Low-complexity code tracking loop with multipath diversity for GNSS over multipath fading channels

Author's Accepted Manuscript

Low-Complexity Code Tracking Loop with Multipath Diversity for GNSS over Multipath Fading Channels Yating Wu, Y.S. Zhu, S.H. Leung, W.K. Wong, Tao Wang

www.elsevier.com/locate/actaastro

PII: DOI: Reference:

S0094-5765(15)00232-5 http://dx.doi.org/10.1016/j.actaastro.2015.05.037 AA5465

To appear in:

Acta Astronautica

Received date: 17 December 2014 Revised date: 13 April 2015 Accepted date: 30 May 2015 Cite this article as: Yating Wu, Y.S. Zhu, S.H. Leung, W.K. Wong, Tao Wang, Low-Complexity Code Tracking Loop with Multipath Diversity for GNSS over Multipath Fading Channels, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2015.05.037 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Low-Complexity Code Tracking Loop with Multipath Diversity for GNSS over Multipath Fading Channels Yating Wu1 , Y. S. Zhu2 , S. H. Leung3 , W. K. Wong3 and Tao Wang1 1. Department of Communication Engineering, Shanghai University 2. Communication and Information Security Lab, Peking University 3. Department of Electronic Engineering, City University of Hong Kong

Abstract A low-complexity noncoherent code tracking loop is proposed for global navigation satellite systems (GNSS) over multipath fading channels. Instead of trying to remove the multipath interference from the received signal or utilizing complex signal processing techniques such as maximum-likelihood estimation of the delay parameters, the proposed scheme collects useful component from the multipath signal by introducing assistant signals into the local reference signals. Cross-correlation from neighboring paths is combined constructively and forms an enhanced effective discriminator characteristic. Multipath diversity is thereby achieved with low computational complexity. The S-curve of the new loop is shown to be bias-free and odd-symmetric. Mean square tracking errors obtained by both linear and nonlinear analyses are used to assess the loop’s tracking performance in terms of accuracy and robustness.

Keywords— Global Navigation Satellite System (GNSS), code tracking loop, multipath channels.

I. I NTRODUCTION Code tracking is an essential function of receivers for global navigation satellite systems (GNSS) such as Global Positioning System (GPS). Through maintaining synchronization between the local pseudorandom noise (PRN) spreading code and the incoming code contained in the received signal, delay lock loop (DLL) provides the estimate of time of arrival and the measurement of pseudo-range. Multipath is a major concern in designing code tracking loops since multipath interference distorts the discriminator curve of the loop and causes a significant performance degradation [1][2]. Over decades, substantial efforts have been devoted to enhancing the code tracking performance in the presence of multipath. Among the improved variants, a family of multipath estimating DLLs based on maximum-likelihood (ML) estimation principle is shown to approach theoretical performance limits [3]-[5]. The idea behind ML estimation is to search for the parameters (i.e., amplitude, delay, phase) of each multipath component and the number of multipath components that maximize the log-likelihood function, which is computationally expensive. To reduce the complexity of ML-based multipath estimating DLLs, an efficient implementation of the Newton iterative likelihood maximization is developed by exploiting the GNSS signal structure and the spreading code periodicity [5]. For closely spaced multipath scenarios, a class of the improved DLLs share the same underlying structure of narrowing the early-late spacing and main lobe of the cross-correlation function [6]-[9]. While in [10], the authors propose a new discriminator function which is insensitive to short multipath signals with delay less than one half the correlator spacing. The latest short multipath mitigation approach is proposed in [11], which incorporates a threshold-based peak detection method and attempts to compensate the multipath error contribution by performing a nonlinear curve fit on the input correlation function. This newly proposed technique achieves superior multipath mitigation performance but requires a good amount of memory and increases the receiver complexity considerably. While Spatial processing provides an alternative way of multipath mitigation [12][13], the hardware solution based on choke-ring antenna design and 2

directive antenna arrays has limitation in terms of size and cost. Rake structure has been combined with DLL to improve the code tracking performance for CDMA systems in multipath environments [14][15]. However, most of such receivers need the knowledge of channel state information to compensate the effect of adjacent path interference. In this paper, we propose a noncoherent code tracking loop with RAKE-like structure to exploit the inherent multipath diversity, but without channel estimation. Assistant signals are introduced into the local correlating signal for each path in order to make full use of the multipath components contained in the received signal. Without extra computation burden imposed, the multipath interference is directly added constructively to form an enhanced effective discriminator characteristic. The rest of the paper is organized as follows. Section II introduces the signal and channel model, and describes the proposed scheme. Key signals of the tracking loop are given along with the S-curve expression. In Section III, linear analysis is carried out to derive the analytical expressions of the mean square tracking error. Section IV shows the simulation results and verifies the analysis results. Finally, conclusions are drawn in Section V. II. S YSTEM

DESCRIPTION AND SIGNAL MODEL

A. Signal and Channel Model The multipath fading channel is assumed to be wide-sense stationary uncorrelated scattering (WSSUS). Using a Tc -spaced or chip-spaced tapped delay line model [16] with L + 1 paths, the low-pass equivalent impulse response of the channel at delay τ and time t can be expressed as hc (τ, t) =

L 

al (t)δ(τ − lTc )

(1)

l=0

where the time-varying complex tap gains al (t) have Rayleigh distributed magnitudes and uniformly distributed phases. The received signal of power P is then given by   L √ jωc t r(t) = 2P Re al (t)d(t − τ − lTc )c(t − τ − lTc )e + n(t) l=0

3

(2)

P I&D

c(t  Wˆ) PRN Code Generator

r (t )

NCO

Loop Filter

c(t  Wˆ  'Tc )  c(t  Wˆ  'Tc ) I&D E-L

Fig. 1.

