Analytical Model for GNSS Receiver Implementation Losses*

Christopher J. Hegarty, The MITRE Corporation

intensive and do not provide much insight into the various BIOGRAPHY1 loss mechanisms.

Dr. Christopher J. Hegarty is a director with The MITRE Earlier analytical studies of DSSS receiver SNR losses Corporation, where he has worked mainly on aviation due to quantization include derivations in [1] for applications of GNSS since 1992. He is currently the quantization losses of 1.96 dB and 0.55 dB, respectively, Chair of the Program Management Committee of RTCA, for 1- and 2-bit uniform quantizers. These derivations are Inc., and co-chairs RTCA Special Committee 159 simplified in that they assume that the received signal is (GNSS). He served as editor of NAVIGATION: The sampled, but with independent noise upon each sample Journal of the Institute of Navigation from 1997 – 2006 and ignoring bandlimiting effects on the desired signal and as president of the Institute of Navigation in 2008. He component. A more thorough treatment of bandlimiting, was a recipient of the ION Early Achievement Award in sampling, and quantization effects in digital matched 1998, the U.S. Department of State Superior Honor filters is provided in [2]. The results in [2] were later Award in 2005, the ION Kepler Award in 2005, and the inferred to apply to GNSS receiver losses in [3, 4]. As Worcester Polytechnic Institute Hobart Newell Award in detailed later in this paper, however, the results in [2] are 2006. He is a Fellow of the ION, and co-editor/co-author truly not directly applicable to GNSS or other DSSS of the textbook Understanding GPS: Principles and receivers because the losses derived therein presume Applications, 2nd Ed. coherent integration over only one symbol of the desired signal by the receiver, whereas DSSS receivers coherently ABSTRACT integrate over many spreading symbols.

Global Navigation Satellite System (GNSS) receivers Monte Carlo simulation results for GNSS receiver SNR suffer signal-to-noise ratio (SNR) losses due to losses for a variety of modulation types, receiver bandlimiting, quantization, and sampling. This paper configurations, and in the presence of both white noise presents an analytical model for GNSS receiver losses and non-white interference are provided in [5] and [6]. applicable to a wide variety of hardware configurations. These results, although useful, have several limitations. The model addresses digitization of the received signal by First, if loss results are required for a scenario not yet a uniform quantizer with an arbitrary (even or odd) investigated, additional simulations must be run, and integer number of output levels. The model provides SNR these can be extremely time intensive. A simulation tool loss values for GNSS signals in the presence of both developed by the second author of [5] and augmented in additive white Gaussian noise and interference, provided capability by others at MITRE takes over 6 hours on a that the interference can be accurately modeled as a non- desktop personal computer to yield results for a typical white, Gaussian wide sense stationary process. run. Second, simulations provide limited insights into the various mechanisms contributing towards implementation INTRODUCTION losses. Lastly, the loss results presented in [5, 6] are all for receivers using sampling rates that are commensurate Direct sequence spread spectrum (DSSS) receivers, with the desired GNSS signal symbol rates, i.e., with an including those used for the Global Navigation Satellite integer number of samples per spreading symbol. As System (GNSS), suffer signal-to-noise ratio (SNR) losses discussed within this paper, with commensurate sampling, due to bandlimiting, quantization, and sampling. Previous receiver SNR losses are highly dependent on the phasing research into such losses has included: (1) low-fidelity of the sampling epochs relative to the symbol transitions. analytical models that predict SNR loss values with many Well-designed GNSS receivers avoid commensurate simplifying assumptions, (2) extensive simulation sampling rates, because of errors that can arise in campaigns that provide useful loss values, but are time pseudorange measurements when such rates are used [7, 8]. *The contents of this material reflect the views of the author. Neither the Federal Aviation Administration nor the Department This paper presents an analytical model for GNSS of the Transportation makes any warranty or guarantee, or receiver losses applicable to a wide variety of hardware promise, expressed or implied, concerning the content or configurations. The model addresses digitization of the accuracy of the views expressed herein. received signal by a uniform quantizer with an arbitrary © The MITRE Corporation. All rights reserved.

r()()()() t Cs t ej  n t  i t

r r q (kM 1) 1 z n Complex n 1 k Lowpass Filter ×  Quantizer M Sampler n kM (fs Hz) se*  j n Figure 1. GNSS Receiver Signal Processing Model

(even or odd) integer number of output levels. The model effective signal power and effective noise density at the includes the effects of sampling and bandlimiting, without output of the correlator, which when combined yield SNR limitations on the allowable sample rates or filter transfer losses. functions. Both commensurate and non-commensurate sampling rates are considered, although as mentioned Signal Model above the former are mostly of interest in comparing results with those reported previously in earlier studies. The baseband normalized desired signal, s(t), is modeled very generally as: As observed in the previous literature, losses are  s t  c p()ll t lT  j  d q() t lT dependent on the desired signal modulation type. () l (c ) 1 l ()c (1) ll  Furthermore, different losses are incurred for the three where  is the fraction of power in the signal‟s in-phase components of the received signal considered: (1) desired component, Tc is the symbol period, {cl} and {dl} are GNSS signal, (2) noise, and (3) Gaussian non-white independent, white pseudorandom sequences from the interference (e.g., inter-/intra-system interference from a (l) (l) binary alphabet [-1,+1], and p (t) and q (t) are, number of GNSS signals of the same or different respectively, the l-th spreading symbols used for the modulation type). The analytical formulation provides inphase and quadraphase components. The spreading insight into how implementation losses result from a symbols are real-valued and selected from finite sets, combination of attenuation of the desired signal, and p()l ( t ) p ( t ), , p ( t ) , q()l ( t ) q ( t ), , q ( t ) with attenuation and/or enhancement of the noise/interference.  1 N p   1 Nq 

