDh = -3597.3957 m/HzavL aVL

SD. = 0.1903 m/count, SD,, = 0.2442 m/countSN, SN,

SD_ = 0.0303 m/rad, SD,, = 0.0389 m/rad

SD. = -5470.9 m/s, SD,, = -7021.0 m/sST ST

A comparison of the values of the partials for D and J indicate that, in all cases, the greatest con-tribution to the observation of a range difference will come from the D's not the J's.

Equations 19a and 19b are equivalent; therefore, the range difference observation can be ob-tained from either. However, inspection of Equations 20a and 20b shows that, for a given error in

-. any of the quantities in the braces, the resulting error in J. will be less than in J,,. This is because of

15

I

V7

the different multiplying factors involved. Also, Equations 22a and 22b indicate that an error in thecount N, time interval T, or in the phase noise tp. will be less in D. than in Db. Again, this is becauseof the multiplying factors. This conclusion is also apparent in the values of the partials listed above.In all cases, the partials of D. and J. are less, respectively than those of D and J,. However, the er-ror propagated into the ionospheric corrected range difference observation due to errors in Nor 1P isthe same irrespective of whether D. or Db is used as the basis of the measurement.

This can be illustrated in the case of an error in N by employing the appropriate partials. Thepartials derived from Equations 19a and 20a are

Ar a D. AN,+ 8 J. AN. + a J. AN,aN, aN , aN 2

2Ar= C AN, + cq2 AN, + (-cq2) AN 22 2 )(V..2

q,(v0, + v,) (ql - q2)(v.+ v,)q, (ql -q)(v. + v,)

Assurning that AN, = AN2 , this results in

Ar = c [( q ,-q 2 ] ANIq ' - q2)(00, + V)I

A similar expression can be written from Equations 19b and 20b:

Ar aDh AN,+ a4 AN,+ a J1 AN2aN2 SN , aN 2

Ar= c AN 2 + cql AN,- cq2 AN,q2 (vO + v,) (qi 2 q2)(V. + v) (q - )(. + v,)

Assuming that AN, = AN2 , this also results in

Ar = c[ q , - q 2 AN( 2 2 q )(Vo, .

A similar result is obtained when the error parameter is ip. Consequently, if the primary error source is dueto uncertainties in N or ip. either expression produces an equivalent error in the final range differencecalculation. Since this is the case, the next section will dwell on evaluating the errors from D. and J..

EXPECTED OBSERVATIONAL ERROR

An estimate of the error in each observation can be made by using the preceding partialderivatives and the expected errors in each of the parameters considered. These expected errors willbe based upon the considerations discussed in the following paragraphs.

16

The frequency offset v, will be expected to drift at a rate of about 4 x 10-'8 parts/s. Using thenominal frequency for v,,,, this is about 4.1 x 10-" Hz/s, or 2.5 x 10-9 Hz in a 60-s observationperiod. The frequency offset v, will be expected to vary less than 1.2 x 10-' parts of Vo,, or 3.5 x10-'0 Hz. This latter number is from the HP5061A-004 specification.

The Doppler count accuracy is limited by the resolution of the fractional time interval counter.At the nominal offset frequency of 28750 Hz, the ± 10-ns resolution of the time interval counter isequivalent to approximately 2 x 10-8 x 28750 of a count. This is 5.75 x 10 4 cycle, or 3.6 x 10-3 rad.The white noise from the receiver is on the order of 0. 1 rad rms, so the limiting factor is not thecounter, but this receiver noise. Thus, the Doppler count error will be taken as 0.1 rad, or 0.016count, in the rest of this development. This error source should decrease with increasing signalstrength.