Block diagram of the traditional dot-product DLL.

where d(t) =

∞

n=−∞ dn Pd (t

− nTd ) represents the navigation data signal, with d n and Pd (t)

denoting the data symbol and the unit amplitude nonreturn-to-zero (NRZ) pulse shape of duration  Td , respectively. c(t) = ∞ n=−∞ cn Pc (t − nTc ) is the pseudorandom modulation waveform with chip duration Tc , where cn is the nth chip of the spreading code sequence, and Pc (t) is the pulse shape such as filtered NRZ pulse and binary offset carrier (BOC) [17] pulse for modernized GPS and Galileo systems. In this paper, we consider the square root raised-cosine (SRRC) pulse, and the results can be extended to other shaping pulses such as the optimum pulse design proposed in [18]. The additive noise n(t) has the bandpass representation n(t) =

√

2[nc (t) cos(ωc t) − ns (t) sin(ωc t)]

(3)

where nc (t) and ns (t) are two statistically independent low-pass zero-mean Gaussian processes with double-sided power spectral density (PSD) of N0 /2 W/Hz. B. The Proposed Code Tracking Loop Fig. 1 depicts a traditional DLL architecture with noncoherent dot-product type discriminator for the generation of error signal. The prompt (P) and early-minus-late (E-L) correlator outputs are then filtered by the integrate-and-dump (I&D) filter and mixed to produce the error signal. 4

P

I&D

s (t  Wˆ) L o

I&D

s (t  Wˆ) 0 o

Tc

yoL (t , [ )

yo0 (t , [ )

Tc PRN Code Generator

r (t ) Tc

NCO

Tc

Loop Filter

s'0 (t  Wˆ)

y'0 (t , [ )

I&D

e0 (t , [ ) +

s (t  Wˆ) L '

y (t , [ ) L '

I&D

e(t , [ ) +

e L (t , [ )

E-L

Fig. 2.

The proposed code tracking loop.

To collect useful signals from the multipath interference, the proposed noncoherent code tracking loop shown in Fig. 2 applies a (L + 1)-finger RAKE-like structure to the traditional DLL, with each finger equivalent to a sub-DLL for each path. The upper and lower part of Fig. 2 consist of L + 1 prompt and E-L correlators, respectively. Since a good part of the DLL structure is retained, the new loop is easy to implement. Let τˆ denote the estimated delay and define ξ = (τ − τˆ)/Tc as the normalized tracking error. The overall error control signal e(t, ξ) is obtained by combining all the individual error signals e l (t, ξ) (l = 0, 1, . . . , L) from the subDLLs, which are the products of the correlator outputs from the lth prompt and E-L branches, l (t, ξ). The time-independent average response of the error signal e(t, ξ) gives i.e., yol (t, ξ) and yΔ

the loop’s discriminator characteristic QS (ξ), which is also called S-curve due to its S-like shape when plotted versus the tracking error ξ. QS (ξ) is a function of ξ, while in turn it drives NCO which subsequently corrects ξ. 5

For the lth sub-DLL, the received signal is cross-correlated with the on-time signal s lo (t− τˆ) on the prompt branch and the early-late difference signal slΔ (t − τˆ) on the E-L branch, respectively. The trick of the new loop lies in these two local signals which are given in detail by

slo (t − τˆ) = c(t − τˆ − lTc )

(4a)

slΔ (t − τˆ) = c(t − τˆ − lTc − ΔTc ) − c(t − τˆ − lTc + ΔTc ) +c(t − τˆ − lTc + Tc + ΔTc ) − c(t − τˆ − lTc − Tc − ΔTc )

(4b)

where Δ is the early-late spacing and is set as 1/2 in the sequel. It is important to point out that in (4b), besides the usual early-minus-late reference signal of the lth path used in the conventional DLL, i.e., c(t − τˆ − lT c − ΔTc ) − c(t − τˆ − lTc + ΔTc ), we introduce another two terms which are the early and late signals of the neighboring (l − 1)th and (l + 1)th paths, respectively. These two assistant signals are used to generate additional residual cross-correlation from adjacent paths that can be exploited to enhance the S-curve, as we shall see later in this section. Instead of being eliminated from the individual discriminator of each path, the multipath interference is added constructively to produce an overall enhanced S-curve. Let hID (t) denote the impulse response of the I&D filter and h(t) denote its baseband Δ

equivalent. Define the code correlation function as Rc (ξ) = E[c(t)c(t+ξTc )], where E[·] denotes the expectation operator. In most cases of practical interest, where the loop bandwidth is much less than the code chip rate, the effect of code self-noise [c(t)c(t + ξTc ) − Rc (ξ)] on loop performance can be neglected [19]. Therefore, with ⊗ referring to the convolution operator, the prompt correlator output of the lth sub-DLL is given by yol (t, ξ)=[r(t)slo (t − τˆ)] ⊗ hID (t) L   √ am (t)d¯m (t − τ )Rc (ξ + m − l)ejωc t = 2P Re +

√

m=1

2[¯ nlco (t) cos(ωc t)

−n ¯ lso (t) sin(ωc t)]