probabilities P p()l ()() t p t P and This paper is organized as follows. The analytical model ipi is first derived and described in the following section, ()l P q()() t q t P . This definition is broad enough to with certain cumbersome steps in the derivation relegated iqi to appendices. The subsequent section provides apply to simple GNSS signals such as the GPS C/A code, implementation loss results from the analytical model and for which there is only an in-phase component with all compares pertinent results against those found in the symbols taking on the same, rectangular shape, or to more previous literature. The final section provides a short complicated GNSS signals such as the GPS L1 civil summary and conclusions. signal (L1C) that include in- and quadra-phase components with a mixture of different symbols (the L1C ANALYTICAL MODEL pilot component uses two different symbol types that are time multiplexed). A simplified block diagram of signal processing within a GNSS receiver is shown in Figure 1. The received signal, The signal defined in equation (1) is cyclostationary, with r(t), is modeled as the sum of a desired signal, noise n(t), a two-parameter autocorrelation function that is periodic and interference i(t). The baseband normalized (unity with time, t, with period Tc: power) desired signal is denoted s(t), with received phase  (radians) and received power level of C watts. Note R t E s t s* t s ;()()   that Figure 1 is highly simplified and omits certain N p  receiver processing functions that are considered lossless  P p()() t lT  p t  lT (2) pn n c n c this analysis, e.g., down-conversion of the original L-band nl1  received signal to its in-phase (I) and quadra-phase (Q) Nq  1 P q()() t lT  q t  lT components (treated here as a single complex signal).  qn n c n c nl1  The following subsections provide statistical models for and its time-average may be defined as the signal, noise, and interference, and then follow these received signal components through the processing shown in Figure 1. The final subsection derives expressions for © The MITRE Corporation. All rights reserved.

T 1 c It is assumed that within the receiver passband, the power Rss   R(;) t dt of the desired signal is much lower than the combined T  c 0 (3) power of the noise and interference, e.g., with ideal low- NNpq pass filtering,  PR    1   PR    pn p n qn q n nn11 BB where C Sfdf()  NBI  Sfdf() (9) T sI0 c 1 f B f B R   p()() t p t dt pn T  n n c 0 (4) T where Sf()is the Fourier transform of R () . 1 c s s R   q()() t q t dt qn T  n n c 0 Sampling

Noise and Interference Models The low-pass filtered received signal is uniformly sampled at rate fs. After sampling, the received signal may The noise, n(t), and interference, i(t), are assumed to be be expressed as: mutually independent, complex Gaussian wide sense rnn r() t stationary random processes. The noise is further assumed (10) r() nTs T0 to be white with power spectral density of Sn () f N0 where Ts = 1/fs is the sampling period, and T0 is a constant watts/hertz (double-sided), whereas the interference has timing offset. I S f I an arbitrary power spectral density I (), where is the received interference power level (watts), and SfI () The autocorrelation of the sampled received signal may is the normalized (unity power) interference power be expressed as: spectrum. * Rr []()() m E r tn m r t n  Filtering  E r r * (11) n m m The low-pass filter in Figure 1 is specified by an arbitrary Rn[][] m Ri m transfer function, H(f). For some of the later results, an where the last line follows from (9) and ideal low-pass filter with transfer function:

 1, fB / 2 Hf()  (5) Rii[]() m R mTs 0, else   j fmT (12) is presumed, where B is the two-sided filter bandwidth.  S() f e2 s df  i The filter outputs corresponding to each received signal  component are denoted by tildes (e.g., s( t ), n ( t ), i ( t ) ). The power spectral density of the sampled received signal is the discrete-time Fourier transform of (11), which can For the transfer function specified in equation (5), after be expressed as a function of the continuous-time power filtering the noise power spectral density is given by: spectral densities of the noise and interference components of the received signal, equations (6) and (8), N0 , f B / 2 Sfn ()  (6) respectively, as  0, else  j jm and its autocorrelation by: Srr e    R[ m ] e R E n t n* t m n ()()() (13)  ff (7) 1 ss    sin B    Sn kf s   Si   kfs   NB T 22    0 B  s k  where the approximation in the second line follows from the assumption stated in equation (9) that the noise and With ideal low-pass filtering as described by equation (5), interference power are much larger than the desired signal the interference power spectral density is given by: power. I SI ( f ), f B / 2 Sfi ()  (8)  0, else Quantizer

For an arbitrary low-pass filter with transfer function H(f), The complex quantizer in Figure 1 is modeled as shown equations (6) and (8) may be replaced by the respective in Figure 2, where Re(·) and Im(·) are the real and input noise and interference power spectral densities 2 imaginary operators, respectively. x and y are multiplied by |H(f)| . n n introduced as the inphase and quadraphase components of the received signal, respectively, and their quantized © The MITRE Corporation. All rights reserved.

versions are denoted by the same notation with a y(x) y(x) superscript „q‟. Q Q/2 -Q/2 Complex Quantizer x x -Q/2 Q/2 x xq n Real n -Q Re() Quantizer N = 2 N = 3 r x jy rq x q jy q (1 bit) (1.5 bit) n n n n nn y(x) + y(x) 2Q 3Q/2 y yq n Real n Q Im() j Quantizer Q/2 -Q/2 -Q Q x -3Q/2 Q/2 3Q/2 x -Q/2 Figure 2. Complex Correlator Model -Q -3Q/2 Each of the two real quantizers is modeled as an odd -2Q symmetric, memoryless nonlinearity whose output, y(t), at N = 4 N = 5 (2 bit) (2.5 bit) arbitrary time, t, is only a function of the input voltage at that instant, x(t), following the input-output Figure 3. Examples of Uniform Quantizers characteristic [9]:

y()() x K f x As derived in Appendix A, the autocorrelation of the K f()() x  f  x (14) output signal of the complex correlator may be related to  the autocorrelation of the input signal utilizing results N /2  from [9, 10]. The result is: f() x  pp u x a  kk   p1 ReRrr [ m ]    ImR [ m ]   Rq [ m ] R [0] b j (18) N u(x) r rk     where is the number of output levels, is the unit k  RR[0] [0] 1 rr    step function, K is a gain constant and  is the floor where operator whose output is the greatest integer less than or 22 equal to its argument. bkk K h0 /! k (19)