The phase noise tp comes primarily from the frequency standards. Equation 10 shows that it is acomposite of contributions from the satellite and local references. The satellite will be assigned anAllen variance over a minute of 2 x 10-" (Reference 4), while the local reference will be assigned I x10-'2.The phase variation over I min can be estimated from Equation 10 in the same manner asbefore. This expression when evaluated becomes

q,1 7,(t + T) - r7,(t) ] = 2n(2 x 10-12) (154) (10.23 x 106) (60) = 1.19 rad

In a similar fashion, the result for the local clock is

Q,['7(tR + T) - rT,(t,)] = 2Tr(I x 10 -12) (3.15.08) (5 x 106) (60) = 0.59 rad

From both of these, the magnitude of Atp = 1.33 rad.

The accuracy of the Doppler count interval is correlated with the count itself, since the intervalis triggered to begin and end by the zero crossings of the Doppler count. Since this signal has noiseattached to it, there will be a corresponding uncertainty in the duration of the interval. Using the0. l-rad figure as representative of the noise and the duration of the nominal Doppler cycle as 3.5 x10-5 s, the time corresponding to 0.1 rad of this frequency is 5.5 x 10-' s.

Finally the error in the observation will be calculated using the total derivative as in Equation21.

A. = a J4 Av+ ' J Av,+ a J AN,+ a J, AN2+ a J, Alp,+ a4 Ai), + M, ATSv, Siv, SN, TN, Sip, a42 a T

The magnitude of AJ,, is obtained from the square root of the sum of the squares of each of theterms above. A similar expression will be used for AD,.

AD,, = D,, Av, + aD,, A,, + SD, AN, + SD,, At, + SD, ATSiV, Or, aN, SiP, S T

The results of the calculation, term by term, follow:

J,,~~~~ ~~ v,=-.x10' , D,,A', =-2.5 x 10"m SD Av, -4.4 x 10-6 m

Vr, Si,

__J, Av, = 1.0 x 10-'' m D,, Av, -1.3 x 10-" i43r, av,

17

-- - -- . 1- . 1 - -A - -,----,.,

ia i,, AN W•W In '.

.. A.\' 0.()4- ii. ,31 A\ ()(03))(in

aN, J\

a J,, AN' -0.(X)6) i. D A )a N,

a J, A, = a4 04l AI = -8 × I( 0i, A4), = 0.04( iai, Dp, or7 jP,

a J- Ay), = a 4l, 2 Ar = -8 x 10- in, a A4, = 0SIpz at4)2 517 Sat.,

aJ- AT = 0.0013 I , ')- AT = -0.(030 ma T a T

It is interesting to see that the accuracy of the ionospheric correction is limited not by phase noisefrom the frequency standards, but by tile errors in determining the Doppler counts N, and N,. Thisnoise comes primarily from randon phase fluctuations in the receiver. If N, and N, are considered to beindependent, then the magnitude of the noise expected on the ionospheric correction observation is

\SN a - , + j]AV

AJ,, 0.0076 ill

The contributions from the reinining parameters arc small enough to ignore.

The error in tile ralge difference ohscrxeation, 4.02 cin, is dominated by the frequency standardphase noise. If this error source \%ere reduced by a factor of 20 or more, then the limits would be setby the error in determination of file count and tile exact interval of the count. This factor of 20 isequivalent to improving tile Allen variance from 10-" to 5 x 10 '" o\er the interval Tand would becomparable to the pertoriance expected troni a Hydrogen Maser.

The levels of error in D, and J, predicted by the preceding analysis have been empiricallyverified by actual data. NAVSTARS I through 4 (SV's ,, 7, 6, and 8) were operational while theNGRS was at Yuma in 1979. The Doppler data that were obtained from all four satellites were pro-cessed so that 20-min segments were fitted to a third-degree polynomial using least squares. Theresiduals of this fit were used to determine ati rms that is considered to be representative of the ran-dom fluctuations of the Doppler observations. Uncorrected L, Doppler data corresponding to D,and the iono;pheric correction to L.,, corresponding to J,, were processed in the same way. The rmsvalues of the residuals for D. are plotted vs date in Figures 4 through 7, and the residuals for 4, areplotted in Figures 8 through II.