6

(5)

where Δ d¯m (t − τ ) = d(t − τ − mTc ) ⊗ h(t)

n ¯ lxo (t) = [nx (t)slo (t − τˆ)] ⊗ h(t)

x = c, s

(6) (7)

and Rc (ξ) = Pc (ξ) ⊗ Pc (ξ) is the raised-cosine pulse shape with α denoting the roll-off factor, i.e., Rc (ξ) =

sin(πξ) cos(παξ) πξ 1 − 4α2 ξ 2

(8)

In view of the fact that Rc (ξ ± k)  0 for |ξ| ≤ 1 and integer k ≥ 2, we can rewrite (5) as yol (t, ξ) =

√

2P Re{ul (t, ξ)ejωct } +

√

2[¯ nlco (t) cos(ωc t) − n ¯ lso (t) sin(ωc t)]

(9)

where Δ ul (t, ξ) = al d¯l Rc (ξ) + al+1 d¯l+1 Rc (ξ + 1) + al−1 d¯l−1 Rc (ξ − 1)

(10)

In the above equation, the time variable associated with a l and d¯l has been dropped for notational simplicity. In the same manner, the lth E-L correlator output can be obtained as l (t, ξ) = [r(t)slΔ (t − τˆ)] ⊗ hID (t) yΔ

=

√

 2P Re [v l (t, ξ) + w l (t, ξ)]ejωc t

√ l + 2[¯ ncΔ (t) cos(ωc t) − n ¯ lsΔ (t) sin(ωc t)]

(11)

where



 Δ v l (t, ξ) = al d¯l Rc (ξ − Δ) − Rc (ξ + Δ) + al+1 d¯l+1 Rc (ξ + Δ) − al−1 d¯l−1 Rc (ξ − Δ)

(12)

Δ

w l (t, ξ) = al−1 d¯l−1 Rc (ξ +Δ)−al+1 d¯l+1 Rc (ξ −Δ)+al−2 d¯l−2 Rc (ξ −Δ)−al+2 d¯l+2 Rc (ξ +Δ) (13) n ¯ lxΔ (t) = [nx (t)slΔ (t − τˆ)] ⊗ h(t)

7

x = c, s

(14)

The error signal of the lth sub-DLL is produced by mixing the correlator outputs y ol (t, ξ) and l (t, ξ). Combining (9) and (11) and ignoring the double frequency terms, we have yΔ l (t, ξ) el (t, ξ) = yol (t, ξ)yΔ

= P Re{z1l (t, ξ) + z2l (t, ξ)} + nl1 (t) + nl2 (t, ξ) + nl3 (t, ξ)

(15)

where

nl2 (t, ξ) = nl3 (t, ξ) =

√



∗ z1l (t, ξ) = ul (t, ξ) v l (t, ξ)

(16)



∗ z2l (t, ξ) = ul (t, ξ) w l (t, ξ)

(17)

nl1 (t) = n ¯ lco (t)¯ nlcΔ (t) + n ¯ lso (t)¯ nlsΔ (t)

(18)

√

P Re{ul (t, ξ)}¯ nlcΔ(t) +

P Re{v l (t, ξ) + w l (t, ξ)}¯ nlco(t) +

√ √

P Im{ul (t, ξ)}¯ nlsΔ (t)

(19)

P Im{v l (t, ξ) + w l (t, ξ)}¯ nlso(t)

(20)

[z1l (t, ξ) + z2l (t, ξ)], nl1 (t) and [nl2 (t, ξ) + nl3 (t, ξ)] correspond to the signal × signal, noise × noise and signal × noise components of the product of the two correlator outputs, respectively. Substituting (10) and (12) into (16), we have

 z1l (t, ξ) = |al |2 d¯2l Rc (ξ) Rc (ξ − Δ) − Rc (ξ + Δ) + |al+1 |2 d¯2l+1 Rc (ξ + 1)Rc (ξ + Δ) − |al−1 |2 d¯2l−1 Rc (ξ − 1)Rc (ξ − Δ) +al a∗l+1 d¯l d¯l+1 Rc (ξ)Rc (ξ + Δ) − al a∗l−1 d¯l d¯l−1 Rc (ξ)Rc (ξ − Δ)

 +a∗l al+1 d¯l d¯l+1 Rc (ξ + 1) Rc (ξ − Δ)−Rc (ξ + Δ)

 +a∗l al−1 d¯l d¯l−1 Rc (ξ − 1) Rc (ξ − Δ)−Rc (ξ + Δ) +a∗l−1 al+1 d¯l−1 d¯l+1 Rc (ξ − 1)Rc (ξ + Δ)−al−1 a∗l+1 d¯l−1 d¯l+1 Rc (ξ + 1)Rc (ξ − Δ)(21) Define the following correlation coefficients of the multipath amplitudes Δ

A0 =

 ∀l

8

|al |2

(22a)

Δ

A1 =

 ∀l

Δ

A2 =



Δ

A3 =

Re{al a∗l+2 } =



Re{al−1 a∗l+2 } =

Re{al−1 a∗l+1 }

(22c)

E[d¯2l (t



Re{al+1 a∗l−2 }

(22d)

∀l

and denote D=

(22b)

∀l

∀l

Δ

Re{al a∗l−1 }

∀l

∀l





Re{al a∗l+1 } =

 − τ )] =

∞

−∞

Sd (f ) |H(f )|2 df

(22e)

where Sd (f ) is the PSD of the data signal and H(f ) is the baseband-equivalent transfer function of the I&D filter. In practical situations, the I&D filter bandwidth is much less than the chip rate 1/Tc . Hence, Δ

Rd¯(ε) = E[d¯l (t − τ )d¯k (t + ε − τ )] 



∞ −∞

Sd (f ) |H(f )|2 ej2πf ε df

(23)

