Enhanced Differential Detection Scheme for Weak

Enhanced Differential Detection Scheme for Weak GPS Signal Acquisition Surendran K. Shanmugam1 , John Nielsen2 and Gérard Lachapelle1 1 Position, Location and Navigation (PLAN) Group, Department of Geomatics Engineering 2 Department of Electrical and Computer Engineering, Schulich School of Engineering University of Calgary, Alberta, Canada. {sskonava, nielsenj, lachapel}@ucalgary.ca BIOGRAPHY Surendran K. Shanmugam is a PhD candidate in the Department of Geomatics Engineering at the University of Calgary. He received his MSc in Electrical Engineering at the same university in 2004 following the bachelor’s in Electronics and Communication Engineering, at the Anna University (India) in 2001. His research interests include high-sensitivity GPS, ground based wireless location and generally in the areas of communications theory and statistical signal processing. Dr. John Nielsen is a professor in the Department of Electrical and Computer Engineering at the University of Calgary. His research interest includes physical layer wireless communications as well as applied information theory. Currently, he is involved in the development of ultra wideband systems for a number of wireless applications. He is also involved in the development of various hybrid cellular positioning systems based on terrestrial CDMA signals combined with GPS. Dr. Gérard Lachapelle is a professor in the Geomatics Engineering at the University of Calgary. He holds a CRC/iCORE chair in wireless location. He has been involved with GNSS research, developments and applications for the past 28 years and has authored/coauthored numerous related publications, software and licenses. His primary research interests are in the areas of positioning, location and navigation. For more information visit: http://plan.geomatics.ucal gary.ca ABSTRACT In this paper several detector schemes for processing weak GPS signals in an unaided acquisition scenario are described and analyzed. Fundamental theoretical considerations based on the generalized likelihood ratio test (GLRT), as applied to GPS signal detection, are discussed. It is shown that the asymptotic version of the GLRT is equivalent to an estimator correlator (EC). For assumed deterministic signals the GLRT further reduces to a matched filter. This implies that the navigation data

code phase and carrier parameters are known. In this paper, we unify the well-known post-correlation noncoherent detection and the newly proposed postcorrelation differential detection in terms of GLRT and EC formulation. As well the generalized post-correlation differential detector scheme, which is a hybrid of postcorrelation, non-coherent integration and differential detectors is analyzed. Simulation results, as well as hardware based experimental measurements, are given to validate the claims of the acquisition sensitivity improvements of the proposed detection scheme.

INTRODUCTION The most difficult aspect of a GPS receiver operation is the synchronization process that involves the initial detection and the subsequent tracking owing to the low received power and moderate bandwidth of GPS signals. For nominal signal conditions, the initial synchronization process incurs minimum computational burden as the GPS signal can be acquired within a few milliseconds. Under adverse conditions, the received GPS signal can be attenuated by 20 dB or more. Consequently, many commercial receivers fail to acquire the GPS signals for C/N0 levels below 35 dB-Hz (Moreau et al. [2000]). In these cases a significant amount of signal processing gain is required to extract the weak GPS signal from the background noise. For instance, increasing the sensitivity by 20 dB or more requires observation (or integration) time of hundred times or even more from the nominal one millisecond integration time (Tsui [2004]). High sensitivity GPS (HS GPS) receivers, unlike standard GPS receivers, rely on extended integration time for significant acquisition sensitivity enhancement (Chansarkar and Garin [2000]). A number of detection schemes have been reported in literature to address the problem of high sensitivity acquisition under nominal and weak signal conditions. These algorithms can be generally categorized under coherent, non-coherent and differential detection schemes.

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The most beneficial approach is to employ coherent detection because of its optimality under additive white Gaussian noise (AWGN) conditions and assumed deterministic signal parameters. The maximum coherent integration time is limited by several factors. For instance, the presence of navigation data modulation typically limits the coherent integration time to less than 20 milliseconds. Advanced receivers such as the Assisted GPS (AGPS) receiver overcome this limitation in maximum coherent integration time through external aiding (Taylor and Sennott [1984]). The notion is to supply the GPS receiver with useful information, such as the navigation data (for data wipe-off), ephemeris data to estimate the satellite vehicle (SV) Doppler. In nominal signal conditions, this information is utilized to significantly reduce the mean acquisition time. Under weak signal conditions, the dwell time is increased considerably with the assistance information to accomplish the necessary acquisition sensitivity (Van Diggelen [2002]). On the down side, AGPS necessitates appropriate infrastructure deployment and the external communication link raises an issue of network delay and communication link reliability under adverse signal conditions. Other factors also limit the maximum coherent integration period besides navigation data and SV Doppler. Longer coherent integration time increases the number of frequency bins to be searched, which in turn increases the search space (in frequency domain) to a significant extent. More importantly, the stability of the reference oscillator in a GPS receiver critically restricts the maximum coherent integration period (Haag and Kelly [2000]). Finally, the time-varying propagation characteristics limits the maximum coherent integration period. In non-coherent processing, the time varying phase introduced by navigation data and that of residual carrier is eliminated through envelope detection. However, envelope detection results in a loss of SNR, which is often referred to as the squaring loss (Lowe [1999]). Therefore, the non-coherent detection is usually executed following an initial coherent observation to minimize the squaring loss. However, the use of longer coherent observation prior to envelope detection necessitates finer residual carrier estimation to minimize the power loss from residual frequency errors. Squaring loss can also be minimized through extended noncoherent accumulations without increasing the initial coherent observation period but the number of noncoherent accumulations to overcome the squaring loss under low C/N 0 conditions is substantial. (Kay [1993a]) Differential detection can be applied without any prior knowledge of carrier phase and thus can be considered as a potential alternative for non-coherent detection (Proakis [2000]). In the differential approach, the decision statistics is the product of uncorrelated samples,