This paper focuses on quantizers with uniform step sizes. with the parameters h0k and K defined as: The input-output characteristic for a uniform quantizer is given by equation (14) with [9] 0, k even  N h  /2 2 (20) Q 0k  ap  /2 pp/2 1 2  e H a, k odd (N even) (15)    p k1  p  a( p 1) Q  p1 p or and 1/2   Q  p K h2 k (N odd). (16)  0k /! (21) a p Q k 1 p ( 1/ 2) H (·) k Examples for various values of N with K = 1 are shown in In equation (20), k is a -th order Hermite polynomial. Figure 3. Efficient methods for numerically computing (20) and (21) are provided in Appendix A. For comparisons with previous studies, it is useful to note that the maximum input threshold, T, for an N level If the interference has a baseband power spectral density S f S f , Rm uniform quantizer is related to the parameter Q as: that is even symmetric, i.e., II()() then r [] will be entirely real for all m and thus: (NQ 2) T  (17) k  2 Rmr [] Rq [ m ] R [0] b (22) r rk  k 1 Rr [0]

The power spectral density of the complex quantizer output may be related to the power spectral density of its input by taking the discrete-time Fourier transform of equation (18): © The MITRE Corporation. All rights reserved.

k  jj where h(t) is the impulse response of the low-pass filter in  S e S e j  rr    Figure 1, equal to the inverse Fourier transform of its Sq e R[0] b  r   rk  k 1 2Rr [0] transfer function, H(f). For the transfer function provided  (23) in equation (5), k S ejj S e  rr     sin 2 Bt   j  h( t ) 2 B (30) 2jR [0]  2 Bt r   If the interference has a baseband power spectral density As mentioned in the introduction, well-designed receivers that is even symmetric, i.e., SII()() f S f , such that jj use sample rates that are incommensurate with the Srr()() e S e , then equation (23) reduces to: spreading symbol rate [7, 8]. With incommensurate

k sample rates, equation (27) is well-approximated by:  j Ser   T S ej  R b  c rq   rk[0] (24) 1 k R [0] Rss ()(;) Rss t dt 1 r T  (31) c 0 which may also be obtained by taking the discrete-time Rh()() Fourier transform of equation (22). s since over the coherent correlation interval, the sampling

epochs tn are uniformly distributed across the spreading Phase Rotation and Correlation symbols. It follows then, that for incommensurate

sampling rates: As shown in Figure 1, the received signal after filtering, sampling and quantization, r q , is phase rotated by an  n R H f S f df ss (0) ( )s ( ) (32) estimate of the incoming signal phase (assumed here to be f  perfect) and then correlated against the complex in general, and conjugate of the discrete-time replica of the desired * B/2 signal, sn , resulting in the complex correlation sum: Rss (0) Ss ( f ) df (33) (kM 1) 1  1 qj*   fB /2 zk   rn s n e (25) M n kM for the ideal filter with transfer function provided in where M is the number of samples per coherent equation (5). T MT correlation interval, I = s. Another case worthy of special note is when the sample As derived in Appendix B, the mean value of the k-th rate is exactly commensurate with the spreading symbol complex correlation sum is: rate:

N f  s (34) E zk   K  h01  C  Rss (0) (26) s Tc

where Ns is a positive integer representing the number of where samples per spreading symbol. Although commensurate 1 (kM 1) 1 sampling is rarely found in GNSS receivers, it has been Rss () Rss(;) t n (27) assumed in previous studies [2, 5, 6] and its treatment in M  n kM this paper has been included to facilitate a comparison of is the time-average (over sample epochs across the results. coherent correlation interval) of the two-parameter cross- correlation function between the filtered and unfiltered Commensurate sampling is illustrated in Figure 5. At the desired signal, i.e., top of the figure is the desired signal component of the baseband received signal with rectangular symbols after R t E s t s* t ss ;()()   filtering. The bottom of the figure indicates the sampling

N p  epochs. Note that there are exactly two samples per P p()() t p t (28) symbol. The phasing of the sample epochs with respect to pn n n nl1  the symbol transitions is captured by the time offset N q  parameter, T0, introduced in equation (10). In Figure 5, T0 1 P q()() t p t  qn n n = 0, i.e., the first of every two samples coincides with the nl1  leading edge of a symbol. with With commensurate sampling, pnn()()() t h t p t 1 Ns 1 qnn()()() t h t q t (29) Rss ()(;) Rss nT s T0 (35) 1 N h()() t  H f  s n0

2.5

2 N = 2

N = 3 1.5 N = 4

N = 5 1 N = 6 Figure 5. Illustration of Commensurate Sampling with N = 7 Two Samples per Spreading Symbol N = 8 Effective (dB) Power Signal Loss 0.5 The variance of the k-th complex correlation sum, as derived in Appendix B, may be accurately approximated N = 16 0 0 1 2 3 4 5 for MTsc T as: Ratio of Maximum Threshold to One-sigma Noise Level

1 M 1 Figure 4. Signal Losses for Quantizers with N Levels var z  Rq [] n R n ks   r   vs. Maximum Input Threshold M nM(  1)  (36) 11 jj Table 1. Minimum Signal Loss Values and Quantizer  Sq ()() e S e d  r s Threshold Settings M 2  Number of Minimum Optimum Optimum Signal-to-noise Loss Quantizer Signal Threshold Maximum Input Levels, N Loss (dB) Parameter, Q Threshold Using equation (26), the effective signal power at the Level, T output of the correlator may be found as: 2 1.961 N/A N/A 2 2 2 C   E zk   K  h Rss (0)  C 3 0.916 1.224 0.612 eff 01  (37)  CL/ c 4 0.549 0.996 0.996 where 5 0.372 0.843 1.265 1 Lc  2 (38) 6 0.272 0.733 1.466 K h R (0) 01 ss 7 0.208 0.651 1.628 represents the loss in signal power due to bandlimiting, sampling, and quantization as can be observed by noting 8 0.166 0.586 1.758 CC that  eff in the limit as the filter bandwidth 16 0.050 0.335 2.345