Ignoring the modulation self-noise [19], we can make the approximation d¯l (t − τ )d¯k (t − τ )  Rd¯(0) = D. Then we have 

Re{z1l (t, ξ)} = (A0 − A1 )DS1 (ξ) + (A2 − A1 )DS2 (ξ)

(24)

∀l

where S1 (ξ) = Rc (ξ)[Rc (ξ − Δ) − Rc (ξ + Δ)] +Rc (ξ + 1)Rc (ξ + Δ)−Rc (ξ − 1)Rc (ξ − Δ) S2 (ξ) = Rc (ξ − 1)Rc (ξ + Δ) − Rc (ξ + 1)Rc (ξ − Δ)

(25) (26)

In a similar way, we obtain 

Re{z2l (t, ξ)} = (A0 − A3 )DS2 (ξ) + (A2 − A1 )DS1 (ξ)

(27)

∀l

which is brought by the extra early-minus-late signals of the neighboring paths we introduced in the second line of (4b).

9

S1(ξ)

0.6

S2(ξ) 0.4

0.2

0

−0.2

−0.4

−0.6 −3

Fig. 3.

−2

−1 0 1 Normalized Tracking Error ξ

2

3

The component curves of the S-curve for SRRC pulse with α=0.22.

The overall error signal e(t, ξ) is the sum of all the individual error signals e l (t, ξ). Define the total noise term as ntotal (t, ξ) =

L 

[nl1 (t) + nl2 (t, ξ) + nl3 (t, ξ)]

(28)

l=0

Then e(t, ξ) can be expressed as e(t, ξ) =

L 

el (t, ξ) = DP QS (ξ) + ntotal (t, ξ)

(29)

l=0

where QS (ξ) = (A0 − 2A1 + A2 )S1 (ξ) + (A0 − A1 + A2 − A3 )S2 (ξ)

(30)

is the normalized discriminator characteristic or S-curve. S1 (ξ) and S2 (ξ) are the constituent parts of the final S-curve. They are plotted in Fig. 3 for SRRC with roll-off factor 0.22 and in Fig. 4 for BPSK (NRZ pulse), BOC(n, n) and composite binary offset carrier CBOC(6,1,1/11) modulations, respectively. The curves of BPSK and

10

0.2 0.15 0.1 0.05 0 −0.05 S1(ξ) BPSK, BOC(n,n)

−0.1

S2(ξ) BPSK, BOC(n,n) S1(ξ) CBOC(6,1,1/11)

−0.15

S2(ξ) CBOC(6,1,1/11) −0.2 −0.5

Fig. 4.

0

0.5

The component curves of the S-curve for BPSK, BOC(n,n) and CBOC(6,1,1/11).

BOC(n, n) coincide with each other. It can be seen from both figures that S2 (ξ), although yielded by residual cross-correlation from adjacent paths (i.e., the multipath interference), can add constructively to the S-curve. Note that although we focus on the SRRC pulse for analysis convenience, the analysis is applicable to a variety of shaping pulse. The loop is locked when the error signal driving NCO averages zero, i.e., at the zero-crossing with positive slope on the S-curve. The ideal lock point is at the origin where the tracking error equals zero. As shown in Fig. 3 and Fig. 4, both components of the S-curve have odd-symmetrical shape and have stable lock point at the origin. Therefore, regardless of the cross-correlation among the channel tap weights, the new tracking loop always holds an odd-symmetrical biasfree S-curve. For the cases where the channel tap gains are uncorrelated, the S-curve can be reduced to QS (ξ) = A0 S1 (ξ) + A0 S2 (ξ)

11

(31)

al 1















r (t )c(t  Wˆ  lTc  Tc  'Tc ) al 1 Rc ([  1)







al 1 Rc ([  1)









r (t ) s (t  Wˆ) l o

al 1













½ °u ° ¾ ° °¿ ½ ° °u ¾ ° °¿

2

al 1 Rc ([  1) Rc ([  ') 









½ ° ° ¾ ° °¿

2

A0 S2 ([ )

, ¦ l











al 1 Rc ([  1) Rc ([  ') 











r (t )c(t  Wˆ  lTc  Tc  'Tc )

Fig. 5.

Illustration of how the assistant E-L signal [c(t − τˆ − lTc + Tc + ΔTc ) − c(t − τˆ − lTc − Tc − ΔTc )] correlates with

the received signal and contributes to the S-curve.

Note that A0 S1 (ξ) represents the S-curve component brought by the E-L signal of the lth path shown in the first line of (4b), while A0 S2 (ξ) represents the component of the S-curve attributed to the assistant E-L signals of the adjacent (l − 1)th and (l + 1)th paths we introduced in the second line of (4b). Fig. 5 illustrates how the assistant signals correlating with the received signal on the E-L branch turn the neighboring multipath interference into an enhancement of the S-curve. As shown in Fig. 5, the early and late correlator outputs are the left-shifted and right-shifted version of the prompt correlator output r(t)s lo (t − τˆ) by (1 + Δ)Tc , respectively. For simplicity, we omit the correlation components associated with a l±2 and terms associated with ai aj (i = j, illustrated by different colors) in the mixing of the correlator outputs. Eventually, 12

the additional residual cross-correlations from adjacent paths form a positive contribution to the S-curve. III. L INEAR A NALYSIS For moderate to high SNR, linear analysis characterizes the loop performance quite well since the tracking error is kept very small and the loop is working within the linear region of the Scurve. Under this condition, QS (ξ) can be conveniently approximated as its linear equivalent Δ

