The computation of PCA is studied and compared with that of FFT-based acquisition. Here we assume that the computations regarding the local sequence are omitted, since it can be calculated in advance. For the 1st-layer process of PCA, the derivation of the phasors of the input sequence as shown in Eq. (4.16) requires K1(M1 additions. In addition, the 1) calculation of G in Eq. (4.25) requires m(1) K K1 1 additions (subtractions) for phase difference and K K1( 1 additions for the sum of phasors. Regarding the computation of 1) the complex phase in Eq. (4.22), CORDIC computing using shifts and additions can be utilized [54]. In CORDIC, let P denote the parameter associated with the required phase 1 resolution in the 1st-layer, i.e.
1
1
1
1 2
tan 2P M
for Eq. (4.28). For example, when P1 , the 8
phase resolution is sufficient for M1 210 . As a result, 1 3K P additions are needed for 1 1
computing the complex phases in Eq. (4.22). Hence, the overall addition in the 1st-layer is
1 ( 1 2 1 2 3 )1
K M K P . For the 2nd-layer process, K K1 2(M2 additions are needed for 1) the input phasor, and K K1 22K K K1 2( 2 additions for computing Eq. (4.32) and Eq. 1) (4.33). In addition, assume that the required phase resolution in the 2nd-layer is
2
1
2
1 2
tan 2P M
, we then need 3K K P additions for the complex phase using the CORDIC 1 2 2
computing. Therefore, K K1 2(M22K2 2 3 )P2 additions are required in the 2nd-layer.
When N is very large, K and 1 K are correspondingly large. The computations for the 2 phase then become relatively insignificant in PCA. In addition, as we consider the case with
1 1
K M N and K2 M2 M1 , approximately 3N additions are required for both the 1st- and the 2nd-layer. Note that the number of computations is almost the same in each layer, which is an inherent advantage of the multi-layer PCA. The comparison of the computational load between the two-layer PCA and the FFT-based method is shown in Table 4.1. The computational burden is significantly reduced in PCA, and the efficiency of PCA is thus clarified.
Table 4.1. Computations of the two-layer PCA and the FFT-based method Multiplications Additions
PCA 0 6N
FFT-base method 2 logN 2 N 2 logN 2N
4.6 Summary
In this chapter, the PCA utilizing complex phasors for the PN sequence acquisition is proposed. Particularly, the PCA requires only complex additions but no complex multiplications. In addition, the acquisition performance can be improved via the use of the multi-layer scheme that also provides an inherent error detection capability. In the
Operation Method
demonstrated case using MLS of length N 220 in the two-layer PCA, the correct 1 segment of the 1st-layer is obtained with probability approaching one when SNR 15dB as shown in Fig. 4.4. In addition, with the correct segment, the acquisition performance with correct probability greater than 0.9 can be attained for SNR 20dB after the 2nd-layer as shown in Fig. 4.8. Note that, because the performance of PCA is determined by the chip error probability, the fading effect as well as the varying SNR can be represented by a nominal SNR that is associated with the actual chip error probability of the sequence, and the analysis results can thus be used appropriately. It is noteworthy that the PCA requires much less computation than the FFT-based approach as discussed in Section 4.5 and Table 4.1.
Chapter 5
Conclusions and Future Work
In this dissertation, the one-bit high-accuracy phase estimation for the tracking process is investigated. The traditional APD and the NB-DPD accurately estimate carrier phase in low and high SNR, respectively. However, the estimation bias becomes significant and their accuracy deteriorates in moderate SNR. Focusing on this SNR range, we propose the SNRaPD using the nonlinear least-square algorithm with the aid of SNR information to improve the accuracy. Because SNR information is critical to the phase accuracy and may be unavailable in many applications, the nonlinear least-square algorithm of SNRaPD is further developed to jointly estimate the accurate phase and SNR. Potential applications for the one-bit estimation method in GNSS and beacon receivers are illustrated regarding the attainable phase and SNR accuracy. It is worthwhile to mention that, owing to the efficient one-bit processing, the range of applications of the proposed method can be easily expanded by increasing the number of data and can accommodate signals with a high dynamic range.
On the other hand, we also propose the PCA method that applies the coherence phase of complex phasors to the PN sequence acquisition. Since the PCA simply utilizes the phase differences rather than the amplitude, the PCA requires only complex additions but no complex multiplications. Segmentation, phasor acquisition and the multi-layer scheme are designed in the PCA to enhance the noise-robustness capability. In particular, the multi-layer scheme also provides an inherent error detection capability. For applications having extra SNR margin, such as the high-SNR applications or the processing of de-noised signals, the use of PCA will require much less computation than the FFT-based method. The superior performance on the computation grants the PCA an efficient method when the length of a
sequence is so large that the FFT-based acquisition is infeasible.
