Frequency Calibration of A/D Converter in Software GPS Receivers

Transcription

1 Frequency Calibration of A/D Converter in Software GPS Receiver L. L. Liou, D. M. Lin, J. B. Tui J. Schamu Senor Directorate Air Force Reearch Laboratory Abtract--- Thi paper preent a oftware-baed method to calibrate Analog-to-digital converter (ADC) ampling frequency in a oftware GPS receiver. Two front-end ytem of a GPS oftware receiver were ued in thi tudy. The ampling frequencie of the ADC were found deviated from the nominal value provided by the manufacturer. Software algorithm wa developed to calibrate the ampling frequencie. The algorithm i baed on the initial-code-phae velocity and the carrier-phae velocity of the GPS ignal at L1 frequency obtained by oftware receiver meaurement. Allan variance analyi of thi frequency characterization method i alo included in thi tudy. I. INTRODUCTION A typical radio frequency front-end of a oftware Global Poitioning Sytem (GPS) receiver include an antenna, amplifier, local ocillator(), mixer, and an analog-to-digital converter (ADC). The GPS ignal were received, amplified, down-converted, and digitized into bae band ample. The bae band ample are then proceed uing oftware routine to generate the receiver poition information. Thee oftware routine typically achieve three operational function: GPS ignal acquiition, tracking, and poition calculation. The accuracy and tability of the frequency ource() in the front-end hardware affect both the IF and ADC ampling frequencie. The uncertainty in the ampling frequency will further propagate into the uncertainty in the carrier and Doppler frequencie. The larger the frequency uncertainty, the more frequency bin need to be earched in order to find the right Doppler. A a conequence, it take longer time to perform data acquiition. During the tracking operation, the frequency ource intability may lead to untable ticking poition between navigation bit which will caue error in ephemeri parameter. A a reult, the receiver poition olution may be either erroneou or completely unachievable. In mot commercial GPS receiver, the ampling frequency of an ADC provided by the manufacturer i of a nominal value. The true ampling frequency i uually reolved or characterized by uing a tandard frequency counter. In thi tudy, we developed a oftware algorithm J. T. Morton Department of Electrical Engineering Miami Univerity, Oxford, OH for the frequency characterization without uing any additional hardware except thoe component decribed in the front-end of the GPS receiver. Thi algorithm i baed on the propertie of the GPS atellite C/A code and of the pecific frequency plan in the ytem' circuit deign. The frequency characterization proce i integrated into the oftware receiver. The frequency calibration technique decribed in thi report i eentially a technique of time and frequency tranfer uing GPS ignal. Thi technique i made poible becaue that GPS preciion time keeping capability i continuouly maintained and the ignal are available globally at mot of the time. While mot GPS time and frequency tranfer ue hardware approach, thi tudy ue a unique oftware approach. The advantage of the oftware approach i it flexibility to implement advanced algorithm for the purpoe of enhancing ignalto-noie ratio or of adopting new ignal when they become available. II. EXPERIMENT Two GPS receiver front-end were ued in thi tudy. One i a cutom-deigned and developed ytem and the other i a commercial-modified ytem. The former contain two high quality frequency ource a hown in Fig. 1. One ource i ued a a local ocillator, and the other i ued to control ampling frequency of an ADC. The firt frequency ource down-convert the GPS' L 1 frequency (f L = GHz) to MHz. The econd frequency ource feed a ampling frequency of 5 MHz into ADC. The chematic frequency plan for thi cutom ytem i hown in Fig. 1. The commercial-modified ytem contain one frequency ource, two mixer, one ADC, and a equence of multiply and diviion circuit a hown in Fig. 2. The primary frequency ource i operating at a nominal frequency of MHz. Through multiply circuit, it feed GHz into the firt mixer and MHz into the econd mixer. Through a diviion circuit, it feed MHz into the ADC a a ampling frequency. The nominal IF and bae-band frequencie are alo hown in Fig /05/$ IEEE. 652