whereas the non-coherent detection utilizes the square of the same signal sample. Hence, the differential detection incurs a SNR loss, which is typically less than the noncoherent squaring loss. Coenen and Van Nee [1992] applied the differential approach for low complexity GPS signal acquisition. Lin and Tsui [1998] later introduced a similar detection method for software GPS acquisition using the differential approach proposed by Tomlinson and Bramwell [1988]. The major limitation of this differential approach is the substantial SNR loss as the differential detection is performed prior to any kind of coherent integration. More recently, Shanmugam et al. [2006] introduced the coherent pre-filtering/Multicorrelation differential detection (PF/MCDD) technique to overcome the SNR loss inherent to the pre-correlation differential detection (PDD). However, a significant number of parallel correlators are required to attain the required sensitivity. An alternate approach, denoted as the post-correlation differential detection (PCDD) method, involves coherent integration followed by differential detection processing. This was initially proposed by Zarrabizadeh and Sousa [1997] for a DS-CDMA PN code acquisition application. Park et al. [2002] adapted the PCDD technique for GPS signal acquisition. Subsequently, the acquisition sensitivity improvements of the PCDD was validated in theory by Schmid and Neubauer [2004] and using live GPS data by Shanmugam et al. [2005]. Elders-Boll and Dettmar [2004] and Schmid and Neubauer [2005] emphasized the viability of fine frequency estimation and PLL application of the PCDD approach. Having reviewed the various detectors utilized for GPS signal acquisition, one can readily discern the inherent limitations of the respective detectors when applied to GPS signal acquisition. For instance, the coherent approach although optimal in terms of noise suppression is crucially limited by the time-varying phase. In contrast, the non-coherent and the differential approaches readily alleviate the problem of time-varying phase but are decisively suboptimal juxtaposed with theoretically optimal detection schemes. Figure 1 summarizes the inherent trade offs of various detection approaches as applied to the problem of GPS signal acquisition. As shown in Figure 1, minimal noise enhancement and utmost resilience to time-varying phase are highly desirable for detectors applied to GPS signal acquisition. Subsequent development of novel algorithms that satisfy the previously mentioned criteria requires a basic perspective GPS signal acquisition in terms of detection theory. As will be discussed further in this paper, the seemingly disparate detectors introduced can readily be related as suboptimal detector implementations of the theoretically optimal detector.

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intensive (multi-dimensional search) process. The GPS signal acquisition is a joint estimation problem as both the code phase and carrier offset need to be estimated.

Figure 1 GPS Signal Detection – Trade offs In this paper, we endeavor to accomplish two imperative goals in regards to enhanced GPS signal acquisition. The first is to relate the detector schemes to the generalized likelihood ratio test (GLRT) processing. The second objective is to propose a novel detector structure, which comprises the more popular post-correlation noncoherent and differential detectors as its specialized forms. The major benefits of the proposed detector includes the much desired noise suppression and the ability to allow for fine frequency estimation. The remainder of the paper is organized as follows. Initially the coherent matched filter is established as the optimal detector based on the assumption of deterministic signal parameters. Subsequently, the decision statistic of the optimal detector is derived for random navigation data and residual carrier offset which is implemented with post-correlation non-coherent and differential detectors. The decision statistic of the proposed generalized post-correlation differential detector is then derived. The effect of navigation data and that of residual carrier on the developed detector is verified in terms of theory and numerical simulations. Moreover, a novel residual frequency offset estimation algorithm using the developed detector is introduced. Test methodology and acquisition performance analysis using the hardware simulated GPS signal are presented. Salient conclusions are listed in the final section. OPTIMAL DETECTORS FOR GPS SIGNAL ACQUISITION Signal demodulation and subsequent parameter extraction are the fundamental functions supported by the GPS receiver signal processing. The initial acquisition and subsequent tracking ensures the continuous demodulation of the received GPS signal to allow for subsequent parameter extraction. The initial GPS signal acquisition, often, is the most challenging (limited a priori information) and the most hardware

The parameter estimation in GPS signal acquisition is typically carried out as a discretized multi-dimensional search process corresponding to a maximum likelihood estimation (MLE) of the unknown signal parameters of code phase, carrier offset and navigation data resulting in a generic GLRT formulation. Typically the multidimensional search space is discretized into cells each representing an orthogonal hypothesis that requires individual testing to implement the MLE of the unknown parameters. Within each cell, ( H1 ) implies a signal present and H 0 implies no signal. The detector decides on H1 if,

LG (x ) =

p x;!ˆ1 , H1

(

)> "

p (x;! 0 , H 0 )

(1)

where !ˆ1 is the maximum likelihood estimate (MLE) of parameters for the cell under the hypothesis that the signal is present and ! 0 is the parameter vector for the cell under the null hypothesis case. Also ! is the threshold related to the probability of false alarm associated with false detection of the individual cell given implicitly by the relation

PFA = #

{x:LG (x )>! }

p (x;" 0 , H 0 )dx

(2)

A detailed exposition of the nuances of (1) and (2) as well as the acquisition algorithms and related performance are beyond the scope of this paper. The readers are referred to O’ Driscoll [2007] for a more detailed analysis of acquisition search process. The derivation of decision statistics for the optimal detectors in this section essentially follows Kay [1993a], but is carried out in the context of GPS signal acquisition.

Known Signal Parameters The structure of the optimal detector is developed by considering the detection of known PRN code signal in AWGN. It is assumed that the detector is positioned at the correct cell or the H1 cell to allow for a binary hypothesis testing. In the presence of perfect parameter estimate, the detection problem in GPS signal acquisition reduces as follows:

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H 0 : x (k ) = s!0 (k ) + w (k ), k = 0,1, 2,K , N s " 1 H1 : x (k ) = s!1 (k ) + w (k ), k = 0,1, 2,K , N s " 1

(3)

where s (k ) and w (k ) are the received GPS PRN code signal, and AWGN respectively. w (k ) is assumed to be complex zero mean AWGN process with variance ! w2 = N 0Ts"1 , where N 0 is the noise power spectral density and Tc is the basic chip duration. N s is the number of samples pertaining to the observation period T (i.e. N s = TTs!1 , where Ts!1 is the sampling period). The complex baseband equivalent of the low-pass GPS signal is given by,

s (k ) = Cd (k $ " )c (k $ " )e j! (k )