QS (0)ξ, where QS (0) = dQS (ξ)/dξ|ξ=0 is the slope of the S-curve at the origin and can be obtained from (8) and (30) as QS (0) = (A0 − 2A1 + A2 )

 4α sin(πα/2)

+

1 − α2 4 cos(πα/2) cos(πα) +(A1 − A3 ) π(1 − α2 )(1 − 4α2)

8 cos(πα/2)(1 − 3α2 )  π(1 − α2 )2 (32)

Assuming a flat PSD for the total noise within the loop bandwidth, the mean square tracking error can be shown to be [22], [27] σξ2 = where Sntotal (f, ξ) =

∞ −∞

2Sntotal (0, 0)BL [DP QS (0)]2

(33)

Rntotal (ε, ξ)e−j2πf εdε and Rntotal (ε, ξ) = E[ntotal (t, ξ)ntotal (t + ε, ξ)]

is the PSD and the autocorrelation function of the total noise n total (t, ξ), respectively. BL = ∞ |Hc (f )|2 df is the loop bandwidth and H c (f ) is the closed-loop transfer function of the 0 linearized model. From the Appendix, it is shown that  ∞ Rntotal (ε, 0)dε Sntotal (0, 0) = −∞

 = (L + 1) 2 − Rc2 (Δ) N02



∞

−∞

|H(f )|4 df

 + 2A0 − 3A1 + 2A2 − A3 + (A4 − A0 )Rc2 (Δ) P N0

13



∞ −∞

Sd (f ) |H(f )|4 df (34)

To give a clear representation of the tracking performance in terms of fundamental system parameters, we introduce two constants 

Δ

∞

|H(f )|4 df C1 = Tb −∞  ∞ Δ C2 = Sd (f ) |H(f )|4 df

(35a) (35b)

−∞

Δ

Also, we define the bit signal-to-noise ratio (SNR) in data bandwidth as γ d = P Tb /N0 and the Δ

ratio of the data rate over the loop bandwidth as ζ 0 = 1/BL Tb . σξ2 is finally reduced to

σξ2

2 = [DQS (0)]2 ζ0 γd



 (L + 1) 2 − Rc2 (Δ) C1 γd

 + 2A0 − 3A1 + 2A2 − A3 + (A4 − A0 )Rc2 (Δ) C2 IV.

 (36)

NUMERICAL RESULTS

In this section, the performance of the proposed tracking loop over multipath fading channels is presented. Specifically, we are concerned with the S-curve and root-mean-square (rms) tracking error performances. The S-curve provides a quick measure of the loop’s performance. Important features such as the lock point and tracking range can be seen directly from the S-curve. The overall tracking range is the region where the detector produces a usable control signal to drive the VCO through the loop filter [21]. On the S-curve, the overall tracking range is explicitly given by the distance between the two zero-crossing points located on the left and right sides of the lock point. Outside the overall tracking range, the loop would be considered “out of lock” and code reacquisition has to be triggered. Therefore, the overall tracking range is an important indicator of the loop’s pull-in capability. RMS tracking error and mean-time-to-lose-lock (MTLL) are generally used to quantify the accuracy and robustness of the tracking loop, respectively. However, since the MTLL is usually very large for SNRs of practical interest, only the rms tracking error will be considered. msequence of length 127 is employed as the PRN code. The roll-off factor α of the raised-cosine 14

1 0.8 0.6

Case 1 with assistant signal Case 1 without assistant signal Case 2 with assistant signal Case 2 without assistant signal

0.4

S−curve

0.2 0 −0.2 −0.4 −0.6 −0.8 −1

Fig. 6.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 Normalized Tracking Error ξ

0.3

0.4

0.5

Two illustrative examples of the S-curve enhancement brought by the assistant neighboring E-L signals.

pulse shape is set as 0.22 and the bit rate to loop bandwidth ratio ζ 0 is 100 unless otherwise specified. We assume binary NRZ data modulation and ideal I&D filters with cutoff frequency of 1/Tb . Under the above conditions, it can be shown that D = 0.902, C 1 = 2, and C2 = 0.902. A. S-curves Fig. 6 shows example S-curves before and after adding the assistant E-L signals introduced in (4b), respectively. For case 1, three consecutive paths are assumed with complex gain a 0 = √ √ 1, a1 = 0.5ejπ/4 , and a2 = 0.1e−jπ/6 , respectively. For case 2, four paths are assumed with √ √ √ a0 = ejπ/3 , a1 = 0.5ejπ/6 , a2 = 2e−jπ/4 , and a3 = 0.1ejπ/5 . With the help of the assistant signals, evident enhancement of the S-curve is achieved and extension of the overall tracking range is also exhibited. For example, for case 2 with assistant signals, the overall tracking range is extended from [−0.44Tc , 0.44Tc ] to [−0.58Tc , 0.58Tc ] (a 32% increase) as compared to the case without assistant signals. For a given tracking offset ξ, the proposed scheme has a higher error signal voltage driving the NCO, which indicates that the loop will respond more quickly. 15

Fig. 7. S-curves of the proposed tracking loop over multipath fading channels with five independent propagation paths (L = 4) and a maximum Doppler frequency of 83 Hz.