B , the sample rate fs , and the quantizer levels 32 0.015 0.188 2.820 N  qj such that in Figure 1, rn  Csn e  n n  i n . 64 0.005 0.104 3.224 Signal losses for a variety of quantizer levels, N, and as a 128 0.001 0.057 3.591 function of maximum input threshold level, T, are shown in Figure 4 and minimum signal loss values and associated quantizer thresholds are summarized in Table  Ts jj 1. In both Figure 3 and Table 1, the results presume that N   Sq ()() e Ss e d (39) 0 eff 2  r the receiver bandwidth is sufficiently wide and the  sampling frequency is sufficiently high so that the desired since in the presence of desired signal plus noise only, as signal component of the received signal is negligibly is well-known, the correlation sum variance must equal NT0 / I and thus (36) must equal NTNMT00//()Is in attenuated, i.e., Rss (0) 1. If this condition does not hold, an additional loss due to bandlimiting must be applied, the limiting conditions ( , , ). utilizing equation (32) for incommensurate sampling or equation (35) for commensurate sampling. RESULTS AND VALIDATION

Rectangular Symbols, Ideal Filtering, Nyquist Sample The effective noise plus interference power density is Rate, Noise-only given by:

bandwidth, fs = B, only noise is considered, and ideal conditions used to produce Figure 5, whereas the filtering as described by equation (5) is employed. In analytical model indicates an SNR loss of 2.42 dB. these conditions, the noise component of the sampled 2.5 received signal is white, i.e., Rnm[] m N0 B . From equation (22), the output of the complex quantizer for any N = 2 number of levels has the same autocorrelation function, 2 j Rq m N B S e N f r [] 0 m . It follows then that rq () 0 s 1.5 and substitution of this result into equation (17) yields N = 4 NN 00 eff The SNR loss, thus, under these conditions 1

is entirely attributable to the effective signal loss provided Loss (C/N0)eff (dB) by equation (18). N = 8 0.5 N = 16 Figures 6 and 7 present SNR loss results from the N = 25 analytical model for a desired signal using rectangular 0 symbols, ideal filtering as described in equation (5), 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Ratio of Maximum Threshold to One-sigma Noise Level Nyquist rate sampling, and thermal noise only. Figure 6 shows losses when the two-sided receiver bandwidth is Figure 7. SNR Losses – Rectangular Symbols, Noise equal to twice the chipping rate, B = 2/Tc. This case is Only, Two-sided Receiver Bandwidth Equal to Ten applicable, e.g., to a C/A-code receiver with a 2.046 MHz Times the Chipping Rate, Nyquist Sample Rate two-sided bandwidth or a L5 receiver with a 20.46 MHz two-sided bandwidth. In this case, there are exactly two Upon further study, it is clear that the SNR loss results in samples per spreading symbol in the desired signal. The [2], without extension, are not truly applicable to GNSS sampling time offset, T0, in equation (8) was selected to receivers. The reason for this is that the results in [2] are be Tc/4, so that the two samples per symbol were centered focused on a communications receiver that only with respect to the symbol minimizing signal loss. coherently integrates over one symbol. GNSS receivers typically coherently integrate over many spreading 3.5 symbols. For example, a C/A-code receiver typically integrates over at least 1023 spreading symbols (chips) for 3 all modes of operation. N = 2 2.5 The method used in [2] to compute SNR losses is to first 2 N = 4 determine the probability mass function for each quantized sample of the received signal. With Nyquist 1.5 sampling, as explained at the beginning of this section, (C/N0)eff Loss (C/N0)eff (dB) N = 8 noise samples are independent of each other. Thus, with 1 N = 16 an integer number of samples per symbol in a digital

0.5 matched filter, it is possible to determine the probability mass function of the sum of quantized samples as N = 25 0 integrated over each symbol by convolving the per- 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Ratio of Maximum Threshold to One-sigma Noise Level sample probability mass functions. The method in [2] can

Figure 6. SNR Losses – Rectangular Symbols, Noise be extended for a receiver that coherently integrates over Only, Two-sided Receiver Bandwidth Equal to Twice many symbols simply by further convolving the the Chipping Rate, Nyquist Sample Rate probability mass function obtained for one symbol over as many symbols as are included within the coherent Figure 7 plots losses when the two-sided bandwidth is ten integration period. times the chipping rate, B = 10/Tc, e.g., a C/A-code receiver with a 10.23 MHz two-sided bandwidth. Nyquist Figure 8 shows SNR losses as a function of the number of sampling was used resulting in ten samples per spreading symbols within the coherent integration period for the conditions used to produce Figure 6. Figure 8 was symbol with T0 = Tc/20, so that the ten samples in each symbol are centered with respect to the start and end of produced using the extension of the method in [2] the symbol. outlined above and with the quantizer thresholds shown in Table 1. As the number of symbols in the coherent The results in Figures 6 and 7 were originally of concern, integration period is increased, the SNR loss results since they are significantly different than the results for asymptote to the values predicted by the analytical model the same scenario as reported in two prominent texts [3] (which presumes a coherent integration period that is and [4], which both provide SNR loss values that much longer than the spreading symbol period). Tables 2 originate from [2]. As an example of the differences, [2] and 3 summarize a comparison of SNR loss results from indicates a 3.47 dB SNR loss with a 1-bit quantizer for the this extension of the method from [2] vs. results from the analytical model described in this paper. Note the © The MITRE Corporation. All rights reserved. excellent agreement, with all results between the two It should be noted that when commensurate sampling is vastly different methods equal to within 0.01 dB. used with a small number of samples per spreading symbol, SNR losses are highly dependent upon the 3.5 relative phasing between the sample epochs and desired

3 signal symbol edges. Figure 9 illustrates the sensitivity by N=2 plotting loss results vs. the timing offset T0 for a one-bit 2.5 (N = 2) quantizer, Nyquist sample rate, rectangular

2 symbols, and ideal filtering with two sided bandwidth of twice the chip rate. These results can best be understood 1.5 by viewing Figure 5. With two samples per spreading N=4

(C/N0)eff Loss (dB) Loss (C/N0)eff symbol, minimum signal power losses occur when the 1 two samples are centered with respect to each symbol. If N=8 N=16