! (k ) = 2#%Fk + !0

(4)

where C is the instantaneous transmitted power. The BPSK modulated data symbols and the PRN (or C/A) code are denoted by d (k ) and c (k ) respectively. The data symbol duration, Tb , is related to the chip duration as

Tb = 20 N cTc where N c = 1023 the number of chips per period of c (k ) which is denoted as Tr = N cTc . The phase

! (k ) is the instantaneous carrier phase with the residual frequency and phase offset !F and !0 respectively. The optimum detector based on the Neyman-Pearson criterion for a known PRN signal in AWGN is the replicacorrelator or the matched filter (Kay [1993a]). Following the approach reported in Kay [1993a], we can finally derive the decision statistics for the matched filter as,

TMF (x ) =

N s #1

% x (k )sˆ (k ) $

k =0

(5)

sˆ$ (k ) = c (k # !ˆ )dˆ (k # !ˆ )exp (# j" (k )) From the above equation, we can infer that the detection performance of the matched filter critically depends on the corresponding estimation of the received signal parameters such as the delay (! ) , data (d (k )) , residual frequency and phase offset ("F , !0 ) . From (5), we also see that the data wipe-off process also involves the matched filtering. In reality, the GPS receiver is usually provided with an estimate of the PRN code signal parameters. Besides, the extent of the a priori information on the transmitted GPS signal parameters depends on the type of GPS receiver technology. For instance, in AGPS technology, the GPS receiver is supplied with the estimate of navigation data along with

good timing and Doppler estimates (Bryant [2005]). However, the receiver clock offset still needs to be estimated depending on the frequency stability of the receiver oscillator. The influence of the residual signal such as the residual carrier and that of navigation data on the matched filter implementation can readily be modeled and is further addressed in O’ Driscoll [2007].

Unknown Signal Parameters In the matched filter implementation, we were able to detect the signal in the presence of AWGN by detecting the change in the mean of the decision statistic. This is possible only when the GPS receiver is supplied with or is able to estimate the transmitted PRN code signal parameters. For example, non-AGPS receivers are readily limited by the presence of unknown navigation data and residual frequency errors. Even AGPS receivers can be limited by residual carrier due to the receiver clock offset. Hence, the presence of unknown navigation data and residual carrier can be more aptly modeled as a random process even in the presence of known PRN code signal. Accordingly, the optimum detector based on NeymanPearson theorem for a known PRN code signal with unknown parameters can be obtained using (1) (Kay [1993a]). The application of such a detector implementation can be found at the pre-correlation level in Coenen and van Nee (1992) and in Lin and Tsui (1998). The generalization of a similar structure can be found in acquisition scheme introduced by Shanmugam et al. (2006). In contrast, this paper introduces the application of structure introduced in Shanmugam et al. (2006) at the post-correlation level. Therefore, we first model the matched filter output as,

y (n ) =

(n +1)Nc #1

$ k =n

x (k )d (k # !ˆ )c (k # !ˆ )exp # j"ˆ (k ) ,

(

)

n = 0,1, 2,K , N # 1.

(6)

where, N is the number of non-coherent accumulations and is related to the total observation period as N = TTr!1 .

d (n ) models the navigation data. y (n ) is the coherent matched filter output or the correlation output that is sampled at nTr seconds and is given by,

y (n ) = Cd (n )d (k $ !ˆ )R! err (n )" #Ferr (n ) + w (n ) (7)

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The noise at the output of matched filter is denoted by w (n ) . R! err (n ) , and ! "Ferr (n ) are the time and

N !1

TPCND (y ) = # y (n ) y " (n ) n=0

frequency ambiguity functions. Therefore, we have,

d (n ) = d (n ) =

(n +1)Nc %1

3 k =n

(n +1)Nc %1

3 k =n

R! err (n ) =

N !1

TPCND (y ) = TPCND (y ) = # y (n ) n=0

d (k )

In the above equation, we see that the Neyman-Pearson detector correlates the received signal with a noisy estimate of the received signal, whose structure is similar to that of estimator-correlator. The expression in (12) represents the envelope detection utilized in postcorrelation non-coherent detection (Kaplan and Hegarty [2006]). Interestingly, one can utilize the periodicity of the underlying PRN code to implement a slightly different structure as follows:

w (k )c (k % !ˆ )exp % j"ˆ (k )

(

)

(n +1)Nc %1

3 c (k % ! )c (k % !ˆ ) k =n

# &Ferr (n ) =

(12) 2

(n +1)Nc %1

3 k =n

exp ( j (2$&Ferr k + "0 ))

' = sinc ($&Ferr nTcoh )exp + 1

' (( N % 1* ) j + 2$&Ferr - nN c + c . Tc + "0 , , 2 0 / 1 22 (8)

H 0 : sˆ (n ) ! y * (n " Tm ) , Tm = mTcoh H1 : sˆ (n ) # y * (n " Tm )

(13)

where !Ferr = !F " !Fˆ . Once again, we model the output of matched filter using the following binary hypotheses:

Hence, the receiver based on (13) correlates the received signal with a noisy and delayed estimate of the received signal. Note that the estimate based on (13) is no longer the maximum likelihood estimate of s (n ) in the

H 0 : y (n ) = w (n ), n = 0,1, 2,K , N ! 1. H1 : y (n ) = s (n ) + w (n ), n = 0,1, 2,K , N ! 1.

presence of navigation data and residual carrier. The receiver implementation based on (13) represents the widely utilized differential detector (Proakis [2000]). For example, Zarrabizadeh and Sousa [1997] utilized the aforementioned receiver structure to implement the postcorrelation differential detection (PCDD) for CDMA signal acquisition. Similar applications can also be found in Park et al. [2002], Schmid and Neubauer [2004], and Elders-Boll and Dettmar [2004] for GPS signal acquisition. The final decision statistics for the post-correlation differential detection can be derived as,

where

s (n ) = 2Cd (n ) R! err (n )" #Ferr (n ) ,

(9)

the GLRT

processing would decide H1 if (Kay [1993a]),

p (y; sˆ (0 ), sˆ (1),K , sˆ (N " 1), H1 )

LG (y ) =

p (y; H 0 )

>!