More examples of the S-curves with the proposed tracking loop over multipath fading channels are plotted in Fig. 7. The results are obtained by 100 independent simulation runs. The S-curve of the traditional DLL is not given here, since it has been extensively reported as randomly biased and distorted by the multipath interference in the literature [23]-[27]. The simulated Rayleigh fading channel is composed of five successive paths with equal power, and each path is produced by an independent Jakes’ fading model with maximum Doppler shift of 83 Hz. It can be seen that the proposed scheme always has odd-symmetric bias-free S-curves with a static lock point at the origin ξ = 0 and a minimum tracking range of [−T c /2, Tc /2]. As shown, the new loop achieves a much more robust pull-in capability with zero tracking bias under frequency-selective fading channels. B. RMS (Root-Mean-Square) Tracking Error Fig. 8 shows a performance comparison of the proposed scheme with traditional DLL in terms of rms tracking error obtained by nonlinear analysis. The upper bound performance of multipath 16

0

10

RMS Tracking Error,σξ

Traditional DLL Noncoherent combination Proposed scheme MEDLL lower bound

−1

10

−2

10

Fig. 8.

−5

0

5 SNR, γd (dB)

10

15

Performance comparison of the proposed scheme with traditional DLL and MEDLL in terms of rms tracking error. √ 0.5ej2π/3 .

a0 = 1 and a1 =

estimating DLL (MEDLL) [3][4] with perfect multipath estimation and only noise being the limiting factor is also given for comparison. The nonlinear analysis follows the procedures outlined in [29] and [30], where the high SNR assumption is removed by using the stochastic differential equation to find the probability density function of the tracking error. The mean square tracking error is then obtained via numerical integration. Since traditional DLL behaves unstable and its performance varies greatly under different multipath scenarios, we only present √ a single case where two paths are assumed with a0 = 1 and a1 = 0.5ej2π/3 . As an illustrative example, it is clear from Fig. 8 that the new scheme outperforms traditional DLL significantly under multipath fading and its performance is comparable to the upper bound performance of MEDLL. The performance enhancement of the proposed scheme is attributed to both the effects of the noncoherent combination of the L + 1 paths and the introduction of additional cross correlation term in the discriminator. The performance with noncoherent

17

0

10

RMS Tracking Error,σξ

Linear Analysis Nonlinear Analysis

ξ =20 0

−1

10

ξ0=100 ξ0=500

−2

10

−9

Fig. 9.

−5

0

5 SNR, γd (dB)

10

15

Linear and nonlinear analyses results for the proposed scheme on rms tracking error versus γd for different ratio of

the bit-rate to loop bandwidth. L = 2 and α = 0.22.

combination alone, i.e., without assistant signal, is also given in Fig. 8. As we can see from the figure, apart from the non-coherent combination, the assistant E-L signals play a critical role in reducing the rms tracking error. An irreducible error floor can be observed for traditional DLL due to the presence of multipath signals. The multipath interference remains a dominant error source for the traditional DLL, and will degrade the system performance. In contrast, the rms tracking error of the proposed loop converges towards zero as the noise vanishes. In the following simulations, we present the rms tracking error performance of the new loop averaging over 2,000 simulation runs. Three-path Rayleigh fading channels are assumed unless otherwise specified, with each path simulated by Jakes’ model with maximum Doppler shift of 83 Hz. Fig. 9 provides the linear and nonlinear analyses results of the rms tracking error with the proposed tracking loop. The rms tracking error is plotted versus SNR γ d for different values of data rate over loop bandwidth ratio ζ0 . As expected, for SNRs below certain threshold values 18

0

10

RMS Tracking Error,σξ

α=0.1 α=0.5 α=0.9

−1

10

L=1 L=4 −2

10

−3

10

Fig. 10.

−9

−5

0

5 SNR, γd (dB)

10

15

RMS tracking error performance of the proposed scheme for different values of α and L.

which drop as ζ0 increases, the linear results deviate from nonlinear ones because the assumption that the loop is working within the linear region of the S-curve no longer holds under low SNR. For SNRs beyond the threshold, the linear results coincide very well with the nonlinear ones and the simpler linear analysis can be used to evaluate the rms tracking error performance of the loop conveniently. Fig. 10 shows the average rms tracking error of the loop for different values of α and L, respectively. The results are obtained by nonlinear analyses. It can be seen from both figures that the roll-off factor has a trivial effect on the loop’s performance. The performance is slightly better for higher α values, due to the increased S-curve slope as a result of a sharper pulse shape. It is also worthy of note that the loop performance improves as the number of paths increase, since the proposed scheme achieves multipath diversity and is able to collect energy from the multipath signals.

19

V. C ONCLUSION In this paper, we have proposed a low-complexity code tracking loop with multipath diversity and enhanced discriminator characteristic for GNSS systems over frequency-selective fading channels. The new loop contains a number of sub-DLLs for retrieving useful signal from each multipath component. Each sub-DLL exhibits almost the same structure of the conventional DLL, except that more early and late signals of the neighboring paths are added in the local reference signals. The resultant residual cross-correlation from adjacent paths is added constructively to form an enhanced bias-free S-curve. Simulation results show that the new loop achieves multipath diversity without paying extra computational complexity. It outperforms the traditional scheme greatly in terms of S-curve and RMS tracking error. ACKNOWLEDGMENT This work was supported by research grants from the National Natural Science Foundation of P. R. China (No. 61201221) and Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20123108120017). A PPENDIX DERIVATION OF

(34)

To obtain the analytical expression of S ntotal (0, 0), we start by deriving Rntotal (ε, 0) first. Assuming random zero-mean data signals, it is easy to show that n l1 (t) are uncorrelated with both nk2 (t, ξ) and nk3 (t, ξ) (∀ l, k). Hence we have L  L 

Rntotal (ε, 0) = E nl1 (t)nk1 (t + ε) + nl2 (t, 0)nk2 (t + ε, 0) l=0 k=0 +nl3 (t, 0)nk3 (t