0.5 T0 is zero, the first of the two samples is aligned with the

N=32 symbol leading edge, which is zero half the time (every

0 0 1 2 3 10 10 10 10 time the preceding symbol has the opposite polarity). The Coherent Integration Time (spreading symbols) net result is that on average 1 of 4 samples is devoid of Figure 8. SNR Loss Results Using Extension of energy from the desired signal component and the Method in [2]; Rectangular Symbols, Noise Only, receiver suffers an approximate 1.3 dB additional loss for Two-sided Receiver Bandwidth Equal to Twice the T0 = 0 vs T0 = Ts/2. Optimum sample phasing was Chipping Rate, Nyquist Sample Rate explicitly utilized in [2] and in the results presented in Tables 2 and 3, whereas worst-case phasing is apparent Table 2. Comparison of SNR Loss Results from from the loss results presented in [5,6]. Extension of Method in [2] vs. Analytical Model; Rectangular Symbols, Noise Only, Two-sided Receiver 4 Bandwidth Equal to Twice the Chipping Rate, Nyquist Sample Rate 3.5 Number of Minimum Loss Minimum Loss Quantizer per Exension of per Analytical Loss (dB)

Levels, N Method in [2] Model (dB) eff (dB)* 3 (C/N0) 2 2.43 2.42 4 1.01 1.01 2.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 8 0.63 0.63 Ratio of Sample Time Offset to Sample Period, T / T 0 s 16 0.51 0.51 Figure 9. SNR Loss Sensitivity to Sampling Epoch Phasing with Commensurate Sampling 25 0.48 0.48 Validation *With coherent integration period = 1000 symbols.

Table 3. Comparison of SNR Loss Results from To validate the analytical model, SNR loss results were Extension of Method in [2] vs. Analytical Model; computed for 17 scenarios and compared with Monte Rectangular Symbols, Noise Only, Two-sided Receiver Carlo simulations using a tool developed originally by the Bandwidth Equal to Ten Times the Chipping Rate, second author of [3] and significantly further developed Nyquist Sample Rate by Dr. Alex Cerruti of MITRE. The scenarios included a wide variety of desired binary phase shift keyed (BPSK) Number of Minimum Loss Minimum Loss and binary offset carrier (BOC) signal modulation types, Quantizer per Exension of per Analytical quantizer levels, receiver bandwidths, and sample rates. In Levels, N Method in Model (dB) the description to follow the notation BPSK-R(n) is used [Chang] (dB)* for a BPSK signal with an n × 1.023 MHz symbol 2 2.01 2.01 (chipping) rate and rectangular symbols. BOC(m,n) denotes a BOC modulation with a m × 1.023 MHz square 4 0.60 0.60 wave subcarrier and a n × 1.023 MHz symbol rate. 8 0.21 0.21 TMBOC denotes the time multiplexed BOC modulation that will be used for the pilot component of the GPS L1 16 0.10 0.10 Civil (L1C) signal that time multiplexes 29 BOC(1,1) 25 0.07 0.07 symbols and 4 BOC(6,1) symbols out of every 33 total symbols. *With coherent integration period = 1000 symbols.

Figure 8 plots the results of the first scenario, which decreased SNR, as demonstrated through additional represents aggregate BOC(10,5) interference to a victim simulation runs for a select number of points. BPSK-R(1) receiver. White thermal noise was also present. The interference-to-noise density was varied over One scenario involving aggregate cosine-phased 20 values from –infinity to 108 dB. The receiver had a -6 BOC(15,2.5) interference to a narrow-band BPSK-R(1) dB two-sided bandwidth of 2.0 MHz with attenuation receiver resulted in abnormally large differences between provided by a 12-th order Butterworth filter (the cascade the simulation results and the analytical results. of two 6-th order filters with -3 dB bandwidths of 2.0 Debugging of the scenario revealed that the filter transfer MHz), and digitized the received signal, noise, and function that was applied in the analytical model did not interference with a 2.0 MHz sample rate and with one-bit truly match the attenuation that occurred within the quantization. Each simulation result point in Figure 8 is simulation tool. MATLAB was used for the simulations, based upon Monte Carlo simulation of 100,000 complex and the MATLAB filtfilt.m function was found to correlation sums with a 1-ms coherent integration time. have significant dynamic range problems, resulting in the The analytical model was carefully tuned to the scenario, filtered received signal power spectral density being much using the transfer function of the true digital filter that higher than would be predicted using the filter transfer was used in the simulations. The analytical model also function provided by another MATLAB function, used an aliased power spectrum for the input desired freqz.m. This problem only manifested itself in one of signal and aggregate interference since the simulations the sixteen scenarios, because this particular scenario had created these digitally with an original sampling rate of an extremely low degree of overlap between the 163.68 MHz. Nearest-neighbor resampling was used in interference and the desired signal power spectra. Since the simulations to decimate from the original high the objective of the comparison exercise was to validate sampling rate to the final receiver sample rate, and this the analytical model, and the discrepancy in results was was accounted for in the analytical model utilizing an found to be due to problems with the simulation tool, this equivalent signal processing model where the high rate particular scenario was dropped from further analysis. sampled signal is converted to analog using a sample-and- hold, with its well known distortion to the signal 40 Simulation spectrum, and then resampled. Three desired signal 30 Analytical powers were evaluated. The analytical results very closely 20 matched the simulation results with an average error 10 below 0.01 dB, and a peak error of 0.12 dB. C/N0eff (dB-Hz)

0 40 50 60 70 80 90 100 110 120 I/N0 (dB-Hz) Figure2 9. BOC(14,2) on BOC(10,5) Scenario Results ; 35 C/N0eff >= 25 dB-Hz 2-bit1 Quantization,C/N0eff < 25 dB-Hz24.0 MHz Two-sided Receiver Bandwidth, 24.0 MHz Sample Rate 30 0

40 -1 Simulation Sim (38 dB-Hz) 30 25 Sim (40 dB-Hz) in Difference C/N0eff (dB) -2 Analytical C/N0eff C/N0eff (dB-Hz) 50 60 70 80 90 100 110 120 Sim (42 dB-Hz) I/N0 (dB-Hz) Analytic (38 dB-Hz) 20 Analytic (40 dB-Hz) 20 Analytic (42 dB-Hz) 10 C/N0eff C/N0eff (dB-Hz)