(10)

N !1

where sˆ (n ) is the MLE estimate of the signal under H1 .

TPCDD (y ) = " y (n ) y * (n ! Tm ), Tm = mTcoh

Modeling s (k ) as deterministic but completely unknown,

Substituting for y (n ) from (7) and (8), in (12) and (13),

p (x; sˆ (0 ), sˆ (1),K , sˆ (N # 1), H1 ) =

1

(2"!

2 #1 N w coh

T

)

$ 1 exp & # ! 2 T #1 ( w coh

N #1

* x (n ) # s (n ) n=0

(14)

n=0

2

% (11) ' )

From the above equation, we can obtain the maximum likelihood estimate after further manipulation as, sˆ (n ) = y ! (n ) (Kay [1993b]). Substituting (11) in (9) and upon further simplification, we arrive at the final decision statistics as,

we arrive at the signal component of the decision statistics for the post-correlation non-coherent/differential detector as, N #1

TPCND (s ) = ) 2CR!2err sinc 2 ("$Ferr nTcoh ) n=0

N #1

TPCDD (s ) = ) %& 2Cd (n )d (n # Tm )R!2err sinc ("$Ferr nTcoh ) n=0

sinc ("$Ferr (nTcoh # Tm ))exp ( j 2"$Ferr Tm )'( (15)

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The characterization of loss due to navigation data as expressed in (18) only hold true for Tm < Tb . For delays exceeding the data bit duration, the loss expression in (16) can be approximated by,

The index, n , in the correlation output R (• ) is excluded due to the code periodicity over the sampling period. In particular, we notice that the decision statistics of the post-correlation differential detector is still characterized by a phase rotation. Elders-Boll and Dettmar [2004] and Schmid and Neubauer [2005] utilized this phase relationship to realize the fine frequency estimation in their respective implementations. It should be noted here that the estimation performance of the corresponding frequency estimation technique would be critically effected by the C/N0 , the observation period and the differential delay (due to the presence of navigation data). Secondly, we can also notice the decorrelation of the navigation data over the delay Tm . The power loss

2 "$ N !1 % # E ') / d (n )d (n ! Tm )* ( & 0 , (. '-+ n = 0

The differential detector typically utilizes Tm < Tb , so as to retain the phase correlation. Hence, the loss due to navigation data modulation can be expressed as,

due to the navigation data decorrelation for a delay of Tm = 1 ms, is approximately around 0.45 dB as reported by Schmid and Neubauer [2004]. The average loss due to navigation data modulation is given by,

LD (Tm ) =

2 2 "$ N !1 % # " $ N !1 %# E &( / d (n )d (n ! Tm )) ' = & E ( / d (n )d (n ! Tm )) ' + -' , * n = 0 +,&* n = 0

"$ N !1 % # $ N !1 % +Var &( / d (n )d (n ! Tm )) ' . ( / E ", d (n )d (n ! Tm )#- ) + - * n=0 + ,* n = 0

2

(16) in the limit as N ! " . The above approximation in the loss due to navigation data modulation is based on the fact that d (n ) is highly correlated for Tm < Tb . Therefore, we have for N ! " ,

$& N !1 '% Var *( " d (n )d (n ! Tm )) + -/ . , n =0

lim # 0, Tm 2 N 0 1 & N !1 ' ( " E $. d (n )d (n ! Tm )%/ ) , n=0 -

(19)

Tb

(17)

Tb2 2

(Tb ! Tm )

, Tm < Tb

(20)

Figure 2 shows the theoretical loss due the navigation data modulation (Black Square) from (18) and the numerical simulation results for 30 (Red Cross) and 300 (Blue Circle) navigation data bits. In Figure 2, we see that for N = 600 (30 navigation data bits), the theoretical and numerical computed loss increasingly differs for Tm > 12 ms. However, the approximation in (18) is well validated up to Tm = 19 ms in the case of N = 6000 (300 bits). The results shown in Figure 2 utilized the asymptotic assumption to characterize the average loss due to navigation modulation. However, the actual loss due to navigation data modulation for a given observation period can differ substantially as shown by the standard deviation for large Tm . Figure 2 readily shows the inherent limitation of the differential detection approach in the presence of navigation data modulation. For example, the detection performance of PCDD is expected to degrade significantly as Tcoh ! Tb . This does not come as a surprise as the underlying assumption of phase correlation in PCDD no longer holds true for Tm ! Tb due to the random phase transition introduced by navigation data.

Hence, using the rectangular pulse shape assumption of navigation data and for N ! " , we express the loss due to the navigation data modulation in PCDD as, 2 2 N !1 $& N !1 ' % & ' E )+ 1 d (n )d (n ! Tm ), * ( E + lim 1 d (n )d (n ! Tm ), . 0* - N "# n = 0 . /)- n = 0 2

& T ' ( +1 ! m , , Tm = mTcoh , Tm < Tb - Tb . (18) Figure 2 Navigation data decorrelation loss in PCDD Scheme

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Under weak signal conditions, the SNR loss for both post-correlation non-coherent and differential detectors can be significant. While longer coherent integration can alleviate this SNR loss, it is crucially limited it to less than 10 ms, due to the presence of unknown navigation data. Besides, longer coherent integration necessitates finer residual frequency offset estimation resulting in increased computational complexity. In contrast, we address the same constraint by means of a novel detector structure, the generalized post-correlation differential detector (GPCDD). The proposed detector derives its major impetus from the multi-correlation differential detector (Shanmugam et al. [2006]). The benefits of the proposed detector include improved noise suppression and robust fine frequency estimation. GENERALIZED POST-CORRELATION DIFFERENTIAL DETECTOR Recognizing the periodic property of the underlying PRN code, an ensemble of estimates can be obtained for various delays as,

sˆm (n ) = y (n ! mTr ), m = 1, 2,3,K , M .