2 III. FREQUENCY CALIBRATION ALGORITHM 1. C/A code Correlation The civilian GPS ignal were ued in thi tudy. The civil GPS ignal have a carrier frequency at L1 band and are modulated by peudorandom code from the Gold code family, the C/A code. The C/A code i a bi-phae code with a chipping rate of MHz. The period of the code i 1 milliecond. The GPS oftware receiver' front-end generate digitized data. The digital data can then be proceed uing oftware algorithm on a block bai. Each block contain a milliecond of digital ample. A oftware acquiition routine wa developed to correlate the meaured digital data with a locally generated reference ignal coniting of a bae-band CW ignal modulated by a C/A code equence [1]. The reference ignal bae-band frequency wa allowed to vary within a reaonable range o that to cover a poible Doppler value. Fig. 3 how a typical correlation reult. For each milliecond of data, the initial C/A code phae repreented by the initial code index (e.g., the equential order of the data point, and it i not necearily an integer) and a coare Doppler frequency can be obtained from the correlation reult. Thee parameter can be ued to demodulate the input ignal and to obtain an average carrier phae for the one milliecond data block. Thi block-baed proce i carried out for everal econd of data to yield ditribution of the initial C/A code phae v. time and the carrier phae v. time. 2. The initial code phae velocity A typical reult of the initial C/A code phae v. time i hown in Fig. 4. The data how a globally negative lope. Thi lope i defined a the initial code phae velocity. The polarity of the lope can be explained a follow. Let u aume that the true ADC ampling frequency i the nominal ampling frequency. If the Doppler frequency i alo zero, i.e., there in no relative line-of-ight motion between the atellite and the receiver, the initial code phae v. time would how a contant value over the time. That mean that every milliecond of data block would how exactly the ame initial code index. If the Doppler i poitive, i.e., the atellite i approaching the receiver, the carrier received by the receiver i queezed into a dener package compared to the cae of zero Doppler. The initial code phae for one milliecond block of data would appear at a leer value than that of the block precede it in time. Therefore, the initial code phae v. time would how a negative lope. Similarly, if the Doppler i negative, one would conclude that the initial code phae v. time plot would how a poitive lope uing imilar reaoning. Now let u aume that the Doppler i zero. If the deviation of the ampling frequency from it nominal i poitive, i.e., the true ampling rate i larger than the nominal value, the carrier wave will appear to pread out more in time than the cae with zero deviation. One would conclude that the initial code phae v. time lope i poitive. Similarly, if the deviation in the ampling frequency i negative, one would conclude that the lope i negative. Apparently, both the Doppler and the ampling frequency deviation affect the lope in the initial code phae v. time plot. The typical reult hown in Fig. 4 i for the cae where the Doppler frequency i about Hz and the deviation of the ampling frequency i -2.6 Hz. In thi cae, the negative Doppler will reult in poitive initial code phae lope, while the negative ampling frequency deviation will lead to a negative initial code phae lope. A hown in the figure, the latter factor i the dominating factor and the final reult i a negative lope. To characterize the ampling frequency, one need to reolve the effect due to two unknown: ampling frequency deviation from nominal value and the ignal Doppler frequency. We achieved the goal by olving a ytem of two equation with two unknown. One equation i related to initial code phae velocity, and the other i related to carrier phae velocity, which will be dicued latter. 3. Formulation of the initial code phae velocity Let the nominal ampling frequency be f and a 0 = f T 0, where T 0 i 1 milliecond (the period of C/A code). If thi nominal value i the true ampling frequency, then, the meaured Doppler will be the ame a the true Doppler. In thi cae, the initial code phae n(t) v. time t i γ a () = + (1) + d 0 nt n0 t 1 γ d T0 where n 0 i n(0), and γ = f / f (2) d d L Here, f d, f L, and f are the Doppler, L1 carrier, and the receiver ADC ampling frequencie repectively. From (1), n(t) v. t how a poitive lope for a negative Doppler, and a negative lope for a poitive Doppler, a i expected baed on the qualitative dicuion in the previou ection. 653