lh1 rotighot thecse 44 days. tlie local frctIIency reterence beha\ ed norm1ally, ad so tile jumps inilie calctlated r1ts (e.g.. thI,,c ii I ig rc 4 bet weeH da\s 45 and 54) ar-e due to imisbeha\ior of ilie SV4freqtkenc.,tIandaiyd. Note. ho\\e\l-, .ial during these iiIe days there is no e\idence of anything

amis in Iignlrc 8. This ilhltsltrates thll file ionospheric corcc ion i, not ,eisitivc to tihe phase noise ofthe frequency tlatndards. lhis " a,. ie result predicted by lhe anal'.,,i of .1_ Anl e,,pecially graphic

18

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Figure "i. 20-Min L, Range Difference RMS (cm) Vs D)ate,OPS SV-8 , Yuma Proving Grounds, Days 36-80 1979

* 22

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demonstration of this is asailable from SV7 in Figures 6 and 10. The rms of the residuals of D,, forthis satellite were poor throughout the 44-day span; however, the ionospheric correction residuals inFigure 10 are unaffected and are comparable with those from any of the other satellites. This featureis certainly a plus for anyone interested in performing ionospheric measurements, because it meansthat they need not invest in a %ery Io%%-noise, and consequently expensive, local-frequency standard.

Since the error in D. is primarily due to Atp and the error in J,, is due to A,'. they should be in-dependent. Therefore, the predicted error in range difference wkould be equal to

Ar - I(AD,)I + (v "0 112

Using the numbers derived previously, the range difference observation error should be about 4.0cm. This is in excess of what is observed for satellites 4, 6, and 8, but less than that for satellite 7. Ifthe primary error source is due to the random phase fluctuaions of the frequency reference, thenassuming that the phase fluctuations of the local reference are represented by o, , the estimates ofthe satellite values can be made using the method previously outlined. The numerical value for o,will be taken from the Allen variance at T 60 s (Table 2). With this value fixed, a numberrepresentative of the satellite frequency reference can be calculated using the following relations:

A),, = alD" At,

and

At, = Wf1k.(N + ) + I)V + [ + 1 1,00' + H

for this application

Ay, - 2n x N) x 10, I(10.23 x 154 x ) + (5 x 315.07925)' o,

Solution foi 4 and substittion for the aserage A), obtained from I-imures 4 through I I oser the en-

tire 44 days gises the rCut' listed in fable 2.

'able 2. Satellite Phase Noi,,e )cri ed From I)oppler ()bsersation,,

SV Al),, Satellite Phase Noise Assumed I ocal Value(ci1) 01,

4 2.98 1.34 x 10 " I - 1o 2

6 2.57 1.4 I ) I It) "

7 10.61 5.83 It) I - 10

8 3.76 1.85 - 0o I - I0 "

27

CONCLUSIONS

The formulation for converting Doppler counts into range difference observations has beendeveloped for the case of the NGRS. The two frequency data were combined to generate the first-order ionospheric term. When this term is subtracted from the uncorrected L, range difference, theresult is the ionospheric-corrected range difference.

The sensitivity of the observation equation to errors in the observed parameters was in-vestigated by evaluating the partial derivatives and using them in the expression for the totalderivative. Errors in the parameters were estimated and used to determine the error expected in therange difference. These estimates were then compared with examples of real observations obtainedwhen the NGRS was on site at the Yuma Proving Ground. Agreement was reasonable and tends toconfirm that the analysis is correct. Further tests are planned based upon the predictions that can bemade from results developed in this report.

REFERENCES

I. Journal of the Institute of Navigation, Summer 1978, 25, No. 2.2. Hewlett Packard Electronics Instruments and Systems, 1980 Catalog.3. L. R. Gibson, Some Explansions for an Electfrotagttefic W are Prolpagafing Through a

Sphericallv Symnetric Refracting Medium, NSWC TR-3344 (Dahlgren, Va., June 1975).4. H. Hendrickson, "On-Board Clock Data Processing", GPS Reference Ephemeris Data

Analysis Working Group Presentation, 14 November 1979.