Where Tcoh is the coherent integration time. Figure 3 shows the output of the individual differential detectors in the presence of navigation data transition (at 32 ms). While the original data is marked in Red (dashed), the detection outputs for delays Tm = 1 ms and 19 ms are highlighted in Blue (Solid) and Green (dash-dot) respectively. As can be seen in Figure 3, for Tm = 1 ms, the effect of navigation data bit transition is pronounced only over a millisecond (31 ms). For Tm = 19 ms, we see that the same data transition is pronounced over longer period (13 ms to 31 ms). In other words, the residual effect due to navigation data is increasingly pronounced at the output of differential detection for increasing delays. However, our goal is to determine the net effect of data on the final detection output. As the final detection output is formed by summing the individual differential detection outputs, the final expression can be obtained by summing (23) over the range of delays. That is, 2

!M T " LD (M ) = $ (1 # m % , Tm = mTcoh & m =1 Tb '

(24)

(21)

Substituting for sˆm (n ) from (21) in (14), we arrive at the following decision statistic, M N !1

TGPCDD (y ) = "" y (n ) y * (n ! mTr )

(22)

m =1 n = 0

From the above expression, we see that the current matched filter output is correlated with an ensemble of earlier matched outputs. Moreover, we see that the above structure decomposes to a post-correlation differential detector for M = 1 , m = 1 and to that of a post-correlation non-coherent detector for M = 1 , m = 0 . Hence, the structure expressed in (22) is termed as the generalized post-correlation differential detector. It is of interest to characterize the effect of the residual signals such as the navigation data and residual carrier on the abovementioned detector structure.

Figure 3 Effect of Navigation Data on Differential Detection

Effect of Navigation Data

Using the summation identity, we can further simplify the above expression as,

To derive an expression for the effect of navigation data on the proposed GPCDD structure, we ignore the effect of other residual errors and AWGN. Hence, we can rewrite (22) as,

" (M + 1)TCOH # !1 LD (M ) $ %1 ! & , M < TbTCOH 2Tb ' (

M N !1

TGPCDD (y ) = "" d (n )d (n ! Tm ), Tm = mTcoh m =1 n = 0

(23)

2

(25)

where the approximate condition is due to the variance term that was neglected in (16). Figure 4 shows the theoretical power loss (Black Square) due to navigation data modulation as a function of number of branches, M .

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The numerical simulation for the power loss due to navigation data (Blue circle) averaged over 1000 independent trials is also plotted.

Figure 4 Effect of Navigation Data Transition on GPCDD Output It should be noted that the above analysis is based on the coherent summation across the differential detection outputs, which is crucially affected by the presence of residual carrier. In the next section, we characterize the influence of residual carrier on the GPCDD structure. In particular, we show how the aforementioned structure can be incorporated with an FFT processing block to provide an estimation of the residual carrier frequency offset.

Effect of Residual Carrier In the absence of AWGN, we can express the signal component of (22) as,

From (26), we see that the gain achieved by the GPCDD structure can be limited by residual carrier even in the absence of navigation data modulation. Interestingly, the influence of residual carrier on GPCDD can be alleviated with non-coherent or differential combining of the individual branch outputs. Unfortunately, the subsequent noise suppression through non-coherent or differential combining is less significant to that of coherent combining. Nevertheless, the same periodic property of the underlying C/A code can be utilized to allow for finer frequency estimation in the developed GPCDD structure. For example, one can notice the complex phase rotation across the individual branches is related to the residual frequency offset and the corresponding branch delay. Thus, the individual complex phase rotations collectively embody the sampled residual carrier. Hence, for correct code phase, the residual carrier can be estimated by applying a M-pt FFT across the differential detection outputs. Besides, the FFT based combining also accomplishes coherent combining. Figure 5 illustrates the GPCDD structure utilizing M-pt FFT based residual frequency offset estimation and coherent combining. In Figure 5, one can also notice the use of recursive filtering to implement coherent averaging in individual differential detector branches, which works effectively in the absence of residual frequency drifts. The proposed frequency estimation technique involves an additional gain of

10 log10

the frequency estimation methods

proposed by Elders-Boll and Dettmar [2004] and that of Schmid and Neubauer [2005]. As the input sampling rate to the FFT is Tcoh = Tr , the corresponding bandwidth and the resolution of the FFT based frequency estimation is given by,

! BW =

M N "1

TGPCDD (y ) = sinc 2 (!#Ferr Tcoh )$$ e j 2!#Ferr Tm

( M ) over

1 Tr

! RES = ±

1 MTr

(27)

(26)

m =1 n = 0

Figure 5 Generalized Post-correlation Differential Detector Structure with FFT Based Robust Frequency Estimation

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For example, with M = 20, the above estimator achieves a minimum resolution of around ! RES = ±50 Hz. Note that the FFT estimator will be able to correctly estimate residual frequencies that fall between ±Tr!1 or 500 Hz. Thus, a residual frequency offset greater than 500 Hz introduces an integer frequency ambiguity due to spectral wrapping over the input sampling rate. Nevertheless, the complex phase rotations are still compensated even in the presence of this integer frequency ambiguity. A more reasonable choice on the number of branches is around 16 based on the data decorrelation loss and FFT implementation efficiency. For M = 16, any residual frequency offset within ±500 Hz can be estimated with an accuracy of around ±62.5 Hz. In summary, the proposed GPCDD structure accomplishes both noise suppression as well as robust frequency estimation for weak GPS signal acquisition. However, the performance of GPCDD structure can be degraded significantly, for Tm ! Tb due to the phase decorrelation caused by the navigation data.