+ ε, 0) + 2nl2 (t, 0)nk3 (t + ε, 0)]

(A.1)

We proceed to derive each term of (A.1) individually. Since nc (t) and ns (t) are mutually ¯ ksx (t) (x = o, Δ, ∀ l, k) are independent and statistically identical, it follows that n ¯ lcx (t) and n independent and statistically identical. Therefore, from (18), we have nlco (t)¯ nlcΔ (t)¯ nkco (t + ε)¯ nkcΔ (t + ε)] + 2E 2 [¯ nlco (t)¯ nlcΔ (t)] E[nl1 (t)nk1 (t + ε)] = 2E[¯ 20

(A.2)

Note that for zero-mean jointly Gaussian random variables x i , i = 1, 2, 3, 4 [31] E[x1 x2 x3 x4 ] = E[x1 x2 ]E[x3 x4 ] + E[x1 x3 ]E[x2 x4 ] + E[x1 x4 ]E[x2 x3 ]

(A.3)

Hence, nlcΔ (t)¯ nkco (t + ε)¯ nkcΔ (t + ε)] = E 2 [¯ nlco (t)¯ nlcΔ (t)] E[¯ nlco (t)¯ nkco (t + ε)]E[¯ nlcΔ (t)¯ nkcΔ (t + ε)] +E[¯ nlco (t)¯ nkcΔ (t + ε)]E[¯ nlcΔ (t)¯ nkco (t + ε)] (A.4) +E[¯ nlco (t)¯ From (7) and (14), we have nkcΔ (t E[¯ nlco (t)¯

N0 E[slo (t)skΔ (t)] + ε)] = 2



∞

−∞

|H(f )|2 ej2πf ε df

(A.5)

Moreover, from (4), E[slo (t)skΔ (t)] = Rc (l − k − Δ) − Rc (l − k + Δ) +Rc (l − k + 1 + Δ) − Rc (l − k − 1 − Δ) ⎧ ⎪ ⎪ R (Δ), k = l + 2, ⎪ ⎨ c = −Rc (Δ), k = l − 2, ⎪ ⎪ ⎪ ⎩ 0, otherwise. Hence,

⎧ ⎨ ± N0 R (Δ)  ∞ |H(f )|2 ej2πf ε df, k = l ± 2, c −∞ 2 E[¯ nlco (t)¯ nkcΔ (t + ε)] = ⎩ 0, otherwise.

Similarly, using E[slo (t)sko (t)] = Rc (l − k), we have ⎧ ⎨ N0  ∞ |H(f )|2 ej2πf ε df, k = l, 2 −∞ nkco (t + ε)] = E[¯ nlco (t)¯ ⎩ 0, otherwise.

(A.6)

(A.7)

(A.8)

Further, we obtain

⎧ ∞ ⎪ 2N |H(f )|2 ej2πf ε df, ⎪ 0 −∞ ⎪ ⎪ ∞ ⎪ 2 j2πf ε ⎪ 3 ⎪ N df, − 0 ⎪ ⎨ 2  −∞ |H(f )| e ∞ E[¯ nlcΔ (t)¯ nkcΔ (t + ε)] = N0 −∞ |H(f )|2 ej2πf ε df, ⎪ ⎪ ∞ ⎪ ⎪ − N20 −∞ |H(f )|2 ej2πf ε df, ⎪ ⎪ ⎪ ⎪ ⎩ 0, 21

k = l, k = l ± 1, k = l ± 2, k = l ± 3, otherwise.

(A.9)

Substituting (A.4) and (A.9) to (A.2) results in L  L 

E[nl1 (t)nk1 (t



+ ε)] = (L + 1) 2 −

Rc2 (Δ)

 2 N0

l=0 k=0



∞ −∞

2 j2πf ε

|H(f )| e

2 df

(A.10)

Also, from (19), it can be shown that 



∗  nkcΔ (t + ε)] E[¯ nlcΔ (t)¯ E[nl2 (t, 0)nk2 (t + ε, 0)] = P Re E ul (t, 0) uk (t + ε, 0)

(A.11)

Combining (10) with (23) gives

∗ E[ul (t, 0) uk (t + ε, 0) ] = al a∗k Rd¯(ε)

(A.12)

Substitute (A.9) and (A.12) into (A.11), yielding L  L 

 E[nl2 (t, 0)nk2 (t+ε, 0)]

= (2A0 −3A1 +2A2 −A3 )P Rd¯(ε)N0

l=0 k=0

∞ −∞

|H(f )|2 ej2πf ε df (A.13)

Following a similar procedure, we have L  L 

E[nl3 (t, 0)nk3 (t

+ ε, 0)] = (A0 −

A4 )Rc2 (Δ)P Rd¯(ε)N0

+ ε, 0)] = (A4 −

A0 )Rc2 (Δ)P Rd¯(ε)N0

l=0 k=0



∞

−∞

|H(f )|2 ej2πf ε df

(A.14)

|H(f )|2 ej2πf ε df

(A.15)

and L  L 

E[nl2 (t, 0)nk3 (t

l=0 k=0



∞

−∞

Finally, substituting (A.10), (A.13)-(A.15) to (A.1) gives the result  ∞ Rntotal (ε, 0)dε Sntotal (0, 0) = −∞ 