4i

28

7*, e ,

- /

APPENDIX A

Frequency Spectrum of the P Code Mtodulation

4b

Define the phase argument of a cosine function to be 0(t) where

(t)= w, i + 0, + kg(t)

The parameters w_, 0,,, and k are constants, and g(t) is a random gate function of unit amplitude.The random character of ,(t) is best demonstrated by Figure A-I. A mathematical expression forg(t) can be devised by the time translation of an elementary gate function centered on zero as inFigure A-2. Using this concept, we can express the random gate function as the sum of translatedg'(t)'s multiplied by a random variable a,.

g(t) a,g'(t - IT) (A-I)1=-.

The value of a, is either one or zero depending upon the outcome of some random process related tothe subscript /

g(t)

... i i **---1--1 1 I fI~ I.- I llulil ... I -Ti I I I I I

-7r -6r -5'r -4r -3r -2r -r 0 7 2r 3r 4r Sr 6r 7r 8r

Figure A-I. Random Gate Function

-762

Figure A-2. Elementary Gate Function Centered On Zero

I f we choo,e 0,. 0 and A = n. then 0(t) becomes

O(t)= , t + ng(t) (A-2)

Flw31

* . - - y -r

No%% according to the del'inition ot g(t), thle value of' 0(t) canl be only w, I+ Tr or w, t. This is thlehip/nise part of' thle argument. When 0(t) is introduced into a cosine Function, the result is

Cos 0(t) = coskIw, + nig(i)J

which when e.\panded becomes

cos 0(t) =cos u), t cos Tng(t') - sin uw, Isin Trg'(t)

Now%, since ng(t) is either 0 or ri. the second termr vanishes, which leaves just

Cos O(t) c CSW, tCos rn(t) (A-3)

The second cosine f'actor here is either + I or -1 depending onl whether the argument is 0 or n, wkhichinl turn depends upon whether it, is I or 0. Since this is thle case, we canl construct a newv factor toreplace

COS ng(i)

which behaves in the same manner. Let this be m&t) where

In(t) = 2g&) - 1.

Equation A-3 written utsing this new factor is

cos 0(t) = 2g~(t) - Ii Icos w, t. (A-4)

Thle random f'unction ,n(t) represents the pseudo-randomi code modulating cos w, I. The Fouriertranstorm of' the elementary gate function is

Ig( Tr sin WT/2 =-G(w) (A-5)

Employing thle time shilt proper-ty,

FII t- T) I/(w ) e

allows FI'(i - /IT) I to be evaluated fromn Fig'(t0f as shown below%:

- nr~ = s~i ijc (;(T)

t ollo%%,s then that

FJi IJTSitl (LOr/2 (, V G (w) (1,(e (A-6)

32

'%%it h the implication th! ile cont ribut ion fromn any term of the stum with a,= 0 is als /ero.Thshie frequent:\ spect rum ot i , (I) is a summation of the form shown above withi an arbitrary number

oft possihie corntributtr inissinv. The terms missing are speciflied by the outcome 01 the randomn pro-ccss conlt rolIin-1 U

Thie basic frequericY spectrum (Equation A-5) behaves as a Sill V/x function mult11iplied b\ aiseries of' conllple\ hmtncHtl UionI ghenb tile Summation in Equation A-6. The first null', of (c) arc

This frequency is 10.23 MHz for the GPS P code. Thus, the bandwidth between first nulls is 20.46MHz and contains the majority of the transmitted energy. Since energy per unit bandwidth is pro-portional to I G(w)V , a comparison can be made by integrating I G(wH) 2 b'2,tween zero and infinityand again between zero and the first null.*

. 2

whereas

This ratio shox\ s that t lic fract ion of' energy between first nulls is 90 percent of' tilie total.