Future works related to this dissertation involve two parts. First, concerning the carrier phase estimation, we assume that frequencies of the input and local carriers are identical throughout the work of one-bit phase and SNR estimation, which may not be the case in realistic applications. In practice, the Doppler shift may increase the error associated with the carrier phase estimation and the receiver may thus lose track of the incoming carrier signal. In order to broaden the scope of this work, the effect of Doppler shift should be considered and the tolerance of the frequency shift in the phase estimation algorithm needs additional study.
Consequently, the associated influence on accuracy requires further investigation and clarification. Second, although the computational burden is significantly reduced in the PCA comparing with the FFT-based method, the performance of the FFT method is superior to that of PCA, i.e. P , when the SNR is low. Therefore, new algorithms should be designed to D1 enhance its noise-robustness of the method, such that P could approach one in lower SNR. D1 Once the 1st-layer detection is correct, it is almost assured that the detection of the further layer will also be correct.
References
[1] V. P. Ipatov, Spread-spectrum and CDMA: Principles and Applications, Hoboken, NJ:
Wiley, 2005.
[2] E. D. Kaplan and C. J. Hegarty, Understanding GPS: Principles and Applications, Norwood, MA: Artech House, 2006.
[3] M. Dillinger, K. Madani and N. Alonistioti, Software Defined Radio: Architectures, Systems and Functions, Hoboken, NJ: Wiley, 2003.
[4] H. Chang, “Presampling Filtering, Sampling and Quantization Effects on the Digital Matched Filter Performance,” Proceedings of International Telemetering Conference, pp.
889–915, 1982.
[5] B. W. Parkinson and J. J. Spilker, Global Positioning System: Theory and Applications, Volume I, Washington DC: AIAA, 1996.
[6] H. C. Papadopoulos, G. W. Wornell, and A. V. Oppenheim, “Low-Complexity Digital Encoding Strategies for Wireless Sensor Networks,” Proceedings of the IEEE International Conference on Acoustic, Speech and Signal, vol. 6, pp. 3273-3276, 1998.
[7] P. H. Wu, “The Optimal BPSK Demodulator with a 1-bit A/D Front-end,” Proceedings of the Military Communications Conference of the IEEE, vol. 3, pp. 730-735, 1998.
[8] A. Brown and B. Wolt, “Digital L-Band Receiver Architecture with Direct RF Sampling,”
IEEE Position Location and Navigation Symposium, pp. 209-216, 1994.
[9] R. H. Walden, “Analog-to-Digital Converter Survey and Analysis,” IEEE Journal on Selected Areas in Communications, vol. 17, no. 4, pp. 539–550, 1999.
[10] B. M. Ledvina and M. L. Psiaki, D. J. Sheinfeld, A. P. Cerruti, S. P. Powell, and P. M.
Kintner, “A Real-Time GPS Civilian L1/L2 Software Receiver,” Proceedings of the International Technical Meeting of the Satellite Division of the ION, pp. 986-1005, 2004.
[11] M. R. Yuce and W. Liu, “A Low-Power Multirate Differential PSK Receiver for Space Applications,” IEEE Transactions on Vehicular Technology, vol. 54, no. 6, pp. 2074-2084, 2005.
[12] S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, volume 1, Englewood Cliffs, NJ: Prentice Hall, 1993.
[13] D. Rife and R. Boorstyn, “Single Tone Parameter Estimation from Discrete-Time Observations,” IEEE Transactions on Information Theory, vol. IT-20, no. 5, pp. 591–598, 1974.
[14] B. G. Quinn, “Estimation of Frequency, Amplitude, and Phase from the DFT of a time series,” IEEE Transactions on Signal Processing, vol. 45, no. 3, pp. 814–817, 1997.
[15] H. Fu and P. Y. Kam, “MAP/ML Estimation of the Frequency and Phase of a Single Sinusoid in Noise,” IEEE Transactions on Signal Processing, vol. 55, no. 3, pp. 834–845, 2007.
[16] Høst-Madsen and P. Händel, “Effects of sampling and quantization on single-tone frequency estimation,” IEEE Transactions on Signal Processing, vol. 48, no. 3, pp.
650–662, 2000.
[17] O. Dabeer and A. Karnik, “Signal Parameter Estimation Using 1-Bit Dithered Quantization,” IEEE Transactions on Information Theory, vol. 52, no. 12, pp. 5389–5405, 2006.