3 If the nominal ampling frequency f i different from the true true value f by δf, the lope of n(t) v. t can no longer be decribed by (1). The new formulation i the following: γ γd + γγd nt () = n0 + ft (3) (1 γ )(1 + γ ) where d true γ = δ f / f ; and δ f = f f (4) Therefore, the meaured initial code phae velocity can be obtained from (3) : dn() t γ γd + γγd = = f dt (1 γ )(1 + γ ) Equation (5) provide one of the two equation to olve for δf and f d. 4. Formulation of the carrier phae velocity The carrier phae v. time provide the information to calculate the Doppler. Fig. 5 how a typical carrier phae v. time plot. The lope i the carrier phae velocity from which Doppler can be obtained. If the true ytem deviation in the ampling frequency i not zero, the meaured Doppler will be different from the true Doppler. The relationhip between the true and the meaured Doppler i given by: d (5) f = ( f ' + f ')(1 + γ ) f (6) d d c c where f d ' i the meaured Doppler, f c ' i the nominal center frequency, and f c i the true center frequency. The relation between the f c ' and f c can be deduced from the frequency plan of the circuit. A general form i given by: f = f ' mf γ (7) c c where m i a parameter determined from the circuit' pecific frequency plan. There are two receiver in thi tudy. For the cutom-deigned receiver, the relation i given by: f = 1.25MHz 4 f γ (8) c For the commercially modified receiver, Equation (7) become: f = 1.364MHz f γ (9) c Combining (6) and (7), we obtain the relation between the meaured Doppler and the true Doppler: f = ( f ' + f ')(1 + γ ) f ' + mf γ (10) d d c c 5. The olution for δf Solving Equation (5) and (10), we obtain f d and δf. The olution for δf i given by: δ f = γ f ( mf + fc ' f ) f = L (11) 2( mf + fc ' + f d ') 1 4( mf + fc ' + f ')( f ' f L ( f) ) 0.5 {1 [1 d d + + ] } 2 ( mf + fc ' f L ) Subtituting f c ' and m from (8) and (9) for the commercial-modified and cutom-deigned receiver repectively into (11), δf can be calculated. Subtitute the reult into (10), we can obtain the true Doppler f d. IV. RESULTS AND DISCUSSIONS Fig. 6 and 7 how typical reult of the initial code phae v. time for the cutom-deigned and the commercial-modified ytem, repectively. The majority of data in Fig. 6 form an apparent linear line. The cattered data point are due to a weak GPS ignal which give poor correlation reult and can be removed uing oftware routine. The lope of the line which i the apparent code phae velocity, can be obtained uing a bet fit routine. The majority of data point in Fig. 7 form three linear line due to the repetitive nature of the initial code index. Comparing thee two reult, one note that the commercial-modified ytem ha a ignificantly larger initial code phae velocity than the cutom deigned one. Thi i becaue that the ampling rate deviation in the commercial-modified ytem i much larger than the one in the cutom deigned ytem. A will be een latter, the commercial ytem ha a ampling deviation in the order of 10 2 Hz, while the cutom ytem one i only in the order of 10 Hz. Fig. 8 and 9 how typical reult of the carrier phae v. time for the cutom and commercial ytem repectively. The lope at each point i the meaured carrier angular velocity. From which one can calculate 654