PERFORMANCE EVALUATION

subsequently resampled to 5.714 MHz during postmission processing. The Euro-3M card utilized an internal temperature controlled crystal oscillator (TCXO) as its reference oscillator. The IF data resampling, pseudo baseband down-conversion and subsequent GPS L1 C/A code acquisition were realized using dedicated routines in MATLAB. The detection performance of the various acquisition algorithms were then compared using correlation output and also using the deflection coefficient (DC) measure, which is expressed as,

! E (T (x ); H1 )# E (T (x ); H 0 )"% DC = $ Var (T (x ); H 0 )

2

(28)

The deflection coefficient measure is an accurate measure only for Gaussian decision statistics. However, the deflection coefficient provides a reasonable metric for non-Gaussian decision statistics. Monte-Carlo simulations were also carried out to assess the average detection performance. The acquisition results were further validated using IF GPS data collected under controlled C/N0 conditions. Figure 6 illustrates the hardware test setup employed for acquisition performance analysis.

Having derived the decision statistics for different detection structures, we now turn to empirical tests for validating the developed theory. The test predominantly utilized IF samples collected from a Spirent GSS 7700 simulator under emulated signal environments. Besides, numerical simulations were also carried out to assess the average detection performance for low C/N0 conditions. The test methodology utilized a Novatel Euro-3M frontend card in conjunction with a 30 dB LNA for RF signal capturing. The developed test methodology allowed for simultaneous L1 and L2C signal capturing. The L2C signal power level was set to nominal conditions, which was later utilized for validating the results obtained for L1 C/A signal acquisition. In Euro-3M card, the digitized IF samples (2-bit) were centered on 70.42 MHz and has a combined sampling rate of 20 MHz, which were

Figure 6 Test Methodology

Table 1 GPCDD: Fine Frequency Estimation Performance (TCOH = 1 ms) C/N0 (dB-Hz) 48 43 38 33 28 23

T (ms) 20 40 75 150 300 600

Frequency Estimation Error, ! Ferr = "Ferr # "Fˆerr M=1 Mean STD 1.8 1.8 2.4 2.1 3.4 2.8 5.6 4.8 12.6 10.1 26.1 21.8

M=2 Mean STD 1.4 1.5 1.9 1.7 2.3 2.1 3.8 2.9 7.9 5.9 16.0 12.6

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M=3 Mean STD 1.2 1.4 1.6 1.6 2.0 1.8 3.0 2.3 4.6 3.6 12.1 9.8

(Hz) M=4 Mean STD 1.0 1.4 1.4 1.3 1.8 1.6 2.5 1.9 3.9 3.1 10.1 10.6

9

Frequency Estimation Performance Monte-Carlo simulations were performed in MATLAB as an initial step to assess the frequency estimation performance of the proposed detector structure. The frequency estimation technique in the proposed detector essentially averaged the frequency estimates obtained across individual branches for the correct code phase estimation. That is,

# 1 M N "1 $ %& y (n ) y * (n " mTr )' ++ MN m =1 n = 0 ) *Fˆerr (M ) = ( "1 2! mTr

(29)

where ! (• ) is the phase angle estimate. From the above equation, we see that for M = 1, the frequency estimation decomposes to the method introduced in Elders-Boll and Dettmar [2004] and Schmid and Neubauer [2005]. Table 1 summarizes the frequency estimation performance for different M under different C/N0 conditions. The frequency estimation results were averaged over 250 independent simulations for different PRN’s, Doppler’s and code phases. While the coherent integration period was set to 1 ms, the total observation period was increased correspondingly for decreasing C/N0 levels. The proposed detector (for M = 4) demonstrated substantial improvement in terms of frequency estimation under low C/N0 conditions due to its enhanced noise suppression via coherent combining across branches. The FFT based frequency estimation approach was also verified using hardware simulated GPS data for 25 dBHz C/N0 data. The proposed detector utilized 1 ms coherent integration and 400 ms observation period with M = 16. Figure 7 shows the normalized FFT output for the developed GPCDD detector structure for various PRN’s in the case of 25 dB-Hz C/N0 data. The coherent integration period was set to 1 ms with 400 ms total observation period.

Figure 7 GPCDD/FFT Based Residual Doppler Estimation (C/N0 = 25 dB-Hz, TCOH = 1 ms, T = 400 ms, M = 16) (Hardware simulated GPS IF Data) The FFT based frequency estimation performance can potentially be degraded for large M. To illustrate this, we plotted the normalized FFT output for PRN 9 for various values of M in Figure 8 with 1024-pt FFT. For M = 32, the maximum delay utilized by the GPCDD is 32 ms (as Tcoh = 1 ms), which exceeds the navigation bit duration of 20 ms. The net effect of navigation data modulation resulted in the phase reversal (1800) for differential delays exceeding the data bit duration. In Figure 8, the effect of navigation data modulation can be clearly seen in the form of split spectrum for PRN 9 with M = 32. The split spectrum can readily be attributed to the phase reversal occurring for the differential detection branches (due to data bit phase transition) with delays exceeding the data bit duration. However, it should be noted that the PRN 9 could still be acquired with GPCDD detector but only with an ambiguous residual Doppler estimate. Thus, it is desirable to keep the value of M to less than 20 to avoid the undesirable effect arising due to navigation data modulation.

Table 2 Effect of Frequency Drift on Acquisition Sensitivity Freq. Drift Hz/s 0 25 50 100 300

Deflection Coefficient (dB) PCND GPCDD (TCOH = 1 ms) TCOH = 1 ms TCOH = 10 ms M=1 M = 10 Mean STD Mean STD Mean STD Mean STD 12.2 2.9 21.5 2.4 17.3 3.0 26.0 2.3 12.6 3.0 21.2 2.3 16.7 3.6 25.7 2.3 12.4 2.7 20.7 2.2 16.3 3.3 25.1 2.1 12.2 2.9 20.4 2.4 15.7 3.5 24.8 2.4 11.7 2.9 15.6 2.7 15.4 4.0 18.7 3.9

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Figure 8 Effect of Navigation Data Modulation on GPCDD/FFT Based Residual Doppler Estimation (Hardware simulated GPS IF Data)

Figure 9 Effect of Frequency Drift on M for GPCDD Based Acquisition In other words, branches with longer differential delays incur more loss due to the increased residual frequency offset occurring due to the frequency drift. Therefore, it is highly desirable to use small values of M in the presence of large frequency drifts.

