WHITEPAPER: ACHIEVING THE CRAMER-RAO LOWER BOUND IN GPS TIME-OF-ARRIVAL ESTIMATION, A FREQUENCY DOMAIN WEIGHTED LEAST-SQUARES ESTIMATOR APPROACH KENNETH M PESYNA, JR AND TODD E HUMPHREYS THE UNIVERSITY OF TEXAS AT AUSTIN Introduction Recall, that for a coherent code phase detector, the variance of the code phase estimate error t, in seconds can be derived as [] where σ t ELB DLL T C C () EL is the early-late correlator spacing, in chips, B DLL is the bandwidth of the delay-locked loop (DLL), in Hz, and T C is the GNSS signal PN-code chip duration, in seconds C is the carrier-to-noise ratio of the received signal It becomes natural to ask if the estimate of t using a delay locked loop is the best that one could do, ie achieves the smallest σ t The so-called Cramer-Rao lower bound (CRLB) provides us with a lower bound on σ t Any estimation strategy that achieves the CRLB is considered efficient One simple estimation strategy that is known to achieve the CRLB is a batch weighted least squares estimator By batch we mean that the estimator processes an entire chuck of data at once We can define the cross correlation function as Date: October 7, Batch Least Squares Estimator R[ˆt s ] E C [ {C[τj t s ]+n(j)}c[τ j ˆt s ] ] () j k +N k N k jj k {C[τ j t s ]+n(j)}c[τ j ˆt s ] () () Copyright c by Kenneth M Pesyna, Jr and Todd E Humphreys
where E C is the expectation over the GNSS PN-code C is the PN-code sequence n is the front-end white-guassian noise N k is the number of samples over which the average will be performed ˆt s is the code-phase estimate, in seconds t s is the true code-phase, in seconds One simple way to estimate the true code phase offset t s is to maximize R[ˆt s ] over ˆt s We will assume that we have accurate models for satellite and receiver motion, clocks, and atmospheric effects such that t s remains constant throughout the entire batch interval (no dependence on τ j This is not typically the case (hence the need for a PLL and DLL), but it simplifies our derivation It is also assumed that the signal has been shifted to baseband (ie no Doppler frequency remaining) The maximizing value of ˆt s is given only at the resolution of the signal sampling interval To overcome this, we can transform the problem into a more convenient representation using the Wiener-Khinchin theorem But, first lets define the code phase error t t s ˆt s and define a new cross-correlation function in terms of this new term This provides us with a more-familiar notion to us of the error between our a priori guess ˆt s and t s R[ t] R[ˆt s ] E C [{C[τ j t s ]+n(j)}c[τ j ˆt s ]] (5) E C [{C[τ j t s +t s ]+n(j)}c[τ j ˆt s +t s ]] (6) E C [{C[τ j ]+n(j)}c[τ j (t s ˆt s )]] (7) }{{} t E C [C[τ j ]C[τ j t]]+e C [n(j)c[τ j t]] (8) E C [C[τ j ]C[τ j t]]+ñ(j) (9) R C [ t]+ n(j) () R C is the auto-correlation function for the noise-free GPS spreading sequence and n(j) is white Gaussian noise, since multiplying Gaussian noise by a random zero-mean, unit variance sequence is still white Guassian noise with the same mean and variance It follows that, using the Wiener Khinchin theorem: S(f) F [R C [ t]+ñ(j)] S c (f)e jπf t +ñ (f) () where ñ (j)isstill whitegaussiannoisesince thefouriertransformoftime-domainindependent Gaussian noise samples yields frequency-domain independent Gaussian noise samples Due to the time-shifting property of the Fourier transform, t is simply the slope of the spectral phase φ(f) tan (S(f)) πf t+ñ(f) () By dividing the one-sided precorrelation bandwidth f into N sub-intervals of width δf and center frequencies f i where i,,n, we can take measurements of φ(f i ) and use these measurements to estimate t by a weighted least squares fit to the slope of the measurements
6 x 5 x 8 x 6 5 6 Power Spectra 8 6 Spectral Phase, cycles Power Spectra 5 5 5 Spectral Phase, cycles 6 8 Frequency, Hz x 6 Figure Plot showing power spectra and spectral phase for a t chips 5 Frequency, Hz x 6 Figure Plot showing power spectra and spectral phase for a t chips Each phase measurement φ i should weighted by its variance σ φ i which is a function of the average carrier-to-noise ratio in its particular frequency band If carrier-to-noise ratio can be expressed as []: C N o σ IQ where is the coherent integration time of the signal, and σ IQ σφ T () a is the sample-variance, then σ φ i C i N o () where the effective signal power for the i th measurement can be computed as C i f i + δf S c (f)df S c (f i )df (5) f i δf We can stack these measurements up into a vector φ(f ) πf n φ(f ) πf [ ] n t + (6) φ(f N ) πf N }{{} H n N The measurement covariance matrix for a weighted least-squares problem can be given by: R diag [ σ φ(f ),σ φ(f ),,σ φ(f N ) ] (7) The error variance of a weighted least squares estimate is defined as [] P [ H T R H ] (8)
Thus, given the above setup, the error variance of the weighted-least-squares estimate of t can be easily derived as and, in the limit as N σ t P [ H T R H ] i (π) fi i (π) fi 8π σ φ (f i) (9) () () No Ta C i 8π i C ifi () 8π i S c(f i )f i df () f f S c (f)f df Expanding S c (f) such that it is the product of the available signal power and the normalized power spectral density: where we can rewrite σ t as σ t () (5) (6) S c (f) C S c (f) (7) 8π C S c (f)df (8) f f S c (f)f df This expression for σ t is equivalent to the formal definition of the CRLB ([], see appendix B) Thus the weighted least-squares estimator produces an estimate whose variance meets the CRLB (9) () References [] P Misra and P Enge, Global Positioning System: Signals, Measurements, and Performance Lincoln, Massachusetts: Ganga-Jumana Press, revised second ed, [] Y Bar-Shalom, X R Li, and T Kirubarajan, Estimation with Applications to Tracking and Navigation New York: John Wiley and Sons,
[] J Betz and K Kolodziejski, Generalized theory of code tracking with an early-late discriminator part i: lower bound and coherent processing, IEEE Transactions on Aerospace and Electronic Systems, vol 5, no, pp 58 556, 9 E-mail address: kpesyna@utexasedu 5