4 the meaured Doppler frequency. The curvature hown in the figure indicate a varying Doppler a time progree. The commercial ytem ha only one frequency ource and a erie of multiplication and diviion circuit. It ha a large m (=289) value (ee Eq. (9)). Becaue of thi large m value, a mall deviation of δf, e.g., in 10 2 Hz range, will reult in a large deviation in f c. Therefore, it i neceary to can a wide range of poible Doppler frequencie during ignal acquiition tage. The cutomdeigned ytem ha two frequency ource to control the IF and ampling frequency eparately. It ha a mall m (= 4) value (ee Eq. (8)). A frequency deviation from the true f in the order of 10 2 Hz will not affect the deviation in f c ignificantly. Furthermore, when mf and f c ' are much maller compared to f L, a i in the cae of the cutom-deigned ytem, the olution in (11) can be implified to: δ f / f + f '/ f γ = = f 1 mf / f d L L (12) Thi approximation generate δf with value very cloe to the olution obtained from Eq. (11). For the cutomdeigned ytem, the difference in the δf value i within one tenth of 1Hz. Fig. 10 how the frequency deviation from the nominal ampling frequency v. time for the cutomdeigned ytem. It how a time erie with a mean and a variance. The mean approache the true ampling frequency deviation. Thi value can be obtained to an accuracy of le than 1 Hz with le than 0.5 econd of data. The average frequency deviation approache Hz for our cutom ytem baed on our data analyi. Thi value i very cloe to the direct frequency meaurement obtained uing a frequency counter. The conventional two-ample Allan variance analyi wa performed on thi time erie. The reult of the Allan variance v. averaging time in log-log cale i hown in Fig. 11. The lope being cloe to two indicate that the noie proce i of a phae modulation nature in the meaured time frame from 10-3 to 10 1 econd. Fig. 12 how the frequency deviation from the nominal ampling frequency v. time for the commercialmodified ytem. The average frequency deviation i about Hz. The reult of the Allan variance v. averaging time in log-log cale i hown in Fig. 13. The figure how a imilar magnitude to the lope in Fig. 11. It eem that the frequency tability analyi uing thi method yield imilar reult to that of the Allan variance method regardle of the quality of the frequency ource. The reaon may be due to poible dominant noie proce occurring at the peripheral device other than the ocillator. Thoe peripheral device include the high gain amplifier to amplify the weak GPS ignal. The amplifier are known to generate phae modulation in the tochatic proce [2]. ACKNOWLEDGEMENT We would like to thank the upport of the Senor Directorate of Air Force Reearch Laboratory. REFERENCES 1. J. B. Tui, "Fundamental of Global Poitioning Sytem Receiver: A Software Approach", 2 nd edition, John Willey and Son, Inc., New York, D. A. Howe, D. W. Allan and J. A. Barne, "Propertie of Signal Source and Meaurement Method", NIST technical Note 1337, pp. TN-14, Fig. 1, a chematic of the cutom deigned GPS oftware receiver front-end. 655

5 Frequency Plan of the Commercially Modified GPS Receiver mixer mixer ADC F=5.456 frequency frequency x100 x4 multiplier x1/3 frequency multiplier divider Local ocillator Fig. 2 the frequency plan of the commercial modified GPS oftware receiver. The frequency unit i MHz. Fig.3, A typical correlation reult to obtain initial code phae in one milliecond data block. The x-axi i the digitized data equence in one-milliecond block, and the y-axi i the correlation trength. Fig. 4, A typical reult of the initial-code-phae v. time (in milliecond). 656

6 Fig. 5, the carrier phae v. time (in milliecond) with (above figure) and without (bottom figure) the bi-phae due to navigation bit. Fig. 7 the initial C/A code phae v. time in milliecond for the commercial-modified ytem. Figure 6, the initial C/A code phae v. time in milliecond for the cutom-deigned ytem. Fig. 8, the navigation code (upper figure) and the carrier phae (lower figure) v. time in milliecond for the cutom-deigned ytem. 657

7 Fig. 9, the navigation code (upper figure) and the carrier phae (lower figure) v. time in milli-econd for the commercial-modified ytem. Fig. 11, The two-ample Allan Variance of the deviation frequency v. averaging time in econd for the cutomdeigned ytem. Fig. 10, the frequency deviation from the nominal operation frequency (5 MHz) v. time in milliecond for the cutom-deigned ytem. Fig. 12, the frequency deviation from the nominal operation frequency (5.456 MHz) v. time in milliecond for the commercial-modified ytem. The mean deviation i Hz. 658

8 Fig. 13 The two-ample Allan Variance of the deviation frequency v. averaging time in econd for the commercial-modified ytem. 659