It is interesting to see how the gate function pictured in Figure A-2 changes when G(w) is Cut Off'due to the finite bandwidth of the transmission and receiving systems. We can approximate this ef'-feet by defining a gate function in the frequency domain W(w) such that it is unity between ± U), andzero elsewhere. This function (pictured in Figure A-3) is then multiplied by G(w), which Pgives theproduct G(wo) 11'(w). Since mu tlt iplicat ion in one domain is equivalent to convolution in the other do-main, the resulting function i thle time domain is the pulse

p(() f 11V c 4 (~

Since

w(t,) - CO, i u),1 2 (Figure A-4)

and g'(1) is as betore, /ero tor I 11 > T/2 (Figure A-5), the convolution integral is

U ~ in 2(t W) e(1

33

or

2r2o , I" sin w,/12(t -S-n T . w, 1 ,2(t -/

W(W,)

1

WX WX W

Figure A-3. Elementary Gate Function Centered On Zero Frequency

This bandwidth limiting causes appreciable distortion in the original pulse shape (Figure A-6);however, this does not appear to degrade the resulting pseudo-range measurement to any greatdegree. The GPS receiver cross-correlates the received pulses or chips with a locally generated code.The result of the cross-correlation of one local chip with the received bandlimited chip is sho%%n inFigure A-7. This correlation curve is then advanced one-halr chip, Ti 2, and retarded one-half chip.The retarded chip is then subtracted from the advanced chip defining a /cro crossing, which in-dicates the exact time of maximum correlation. The result of these manipulations is shown in FigureA-8. This is a much more sensitive method of obtaining maximurn correlation becaue the actualcorrelation curve (Figure A-7) has a rounded top, whereas the advanced minus retarded curve has awell-defined zero crossing due to the steep slope. This method of finding maximum correlation iscalled a delay lock loop.*

If there were no bandwidth limiting, the chip shown in Figure A-5 would be received withperfect fidelity. Consequently, the cross-correlation function would have a triangular shape (FigureA-9) instead of the smooth curve of Figure A-7. The shape of the curve from Figure A-9corresponding to that of Figure A-8 is drawn as Figure A-10. These two curves are more closely ap-proximated if the bandwidth of the receiver is increased. This result is illustrated in Figures A-I Ithrough A-14 for the case where w, = 2r/T. and in Figures A-15 through A-18 the case where w, =3

1T/T.

R. (C. t)iioi, S/pread S!p/ /linm Sv ms, John \Vile, ond Sol,,i p. 205.

La3

x-2.200 -1.400 -0.6000 0.2000 1.000 1.800

C! C!tt

0

_j

. a -

C3 0a 0A* ,g

-2 .200 -1- 400 -0 .6000 0.2000 I ,000 I .900T U

Figure A-4. Frequency Domain Gate Function Transformed

to *he Time Domain, w,, = PiTAU

35

"a " ';" ' -- - " -- -- . . ..*

-2.-t0o -1.400 -0.6000 0.2000 1.000 1.900

0 0C? C?

0 C,

in Sn

0 10

0 0

C? C?

C0

0 C

o

in S

*-2.200 -1.400 -0.6000 0.2000 1.000 1.000

TRU

I-iguIre A-~5. 1icnicnii a,\ ( ime ILufcI ion in I'imew D omain, TAUJ 97.752 nN

36

-2.200 -1.400 -0.6000 0.2000 1.000 1.800

a a

in i

1=;..

a * *C3*

LU! C; *

InI* 'a cc:* ';

* '?

a * 'a aCc,.a * aI.' a 'aN

1210 'a 00 0 '00 1.0 .0* 'a

Fiur A6.CovoutonofElmetay at Wt TransomdGtw U(Pulse* Shp'fRcieahp uet iieBn.-dh

* 37

-2.200 -1.400 -0.6000 0.2000 1.000 1.900

CD a

0 a LIV m

o0 a

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* 8

x-2.200 -1.400 -0.6000 0.2000 1.000 1.000

!II I I

ICI

N\.Y

' TRU

NM

In* ww0

04

"4 Figure A-8. Half 'Chip Farly Minu% Half C'hip ILate Correlation C'urve,1 o.,, =PI/TALU