0 15 50 60 70 80 90 100 110 75 80 85 90 95 100 105 110 I/N0 (dB-Hz) I/N0 (dB-Hz) 2 Figure 8. BOC(10,5) on BPSK-R(1) Scenario Results Figure 10. BOC(10,5) on BPSK-R(10) Scenario Results; 2-bitC/N0eff Quantization, >= 25 dB-Hz 24.0 MHz Two-sided 1 C/N0eff < 25 dB-Hz Receiver Bandwidth, 24.0 MHz Sample Rate The remaining 16 scenarios were run with only 10,000 1- 0 ms coherent correlation sums per simulation data point 40 due to time considerations. The results for a few of the -1 Simulation 30 Analytical scenarios are shown in Figures 9 – 11. Overall, the in Difference C/N0eff (dB) -2 simulation and analytical model results were in excellent 2050 60 70 80 90 100 110 I/N0 (dB-Hz) agreement, with typical differences less than a few tenths 10 C/N0eff C/N0eff (dB-Hz) of a dB for effective C/N0‟s down to around 20 dB-Hz. 0 This level of error is attributable to the natural variation of 40 50 60 70 80 90 100 I/N0 (dB-Hz) the Monte Carlo simulations with 10,000 correlation sums per data point. At lower effective C/N0‟s, larger Figure2 11. TMBOC on BPSK-R(10) Scenario Results; differences were observed but deduced to be due to the C/N0eff >= 25 dB-Hz 1.5-bit1 (3-level)C/N0eff

22 yR y(0) x (A-4) SUMMARY AND CONCLUSIONS Using equations (A-1) and (A-2), the condition stated in

(A-4) requires that: This paper has presented an analytical model for the  computation of implementation losses suffered by GNSS  1/2 2 2k  2 receivers due to bandlimiting, sampling, and quantization. K  h0kx /! k (A-5) k 1 The model has been shown to accurately predict SNR losses due to bandlimiting, sampling, and quantization. It The power spectrum of the quantizer output signal may be has the distinct advantage over Monte Carlo simulations related to the power spectrum of the input signal by taking in that it allows rapid determination of losses for the Fourier transform of both sides of equation (A-1) scenarios not yet considered. yielding:  k  APPENDIX A: OUTPUT AUTOCORRELATION Sy  f   bk S x  f  (A-6) AND POWER SPECTRAL DENSITY OF A k 1 k COMPLEX QUANTIZER WITH WIDE-SENSE- where the notation Sf  denotes the convolution of STATIONARY GAUSSIAN INPUT x the input power spectrum Sfx   by itself k times. The autocorrelation R   of the output signal y(t) of a y The results of equation (A-1) and (A-6) are readily memoryless nonlinearity driven by a zero-mean, adapted to discrete-time signals. For instance, as applied Gaussian, wide-sense stationary random process x(t) with to the in-phase discrete-time signal, xn , from Figure 2: autocorrelation function Rx   may be generally  expressed as [10]: k R m b R m xq [][] kx (A-7)  k 1 k Ry    bk R x   (A-1) and k 1  k where the coefficients { b } are a function of the specific jj k Sq e b S e (A-8) x   kx   nonlinearity. A variety of methods for determining the k 1 coefficients are provided in [10]. The coefficients for a where it is understood in equation (A-8) that the uniform quantizer with input-output characteristics as convolution is now circular as appropriate for discrete- described in equation (2-14) may be expressed as: time signals, e.g.,

2 S ej  S ej S e j x  x  x   b K22 h/! k kk0  (A-9) 1 j j  0, k even  Sxx e  S e d 2    .  N  h  /2 2 0k  1  apx/ /2 2 e  H a k It is useful numerically to replace the model for each real   k  p k1  p/ x  , odd   x p1 quantizer with an equivalent model shown in Figure A-1. (A-2) In this figure, (a) and (b) are equivalent provided that the quantizer input/output levels are adjusted to maintain 2 constant ratios with respect to the input signal standard where  xx R (0) is the power of the input process and deviations, i.e., if a quantizer parameter Q is used in (a), Hxk () is a k-th order Hermite polynomial defined as: (b) must use the quantizer parameter Q /  x . k k 22d Hk  x  (  1) exp(x / 2) exp(x / 2). (A-3) dxk x xq n n Equation (A-2) is easily derived from a more general Real Quantizer result for the output autocorrelation of a uniform quantizer driven by the sum of Gaussian noise and an (a) unmodulated carrier from [9]. It is noted in [9] that the output autocorrelation function of a uniform quantizer 1 Real driven only by additive Gaussian noise (as used in this  x  x report) was derived earlier by Velichkin, but the reference Quantizer cited in [9] for this prior work could not be located for use (b) herein. For N = 2, the coefficients in (A-2) are a series expansion of the arctangent function [11]. Figure A-1. Equivalent Models for a Real Quantizer

The gain constant K is selected to achieve unity power The equivalence of (a) and (b) in Figure A-1 may be gain for the quantizer, i.e., such that: demonstrated by rewriting equation (A-7) as: © The MITRE Corporation. All rights reserved.

k  X x() t  2k Rmx [] 1 Rq [] m b  (A-16) x  kx 2 X x() t k 1  x 2 (A-10) k It follows then, that by selecting:  Rm[]   2 b x xk 2  Xx1  nm k 1 x (A-17) Xy where 2 m 22k  Then, bbk k x (A-11)  22 k K hk /! k Rqq[][] m b R m (A-18) 0 xy  k xy  k 1 with the parameters h and K defined as: 0k Substituting (A-7) and (A-18) into (A-15) yields: k hhk k x Rm Rm  jR m  jR mRm  00 rq [][][][][]xq xq y q yq x q yq 0, k even  (A-19)  (A-12)  bRmk  jRk m  jRk mRm  k N /2 2  k x [][][][]xy yx y     a /2 2  ep  H a k k1    p k1  p , odd  p1 Based upon earlier assumptions that the noise and and interference are zero-mean and wide sense stationary, and further that these received signal components dominate