[18] O. Dabeer and E. Masry, “Multivariate Signal Parameter Estimation under Dependent Noise from 1-Bit Dithered Quantized Data,” IEEE Transactions on Information Theory, vol. 54, no. 4, pp. 1637–1654, 2008.
[19] A. H. Pouzet, “Characteristics of Phase Detectors in Presence of Noise,” Proceedings of the International Telemeter Conference, pp. 818-828, 1972.
[20] F. M. Gardner, Phaselock Techniques, Hoboken, NJ: Wiley, 2005.
[21] C. F. Chang and M. S. Kao, “High-Accuracy Carrier Phase Discriminator for One-Bit Quantized Software-Defined Receivers,” IEEE Signal Processing Letter, vol. 15, pp.
397–400, 2008.
[22] C. F. Chang, R. M. Yang and M. S. Kao, “Implementation of An Innovative Phase Discriminator for Improved Tracking Performance in One-Bit Software GPS Receiver,”
Proceedings of the National Technical Meeting of the ION, pp. 798-803, 2008.
[23] A. Polydoros and C. L. Weber, “A Unified Approach to Serial Search Spread-Spectrum Code Acquisition — Part II: A Matched-Filter Receiver,” IEEE Transactions on Communications, vol. COM-32, no. 5, pp. 542-560, 1984.
[24] J. K. Holmes and C. C. Chen, “Acquisition Time Performance of PN Spread-Spectrum Systems,” IEEE Transactions on Communications, vol. COM-25, no. 8, pp. 778-783, 1977.
[25] K. K. Chawla and D. V. Sarwate, “Parallel Acquisition of PN Sequences in DS/SS Systems,” IEEE Transactions on Communications, vol. 42, no. 5, pp. 2155-2164, 1994.
[26] W. Zhuang, “Noncoherent Hybrid Parallel PN Code Acquisition for CDMA Mobile Communications,” IEEE Transactions on Vehicular Technology, vol. 45, no. 4, pp.
643-656, 1996.
[27] M. Salih and S. Tantaratana, “A Closed-Loop Coherent Acquisition Scheme for PN Sequence Using an Auxiliary Sequence,” IEEE Journal on Selected Areas in Communications, vol. 14, no. 8, pp. 1653-1659, 1996.
[28] J. G. R. Delva and I. Howitt, “PN Acquisition for DS/SS Using a Preloop Parallel Binary Search Phase Estimator and a Closed-Loop Selective Search Subsystem,” IEEE Transactions on Wireless Communications, vol. 3, no. 2, pp. 408-417, 2004.
[29] J. W. Cooley and J. W. Tukey, “An Algorithm for the Machine Calculation of Complex Fourier Series,” Mathematics of Computation, vol. 19, no. 90, pp. 297–301, 1965.
[30] W. T. Cochran, J. W. Cooley, D. L. Favin etc., “What Is the Fast Fourier Transform,”
Proceedings of the IEEE, vol. 55, no. 10, pp. 1664–1674, 1967.
[31] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice Hall, 1989.
[32] U. Cheng, W. J. Hurd and J. I. Statman, “Spread-Spectrum Code Acquisition in the Presence of Doppler Shift and Data Modulation,” IEEE Transactions on Communications, vol. 38, no. 2, pp. 241–250, 1990.
[33] D. J. R. van Nee and A. J. R. M. Coenen, “New Fast GPS Code-Acquisition Technique Using FFT,” Electronics Letters, vol. 27, no. 2, pp. 158–160, 1991.
[34] M. L. Mao, W. H. Lin, Y. F. Tseng, H. W. Tsao, and F. R. Chang, “New Acquisition Method in GPS Software Receiver with Split-Radix FFT Technique,” the 9th International Conference on Advanced Communication Technology, pp.722-727, 2007.
[35] J. A. Starzyk and Z. Zhu, “Averaging Correlation for C/A Code Acquisition and Tracking in Frequency Domain,” Proceedings of the 44th IEEE Midwest Symposium on Circuit and Systems, pp. 905-908, 2001.
[36] D. M. Lin and J. B. Y. Tsui, “Acquisition Schemes for Software GPS Receiver, ” Proceedings of ION GPS, pp.317-325, 1998.
[37] S. Z. Budisin, “Fast PN Sequence Correlation by Using FWT,” Proceedings Mediterranean Electrotechnical Conference , pp. 513-515, 1989.