The influence of residual frequency drift was also evaluated using numerical simulation. The acquisition sensitivity performance for the developed GPCDD based acquisition is tabulated in Table 2. The deflection Acquisition Sensitivity Performance coefficient results were obtained via 100 independent trials. The C/N 0 was fixed to 25 dB-Hz and the total The noise suppression performance of the various observation period was set to 400 ms. The PCND with 1 detection algorithms was also evaluated in terms of ms coherent integration and GPCDD with M = 1 based deflection coefficient for various C/N0 conditions. The acquisition demonstrated the least degradation in terms of acquisition sensitivity performance was evaluated using frequency drift. PCND based acquisition with 10 ms both numerical simulations as well as live GPS data. coherent integration degraded by 10 dB for 400 Hz/s Table 3 shows the average acquisition sensitivity frequency drift. GPCDD based acquisition with M = 10 performance obtained through 250 independent trials for experienced the worst-case degradation of around 12 dB. different C/N0 levels. The observation period was Figure 9 shows the influence of frequency drift on the increased accordingly with decreasing C/N0 values to acquisition sensitivity as a function of M . In Figure 9, ensure proper detection performance. The GPCDD we can notice the improvement in deflection coefficient based acquisition with M = 10 demonstrated an average with increasing M in the absence of frequency drift. On performance improvement of around 7 dB in comparison the other hand, the presence of frequency drift readily to M = 1. Similarly, the PCND with 10 ms coherent degrades the deflection coefficient for large M . The integration readily resulted in 8 dB improvement over 1 frequency drift essentially manifests into frequency offset ms coherent integration for the same observation period. at the output of the differential detector. Accordingly, the coherent summation (recursive filtering) in Figure 5 incurs a loss due the residual frequency drift. Table 3 Acquisition Sensitivity Performance C/N0 (dB-Hz)

T (ms)

45 40 35 30 25 20

30 40 80 150 300 450

Deflection Coefficient (dB) PCND (dB) GPCDD TCOH = 1 ms TCOH = 1 ms TCOH = 10 ms M=1 M = 10 Mean STD Mean STD Mean STD Mean STD 42.5 1.5 48.5 2.1 48.0 1.5 50.7 2.3 34.1 1.3 41.8 2.1 39.9 1.4 45.3 2.6 27.2 1.7 35.2 1.9 32.9 1.8 39.4 2.7 20.1 1.8 28.5 2.3 25.6 1.9 33.3 2.2 12.2 2.9 21.5 2.4 17.3 3.0 26.0 2.3 3.0 7.5 13.3 3.9 3.4 9.8 16.7 4.4

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B d ( t n e i c i f f e o C

Under low C/N0 conditions, as in the case of 20 dBHz, the detection performance of PCDN with 1 ms coherent integration and that of PCDD ( M = 1) are severely degraded resulting in an increased false acquisition. On the other hand, the noise suppression in PCND with 10 ms coherent integration and that of GPCDD with M = 10 resulted in minimal false acquisition conditions. Besides, the influence of M for noise suppression under various C/N0 conditions was also evaluated. Figure 10 shows the convergence of deflection coefficient as a function of M for various C/N0 levels. As expected, under higher to moderate C/N0 conditions, the convergence is not apparent for large values of M . On the other hand, the use of large values of M is readily apparent for low C/N0 conditions. However, the incremental noise suppression decreases with M due to the presence of navigation data.

Figure 10 Convergence Performance of GPCDD for Different C/N0 Conditions

n o i t c e l f e D

60 PCND (TCOH = 1 ms)

T = 40 ms

PCDD PCND (TCOH = 10 ms)

50

T = 80 ms

GPCDD (M =10)

T = 100 ms

40 T = 150 ms 30

20

T = 300 ms T = 600 ms

10

0

20

25

30

35

40

45

C/N0 (dBHz)

Figure 11 Deflection Coefficient Performance as a function of C/N0 (Hardware Simulated GPS Data)

Figure 12 Normalized Correlation Output for PCND ) Based Acquisition Scheme (TCOH = 10 ms and GPCDD B d with TCOH = 1 ms and M =10) ( t n e i c i f f e o C

The acquisition performance was also evaluated using hardware simulated GPS IF data captured via Novatel front-end card. The IF down conversion and subsequent acquisition, based on the different detection algorithms, were implemented in MATLAB. Figure 11 shows the average deflection coefficient performance for various C/N0 levels in the case of different detection algorithms. The final deflection coefficient results were obtained by averaging the individual deflection coefficient pertaining to the different PRN satellites (8 in total) that were transmitted with equal power levels. Figure 12 shows the normalized correlation output for PCND and GPCDD based acquisition for a successful PRN code detection.

n o i t c e l f e D

60

50

40

30

20

10

0

2

4

6

8

10

12

45 dBHz 40 dBHz 35 dBHz 30 dBHz 25 dBHz 20 dBHz 14 16

M

Figure 13 Convergence of Deflection Coefficient for GPCDD (Hardware Simulated GPS Data) In Figure 12, the correlation output for PCND with 1 ms coherent integration and that of PCDD based acquisition was excluded due to the false code phase acquisition.

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Both the PCND with 10 ms coherent integration and GPCDD based acquisition utilized a total observation period of 600 ms. From Figure 12, the advantage of utilizing GPCDD based acquisition can readily be observed. Figure 13 shows the convergence of deflection coefficient for the GPCDD based acquisition using hardware simulated GPS data. As expected, the convergence plot in Figure 13 closely follows the trend in Figure 10, which was obtained through numerical simulations.

REFERENCES Bryant, R. (2005), Assisted GPS – Using Cellular Telephone Networks for GPS Anywhere, GPS World, pp. 40-46. Chansarkar M. M., and L. Garin. (2000), Acquisition of GPS signals at very low signal to noise ratio, Proceedings of ION NTM 2000 (Session E1, Anaheim, CA, January 26-28), The Institute of Navigation, pp. 731-737.