KK / x the desired signal, the cross-correlation of the received 1/2 signal in-phase and quadraphase components must satisfy  22k certain constraints [12]:  hk0kx /! (A-13) k 1 R m  R m  R  m  xy [][][]yx yx  1/2 2 (A-20)  hk/! Rxy[][] m R m  0k k 1 Using these relationships, (A-19) may be simplified as: The second line of equation (A-10) validates the  equivalence of Figure A-1 (a) and (b) since the k k k k Rmq [][][][][] bRm jRm  jRmRm  summation in this line is seen to be the autocorrelation of r  k x xy yx y  k 1 the output of a real quantizer (see equation A-7) with the  k input signal x /  . The autocorrelation is increased by  b Rkk m j   R m (A-21) nx  k2 x [ ] 1 1  yx [ ]  2  k 1 x after the gain factor of x following the real  kk quantizer in Figure A-1 (b). 2bk R x [ m ] jRyx [ m ] k 1 Using this equivalent model, the power spectral density in where (A-2) was used (specifically, the fact that for a (A-8) may be rewritten as: uniform quantizer bk = 0 for k even) going from the k second to the last line of (A-21).  j Sex   S ej   2 b  xq   xk 2 (A-14) Equation (A-21) may be simplified further by noting that k  1 x the autocorrelation of the input to the complex quantizer, The output autocorrelation function for the complex Rmr [], may be expanded as: quantizer in Figure 2 may be found as: * Rr [] m E rm n r n  qq*  Rq [] m E rm n r n  r  E x  jy x  jy m n m n  n n  (A-22) E xq  jy q x q  jy q (A-15) m n m n  n n  Rx [][][][] m  jRxy m  jRyx m  Ry m R m  jR m  jR m  R m R m jR m xq [][][][]xq y q yq x q yq 2x [ ] 2yx [ ] The first and last terms of the last line of (A-15) are The final result, relating the autocorrelation of the output provided by equation (A-7). of the complex quantizer to the autocorrelation of its input is presented in the main body of this paper as equation The two middle terms of (A-15) can be determined by (18) and the input-output relationships for power spectral observing that the derivation of (A-1) from [10] applies to densities is presented in the main body as equation (23). the cross-correlation between any pair of random variables before and after the same memoryless Numerical Considerations nonlinearity, Kf(x), is applied to each of the two variables. Determining the {bk} coefficients in (A-2) requires X X Y Let 1 and 2 be a pair of random variables and let 1 = numerical computation of Hermite polynomials and k!., Kf(X ) Y Kf(X ) 1 and 2 = 2 . Equation (A-1) results from both of which can lead to overflow problems for large k. selecting: These may be avoided by noting that that although the © The MITRE Corporation. All rights reserved.

numerator and denominator of bk grow exceedingly large 2 N /2 1 2 Q  2 with k, the ratio does not.     2p 1  erf  pQ  erf ( p  1) Q  2 p  1 (A-29) The k-th order Hermite polynomial, Hx(), is defined in k 2 1 N  N   erf  Q  (A-3). The polynomials for k = 0 and k = 1 may be 11   22 readily found as: where it is understood that the summation is zero if the

Hx0 ( ) 1 upper limit is lower than the lower limit, as is the case for (A-23) N H() x x a 1-bit quantizer ( = 2). 1 Higher order Hermite polynomials may be efficiently With the same input process, a real quantizer with an odd computed using the recursion: number of levels has an output variance:

H x x  H x  k  H x N /2 1 2 k ()()()k 12k (A-24) 2 2 21pQ  N 1 2   2Q  2 p 1  erf  Q (A-30) The magnitude of for large x is bounded by p 22 1  1.086 kx !exp(2 / 4) [9]. Both (A-29) and (A-30) were derived by determining the variance for N = 2, 3, 4, 5, 6, and 7 levels and observing To avoid overflow problems in the computation of bk, the patterns that emerged with increasing N. equation (A-2) may be rewritten as: Finally, the alternate expression for is: 0, k even  2 1 N /2 2 K  (A-31) bk    a  2'2 1 px/  /2  K k pe Hk1  a p/ x  , k odd    2   x p1 using (A-29) or (A-30) as appropriate. This result is also useful to determine the degree of convergence obtained (A-25) ' by truncating the series { } to a finite number of terms, where Hxk () is defined as: since the following should hold:

' Hxk ()  Hk  (A-26) 22 k! hk0k /!  (A-32) which may be efficiently computed recursively using: k 1

' Hx0 ( ) 1 APPENDIX B: DERIVATION OF MEAN VALUE AND VARIANCE OF CORRELATION SUMS H' () x x (A-27) 1 1 H' () x  xH' ( x ) k  1 H' ( x ) The mean value of each correlation sum is: k k k 12k (kM 1) 1 1 qj  The quantizer gain coefficient K is defined in equation E z   E r s* e (B-1) (A-13) and represents a required gain factor for a real k M  n n n kM quantizer with unit power, zero-mean Gaussian input to Using the cyclostationary model for the desired signal can provide an output also with unit power. Although as be expanded as: indicated in (A-13), this coefficient may be determined (kM 1) 1 1 qj  as: E z   E E r s s* e k M  n n n  n kM  1/2 (B-2) 2 (kM 1) 1 1 q q  j K hk /! k (A-28) *  0 E E xn s n  jE  yn s n  s n e k        1 M n kM  numerically, this is somewhat challenging since the series q where the inner expectation is with respect to rn , and the { h0k } converges slowly with k, especially for quantizers outer expectation is with respect to sn . (see, e.g., [13, N with a small number, , of output levels. Section 3.2] for a discussion of this expanded form of the An alternative approach to determine K is to examine expectation operator). the probabilities of occurrence for each of the N quantizer q The real and imaginary components of rn each are output levels, similar to the analysis to be presented in constrained to only a finite number of levels (the N real Appendix B, except with the input to the quantizer quantizer output levels) so that the inner expectations in approximated as zero-mean. equation (B-2) involve finite summations, e.g., With unit power, zero-mean Gaussian input a real N /2 Exsq  oPxos q    Px q   os  quantizer with K = 1 and an even number of levels has an n n  p  n p n  n p n  (B-3) output variance given by: p1 where op is the p-th positive quantizer output level given by © The MITRE Corporation. All rights reserved.

p oKpp  (B-4) (kM 1) 1  1 qj*   q1 E z   E E r s s e k M  n n n and the odd symmetry of the real quantizer has been n kM (B-12) utilized. Since the input to each real quantizer is 1 (kM 1) 1 K  h  C  E s s * 01  nn Gaussian-distributed, the probabilities in (B-3) are readily M n kM computed as, e.g.,: which leads directly to equation (26) in the main body of P xq  o s  P a  x  a s this paper. n p n p'' n p1 n The variance of the k-th complex correlator output is: a' C Re  s e j pn  erf (B-5) 2 2  varzk   E zk E zk  (B-13) x 