[38] A. Alaqeeli and J. Starzyk, “Hardware Implementation for Fast Convolution with a PN Code Using Field Programmable Gate Array,” Proceedings of the 3rd Southeastern Symposium on System Theory, pp.197-201, 2001.
[39] A. Jeffrey, Handbook of Mathematical Formulas and Integrals, Amsterdam, Boston: Elsevier, 2004.
[40] E. K. P. Chong and S. H. Żak, An Introduction to Optimization, Hoboken, NJ: Wiley, 2008.
[41] P. A. Bernhardt, C. L. Siefring, I. J. Galysh, T. F. Rodilosso, D. E. Koch, T. L.
MacDonald, M. R. Wilkens, G. P. Landis, “Ionospheric Applications of the Scintillation and Tomography Receiver in Space (CITRIS) Mission When Used with the DORIS Radio Beacon Network,” Journal of Geodesy, vol. 80, no. 8-11, pp. 473–485, 2006.
[42] C. J. Kikkert and O. P. Kenny, “A Digital Signal Processing Based Ka Band Satellite Beacon Receiver” IEEE International Conference on Electronics, Circuits, and Systems, pp. 1-8, 2008.
[43] B. W. Parkinson and J. J. Spilker, Global Positioning System: Theory and Applications, Volume II, Chapter 21, Washington, DC: AIAA, 1996.
[44] O. Montenbruck, M. Garcia-Fernandez, and J. Williams, “Performance Comparison of Semicodeless GPS Receivers for LEO Satellites,” GPS Solutions, vol. 10, no. 4, pp.
249–261, 2006.
[45] C.-W. Huwang, T.-P. Tseng, T.-J. Lin, D. Švehla, U. Hugentobler, and B. F. Chao,
“Quality Assessment of FORMOSAT-3/COSMIC and GRACE GPS Observables:
Analysis of Multipath, ionospheric delay and phase residual in orbit determination,” GPS Solutions, vol. 14, no. 1, pp. 121–131, 2010.
[46] O. Montenbruck and R. Kroes, “In-flight performance analysis of the CHAMP BlackJack GPS Receiver,” GPS Solutions, vol. 7, no. 2, pp. 74-86, 2003.
[47] C.-W. Hwang, T.-P. Tseng, T.-J. Lin, and D. Švehla, B. Schreiner, “Precise orbit determination for the FORMOSAT-3/COSMIC satellite mission using GPS,” Journal of Geodesy, vol. 83, no. 5, pp. 477-489, 2009.
[48] D. Švehla and M. Rothacher, “Kinematic and reduced-dynamic precise orbit determination of low earth orbiters,” Advances in Geosciences, vol. 1, pp. 47-56, 2003.
[49] A. Jäggi, U. Hugentobler, H. Bock, and G. Beulter, “Precise orbit determination for GRACE using undifferenced or doubly differenced GPS data,” Advances in Space Research, vol. 39, no. 10, pp. 1612-1619, 2007.
[50] P. A. Bernhardt, C. A. Selcher, S. Basu, G. Bust, and S. C. Reising, “Atmospheric Studies with the Tri-Band Beacon Instrument on the COSMIC Constellation,” Terrestrial, Atmospheric and Oceanic Sciences, vol. 11, no. 1, pp. 291-312, 2000.
[51] M. Yamamoto, “Digital beacon receiver for ionospheric TEC measurement developed with GNU radio,” Earth Planets Space, vol. 60, no. 11, pp. e21-e24, 2008.
[52] K. F. Davies, Ionospheric Radio, Peter Peregrinus: London, UK, 1990.
[53] C. Rocken, Y.-H. Kuo, W. S. Schreiner, D. Hunt, S. Sokolovskiy, and C. McCormick,
“COSMIC System Description,” Terrestrial, Atmospheric and Oceanic Sciences, vol. 11, no. 1, pp. 21-52, 2000.
[54] J. E. Volder, “The CORDIC Trigonometric Computing Technique”, IRE Transactions Electronic Computers., vol. EC-8, no. 3, pp. 330–334, 1959.
[55] P. Beckmann, Probability in Communication Engineering, New York, NY: Harbrace, 1967.
Appendix
A. Derivation of mean and variance of I-Q channel outputs
According to Eq. (2.4), for k[0,), we have sink 0. In inphase (I) channel, the conditional probabilities are denoted as
k
The associated mean and variance are given by
1
Assume the noise component in each sample is independent. Since ’s are uniformly k distributed over [0,2), we have
Similarly, by the same calculation, the mean and variance for the quadrature (Q) channel are obtained by