CONCLUSIONS In this paper, the generic GLRT detection processing was shown to established a number of subsequent detection schemes practically implementable in GPS receivers for signal acquisition. More specifically, the matched filter receiver is shown to be the optimal receiver when the signal parameters are known. In the presence of unknown signal parameters, the optimal structure is shown to be similar to that of estimatorcorrelator form. Consequently, a new detector structure denoted as the generalized post-correlation differential detector was introduced which was based on combining the post-correlation non-coherent and differential detectors. The major benefit of the developed detector structure stems from its enhanced noise suppression and the ability to allow for fine frequency estimation. Most importantly, adequate noise suppression is achievable without increasing the coherent integration period. The limitations of the proposed detector structure in terms of navigation data and frequency drift was analyzed. Finally, the achievable acquisition sensitivity improvements based on the proposed detection algorithm were demonstrated using both numerical simulations and hardware simulated GPS data. NOTE: A patent related to this new detection approach for GPS signal acquisition described herein has been submitted to the U.S. Patent and Trademark Office. ACKNOWLEDGEMENTS The authors acknowledge the Informatics Circle Of Research Excellence (iCORE) and the GEOIDE Networks of Centre of Excellence for their financial support. The authors would also like to thank Dr. M. L. Psiaki for providing a MATLAB based GPS L1 software simulator.

Coenen, A. J. R. M., and D. J. R.,Van Nee. (1992), Novel Fast GPS/GLONASS Code-Acquisition Technique Using Low Update Rate FFT, IEEE Electronic Letters, Vol. 28, No. 9, pp. 863-865, Apr. 1992. Elders-Boll, H., and U. Dettmar. (2004), Efficient differentially coherent code/Doppler acquisition of weak GPS signals, Proceedings of IEEE Symposium on Spread Spectrum Techniques and Applications, pp. 731735. Sep. 2004. Uijt de Haag M. and J. M. Kelly. (2004), Influence of Oscillator Variations on the Observed Frequency Behavior in Software-Based Navigation Receivers, IEEE Signal Processing Letters, Vol. 11, No. 3, pp. 390392, Mar. 2004. Kaplan, E. D., and C. J. Hegarty. (2006), Understanding GPS: Principles and Applications, 2nd Edition, Artech House Publishers, Boston, MA, 2006. Kay, S. M. (1993a), Fundamentals of Statistical Signal Processing: Detection Theory, Prentice-Hall, Englewood, NJ, USA. Kay, S. M. (1993b), Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice-Hall, Englewood, NJ, USA. Lin, D., and J. Tsui. (1998), Acquisition Schemes for Software GPS Receiver, Proceedings of ION GPS 2000, (Session D1, Nashville, TN, September 15-18), The Institute of Navigation, pp. 317-325. Lowe S. T. (1999), Voltage Signal-to-Noise Ratio (SNR) Nonlinearity Resulting From Incoherent Summations, Tech. Rep., The Telecommunications and Mission Operations Progress Report, TMO PR 42-137. Moreau, M., P. Axelrad, J. L. Garrison, D. Kelbel, and A. Long. (1999), GPS Receiver Architecture and Expected Performance for Autonomous Navigation in Highly Eccentric Orbits, Navigation, Journal of The

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Institute of Navigation, Vol. 47, No. 3, pp. 191-204, Fall 2000. O’ Driscoll, C. (2007), Performance Analysis of the Parallel Acquisition of Weak GPS Signals, Ph.D. Thesis, Department of Electrical and Electronic Engineering, National University of Ireland, Cork, Ireland. Park, S. H., H. Choi, S. J. Lee, and Y. B. Kim. (2002), A Novel GPS Initial Synchronization Scheme using Decomposed Differential Matched Filter, Proceedings of ION NTM 2002, (Session A3, San Diego, CA, January 28-30), The Institute of Navigation, pp. 246253.

Conference on Satellite Systems for Mobile Communications and Navigation, 17-19 October, pp. 232-238. Tsui, J. (2004), Fundamentals of Global Positioning System Receivers: A Software Receiver Approach, 2nd Edition, Wiley-Interscience, Hoboken, NJ, USA. Zarrabizadeh, M. H., and E. S. Sousa. (1997), A Differentially Coherent PN Code Acquisition Receiver for CDMA Systems, IEEE Transaction on Communication, Vol. 45, No. 11, pp. 145-1465, Nov. 1997.

Proakis, J. G. (2000), Digital Communications, 4th Edition, McGraw-Hill Science, NY, USA. Schmid, A., and A. Neubauer. (2004), Performance Evaluation of Differential Correlation for Single Shot Measurement Positioning, Proceedings of ION GNSS 2004, (Session B5, Long Beach, CA, September 21-24), The Institute of Navigation, pp. 1998-2009. Schmid, A., and A. Neubauer. (2005), Adaptive Phase Correction Loop for Enhanced Acquisition Sensitivity, Proceedings of ION GNSS 2005, (Session C1, Long Beach, CA, September 13-16), The Institute of Navigation, pp. 168-177. Shanmugam, S.K., J. Nielsen, G. Lachapelle, and R. Watson. (2005), Differential Signal Processing Schemes for Enhanced GPS Acquisition, Proceedings of ION GNSS 2005, (Session C1, Long Beach, CA, September 13-16), The Institute of Navigation, pp. 212221. Shanmugam, S.K., R. Watson, J. Nielsen, and G. Lachapelle. (2006), Pre-Correlation Noise and Interference Suppression for Use in Direct-Sequence Spread Spectrum Systems With Periodic PRN Codes, Proceedings of ION GNSS 2006, (Session C3, Fort Worth, TX, September 26-29), The Institute of Navigation, pp. 1297-1308. Shanmugam, S.K. (2007), New High-Sensitivity Techniques for GPS L1 C/A and Modernized Signal Acquisition, Ph.D. Thesis, Department of Geomatics Engineering, University of Calgary, Alberta, Canada. Taylor, R. E., and J. W. Sennott. (1984), Navigation System and Method, U.S. Patent (4,445,118). Tomlinson, M., and J. Bramwell. (1988), A digital signal processing modem receiver for the INMARSAT Standard C system, Proceedings of IEEE

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