The mean value of the k-th complex correlator output, zk, a C s e j pn1 ' Re   erf was determined previously in equation (B-12). However,  note that in the derivation of the autocorrelation and x power spectral density of the complex quantizer, where equations (18) and (23), respectively, the mean value of 1 x the input to the quantizer (the desired signal term) was erf x  expt2 / 2  dt (B-6) noted to be very small relative to the variance of the input 2  0 (noise and interference terms) and approximated as zero. and a p ' is an extension of a p  defined in equations To maintain consistency with this earlier approximation, the variance of zk is thus approximated as: (15) and (16): z E z 2  varkk  ap , p  N / 2 desired signal absent a p '   (B-7) (k 1) M  1 (k 1) M  1 pN *  ,  / 2 11q*  jq j  E rn s n e rm  s m e By the low input signal-to-noise ratio assumption in MMn kM m kM (k 1) M  1 ( k  1) M  1 * 1 qq ** equation (9), it follows that Csnx/1  and the  E E r r s, s s s 2 n m  m n m n (B-14) M n kM m kM  approximation (k1)M 1 ( k 1) M 1 1 * Rq [] n m E s s 2 2   r mn ex /2 M n kM m kM erf x  x   erf() x   x (B-8) 1 (k 1) M  1 ( k  1) M  1  R n m R m n  m 2 2 r q [];s   M n kM m kM for x  1may be applied to (B-5) resulting in: Mn 1 M 1 1 R n R m kM n 2 rq [] s ;  N /2 M n( M  1) m0 Exsq   oPaxasPa   '  xas   n n   p  p'' n p11 n  p n p' n  p1 For large M, equation (B-14) simplifies to:

j N /2 1122 C s e  aa 2 Re n p'' x  p1 x  M 1  22 n  op  e e 1 x p1 varz  Rq [ n ] R n  1 (B-15)  ksMM r  j nM(  1)  NN/2  1122/2 1 CRe sn e aap'' x  p1 x  2  22 opp e o e If the coherent correlation interval is large relative to the    (B-9) x pp11 MT T j desired signal spreading symbol rate, i.e., sc, then NN/2  1122/2  CRe sn e aap'' x  p x  2  22 opp e o1 e (B-15) may be further approximated by: x pp12 j M 1 N /2 112 2 C s e  aa 1 2 Re n p x  1 x   o o e22 o e varzks   Rq [] n R n   pp11  r x p2 M nM(  1) j (B-16) C s e N /2 1 2  2 Re n  apx   Ke 2 11 jj  p  Sq ()() e Ss e d x p1  r M 2  K  h  C  Re  s e j 01 n which involves the inner product between the power Using (A-12) and (A-13), (B-9) can also be expressed as: spectral density of the complex quantizer output due to noise and interference and the power spectral density of E xqj s   K  h  C   s e   n n  01 Re n  (B-10) the normalized desired signal. The above derivation Similarly, an expression for the mean value of y q may be closely parallels the development of correlator output n variance with analog signal processing, e.g., see [14]. determined as:

ACKNOWLEDGMENTS [11] Van Vleck, J. H., and David Middleton, “The Spectrum of Clipped Noise,” Proceedings of the The author would like to thank Dr. Alex Cerruti for IEEE, Vol. 54, No. 1, January 1966, pp. 2 – 19. graciously providing the simulation results. This work [12] Proakis, J., Digital Communications, New York: was funded by the Federal Aviation Administration. McGraw-Hill, 1995.

REFERENCES [13] Stark, Henry, and John W. Woods, Probability, Random Processes, and Estimation Theory for [1] Spilker, James J., Jr., Digital Communications by Engineers, Englewood-Cliffs, New Jersey: Satellite, Englewood Cliffs, NJ: Prentice-Hall, Prentice-Hall, 1986. 1977. [14] Hegarty, C., Y. Lee, and M. Tran, “Simplified [2] Chang, H., “Presampling Filtering, Sampling and Techniques for Analyzing the Effects of Non- Quantization Effects on the Digital Matched white Interference on GPS Receivers,” Filter Performance,” Proceedings of the Proceedings of ION GPS 2002, Portland, International Telemetering Conference, San Oregon, 2002. Diego, California, 1982, pp. 889 - 915. [3] Van Dierendonck, A. J., “GPS Receivers,” in B. Parkinson, J. J. Spilker, Jr., Eds., Global Positioning System: Theory and Applications, Vol. I, Washington, D.C.: American Institute of Aeronautics and Astronautics, 1996. [4] Ward, P., J. Betz, and C. Hegarty, “Interference, Multipath, and Scintillation,” in Understanding GPS: Principles and Applications, 2nd Ed., Kaplan, E., and C. Hegarty, Eds., Norwood, MA: Artech House, 2006. [5] Betz, John W., and Nathan Schnidman, “Receiver Processing Losses with Bandlimiting and One-Bit Quantization,” Proceedings of The Institute of Navigation ION-GNSS-2007, Fort Worth, Texas, September 2007. [6] Betz, John W., “Bandlimiting, Sampling, and Quantization for Modernized Spreading Modulations in White Noise,” Proceedings of The Institute of Navigation National Technical Meeting, San Diego, January 2008, pp. 980 - 991. [7] Thomas, J. B., Functional Description of Signal Processing in the Rogue GPS Receiver, National Aeronautics and Space Administration Jet Propulsion Laboratory, JPL Publication 88-15, Pasadena, California, June 1, 1988. [8] Akos, Dennis M., and Marco Pini, “Effect of Sampling Frequency on GNSS Receiver Performance,” NAVIGATION: Journal of The Institute of Navigation, Vol. 53, No. 2, Summer 2006, pp. 85 – 95. [9] Hurd, William J., “Correlation Function of Quantized Sine Wave Plus Gaussian Noise,” IEEE Transactions on Information Theory, Vol. IT-13, No. 1, January 1967, pp. 65 – 68. [10] Davenport, Jr., W. B., and W. L. Root, Random Signals and Noise, New York: McGraw-Hill, 1958.