39

I00 p=,

-7"0

Figure A-9. Cross Correlation Function of Two Gate Functions of Width T

-r *

4

Figure A-10. Half Chip Early Minus Half Chip Iate Correlation Curve

40

VL

x-2.200 -1.400 -0.6000 0.2000 1.000 1.800

A

_J

2.200 -1.400 -0•6000 0•2000 1.000 1.800

TRU

_ Figure A- 11. Frequency Domain Gale Function Transformed to the

Time Domain, , -2 PI/TAU

41

o 0•

-2.200 -1.400 -0.6000 0.2000 1.000 1.600

ol 0

o C

0 CD

in L

0 0l

C:* 0D

CD0CD0

0 Im

0

-220 -140-060 00 1.0 D0T R I-

Figure~ ~ ~ ~ ~ A02 f 'eetr~ t ih rnfridO e . P/ACI (Pls Shp0f eevdCi, ct iieBnwdh42 0

o aak

x-2.200 -1.400 -0.6000 0.2000 1.000 1.500

oVoA

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x-2.200 -1.400 -0.6000 0.2000 1.000 1.900

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o 220 -140- 0 0 0a00100 .0a aU

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a 03

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0 0•

0 0

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en3

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- -,- . -- 4 + +-- 0

@-2.200 -1- 400 -0-6000 0.2000 1 .000 1 .800

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illl C I I W POll'lU D % 1111d]llI (lilt. I llll+.'Illn |l;11) t~N 1., I[ to lilt

45

-'-A .Ndi~- .0

x-2-200 -1.400 -0.6000 0.2000 1 .000 1.800

o.

* 0

0>-0 0 0 0

* o

-o 20- 40-.00 .00I-0 00

czc

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P f

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(Pulse0 Shp0fRcie hp Det iieBnwdh

046

x-2.200 -1.400 -0.6000 0.2000 1.000 1.900

0 CD

* 4,* 4,

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", ,

4, 0

o

o D:D

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-)

o 0 ' ,; -4 -

in LO

00

0 0

-2*200 -1.400 -%.6000 0.2000 1.000 1 .900

TRU

Figure A-17. Cross Correlation of Received Chip With locallyGenerated Chip, w,, 3 PI/TAU

47

- .-- -- ' k a i im, d . -- -. . .... ... -r

x-2.200 -1.400 -0.6000 0.2000 1.000 1.800

o CDI I

Ca)0

0,

0000

LO ,_

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0 0* S

o 2 2 0- . 0 - . 0 0 0 . 0 06.0 00, 0

* D

0 I- 20-140- 600 0 00 1.001.0

Figure A-18. Half Chip Early Minus Half Chip Late Correlation Curve,co,=PII/TALJ148

0I

APPEND)IX B

LOCAL CLOCK CORRE~CTION FROM PSE:UIO-RANCE MEASUREMENT

The single-frequency pseudo-range is used to obtain the local clock correction. As was il-lustrated in Figure 2 of the text, this correction, r,, can be obtained if T, (the satellite offset fromGPS) and T,, (the satellite offset from the local clock) are known. The predicted values for ', can beobtained by decoding the satellite navigation message.* The corresponding value for To can be foundif the true slant range to the satellite is known at the time of each observation. A reference trajec-tory, or the data from the navigation message, can be used to determine the satellite position. At thetime transmission, let this be rt, ). The receiver position in inertial space at the time of reception isr (t,). The slant range r is the difference

r(t) = rt,)- r(t,)

Therefore, the propagation time is T, = r/c. Propagation delays due to refraction are not includedin this calculation, since the errors they contribute are less than the l-p.s accuracy desired.

The pseudo-range measurement is concluded at t, when the satellite epoch is received. Conse-quently, the local offset will be time tagged t, which equals the satellite epoch of transmission t,plus the correction to the satellite clock T, plus the propagation delay r,. Therefore, the local offsetis

rO(/2) t ",(ti + T, + T,) = T ' T,

From Figure 2, the correction to the local epoch T, must be

T, = T, - T 0 = T, - Tp + T,

where r, is obtained from the satellite NAVDATA message and is calculated using the station andsatellite positions at the times of reception and transmission, respectively. The pseudo-range obser-vation is ,.

The data T, and its time argument t, can be collected and fitted by a polynomial whose coeffi-cients were obtained using least squares. The polynomial would have the form

T,(t)= lt

Using this, the local time is corrected by putting the apparent local time in for t and calculating thecorrection U,(tI). Several iterations may be used to improve the accuracy.

'van t)icrcndonck. R,,,,ell. Kopit/kc. Birnhaun. "The (PS Navigation Mesage", Navigation, Vol. 25. No. 2, Summer 1978.

51

--~~~7 7r- -~-- -

APPENDIX C

MAXIMUM DOPPLER AND DOPPLER RATE

4b

This simplified development establishes a worst case configuration for relative satellite-earthstation motion. It is useful in that it sets limits on the maximum expected Doppler frequency (veloc-ity) and Doppler rate (acceleration).

The worst case is presumed to be a satellite at GPS altitude in a retrograde equatorial orbit(Figure C-1). No GPS satellite is ever expected to be in such an orbit, so the results calculated herewill determine an upper limit on the velocity and accelerations along the slant range vector.

We\

Figure C-I. Retrograde Equatorial Orbit Geometry

From the law of cosines

r2 + r.-2r, rk cos0 (C-I)

Then

2r dr 2r, r. sin6 dO_ (C-2)dt di

Assume that w, and w, are constant.

dOdt

55

Ck-AwGia, i.h bj.M L4K-NOI FlJ,%,u

"," 7 . .--. - -. -- -. - .

dr =v=r,rr' wsin0 (C-3)

dv = r,ro r-'cosO dO -sin Or-2 dr (C-4)

Set dv/dt = 0 to find the e, which gives the maximum velocity:

r 2 cos0=r,rsin2

This can be rewritten by substituting for r2 :

2 2r, + rg cos0=l+cos'0 (C-5)r. rx

This result is independent of w. Substituting r, 26560 km, r, 6378 km results in a value for6 760. The combined angular velocity

= 2r + 4n 6n

86164 86164 86164

Evaluating Equation C-3 at 6 - 760 gives the maximum velocity

r,r~w sin 9 = 1395 m/s (C-6)Ir + r.' - 2r, r, cos 0 1 ,'

At the two frequencies, L, = 1575.42 MHz and I.2 = 1227.6 MHz, the Doppler shift is approx-imately 7326 Hz on L, and 5708 Hz on L. The acceleration is obtained by substituting EquationC-3 into Equation C-4:

dv = r,r~Ir-'w cos 0 - sin 2 0 r'r,rwjdt

dr= r,r -2r'[cos 0 - r-2r,r, sin' 01di

By inspection, it is evident that dv /dr will be greatest when 0 = 0. The second term in brackets is zerowhen 0 = 0 and is greater than zero for any other 0. Also, Figure C-I shows that r will be a minimumat 0 = 0, which makes r' and, consequently, dv/dt a maximum. At 0 = 0, Equation C-4 becomes

dr_ 0 .=r,r,w 2 = 0.402 m/s'dt [~ r,2, r- 2r, .] '

56

-, : ;".pP - '7- ..... . . " - " " " '' - ...

The maximum Doppler rate is also at the angle of maxiinum acceleration. At the two GPS frequen-cies, these are 2.1 Hz/s on L, and 1.6 Hz/s on L2 .

The error in the Doppler count over 1 min due to the use of the average frequency can beestimated using these rates. With an L, rate of 2.1 Hz/s, the average frequency can be in error byabout 63 Hz at each end of the 60-s count interval. Assuming a value for the fractional interval equal

to a full cycle (using the zero Doppler frequency 28750 Hz), the count will be in error by 63/28,750= 0.002. This is equivalent to a 0.04-cm error in the delta range measurement at L,. Thecorresponding error at L, is 0.03 cm.

415 57

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