 2 ∞ 2 = (L + 1) 2 − Rc (Δ) N0 |H(f )|4 df −∞

 ∞

 2 + 2A0 −3A1 +2A2 −A3 +(A4 −A0 )Rc (Δ) P N0 Sd (f ) |H(f )|4 df (A.16) −∞

R EFERENCES [1] R. D. J. Van Nee, “Spread-spectrum code and carrier synchronization errors caused by multipath and interference,” IEEE Trans. Aero. Electron. Systems, vol. 29, pp. 1359-1365, Oct. 1993. [2] S. K. Kalyanaraman, M.S. Braasch and J. M. Kelly, “Code tracking architecture influece on GPS carrier multipath”, IEEE Trans. Aero. Electron. Systems, vol. 42, no. 2, pp. 548-561, Apr. 2006. [3] R. D. J. Van Nee, J. Siereveld, P. C. Fenton, and B. R. Townsend, “The multipath estimating delay lock loop: Approaching theoretical accuracy limits,” in Proc. IEEE Position, Location Navigation Symp., vol. 1, pp. 246-251, Apr. 1994.

22

[4] M. Sahmoudi and M. G. Amin, “Fast iterative maximum-likelihood algorithm (fimla) for multipath mitigation in next generation of GNSS receivers,” in Proc. 40th Asilomar Conf. Signals, Syst. Comput., 2006, pp. 579-584. [5] M. Sahmoudi, and M. G. Amin, “Fast Iterative Maximum-Likelihood Algorithm (FIMLA) for Multipath Mitigation in Next Generation of GNSS Receivers,” IEEE Trans. Wireless Communication, vol. 7, no. 11, pp. 4362-4374, Nov. 2008. [6] A. J. Van Dierendonck, P. Fenton, and T. J. Ford, “Theory and performance of narrow correlator spacing in a GPS receiver,” Journal of the Institute of Navigation, vol. 39, no. 3, pp. 265-283, 1992. [7] G. A. McGraw and M. Braasch, “GNSS multipath mitigation using high resolution correlator concepts,” in Proc. National Technical Meeting of the Institute of Navigation (IONNTM’99), pp. 333-342, San Diego, Jan. 1999. [8] G. A. McGraw, “Practical GPS carrier phase multipath mitigation using high resolution correlator techniques,” in Proc. IAIN World Congress and the 56th Annual Meeting of the Institute of Navigation, pp. 373-381, San Diego, Jun. 2000. [9] P. Duraisamy, W. M. Jang, L. Nguyen, “Multipath error mitigation using self-encoded spread spectrum for navigation”, IET Radar Sonar Navig., vol. 7, no. 4, pp. 433-442, 2013. [10] N. Jardak, N. Samama, “Short Multipath Insensitive Code Loop Discriminator”, IEEE Trans. Aero. Electron. Systems, vol. 46, no. 1, pp. 278-295, Jan. 2010. [11] M. Z. H. Bhuiyan, E. S. Lohan, “Advanced Multipath Mitigation Techniques for Satellite-Based Positioning Applications”, International Journal of Navigation and Observation, vol. 2010, Article ID 412393, 2010. [12] J. K. Ray, M. E. Cabbib and P. Fenton, “GPS Code and Carrier Multipath Mitigation Using a Multiantenna System,” IEEE Trans. Aero. Electron. Systems, vol. 46, no. 3, pp. 1171-1184, Jul. 2010. [13] M. T. Brenneman, Y. T. Morton and Q. Zhou, “GPS Multipath Detection with ANOVA for Adaptive Arrays,” IEEE Trans. Aero. Electron. Systems, vol. 37, no. 1, pp. 183-195, Jan. 2001. [14] G. Fock, J. Baltersee, P. Schulz-Rittich, and H. Meyr, “Channel Tracking for Rake Receivers in Closely Spaced Multipath Environments”, IEEE JSAC, vol. 19, no. 12, pp. 2420-2431, Dec. 2001. [15] X. Sun, S. Zhang, Q. Hu, J. Zhang, and Y. Jiang, “Modified Rake architecture for GNSS receiver in a multipath environment”, Acta Astronautica, vol. 79, pp. 1-11, Oct. 2012. [16] J. G. Proakis, Digital Communications, 4th Edition. New York: McGraw-Hill, 2001. [17] J. W. Betz, “Binary offset carrier modulations for radion navigation,” Journal of the Institute of Navigation, vol. 48, no. 4, pp. 227-246, 2002. [18] F. Zanier, G. Bacci, and M. Luise, “Criteria to improve time-delay estimation of spread spectrum signals in satellite positioning,” IEEE J. Sel. Topics Signal Process., vol. 3, no. 5, pp. 748-763, Oct. 2009. [19] M. K. Simon, J.K. Omura, R.A. Scholtz, and B.K. Levitt, Spread Spectrum Communications Handbook, Electronic Edition, McGraw-Hill, 2002. [20] J. J. Spliker, “Delay lock tracking of stochastic signals,” IEEE Trans. Space. Electron. Telemet., vol. SET-9, pp. 1-8, Mar. 1963. [21] A. Wilde, “Extended tracking range delay-locked loop” , in Proc. IEEE ICC’95, pp. 1051-1054, Seattle, Jun. 1995. [22] A. Polydros and C. L. Weber, “Analysis and optimization of correlative code-tracking loops in spread-spectrum systems,” IEEE Trans. Commun., vol. COM-33, pp. 331-339, Jan. 1985. [23] R. L. Bogusch, F. W. Guigliano, D. L. Knepp, and A. H. Michelet, “Frequency selective propagation effects on spreadspectrum receiver tracking, Proc. IEEE, vol. 69, pp. 787-796, Jul. 1981. [24] W.-H. Sheen and G L St¨uber, “Effects of multipath fading on delay-locked loops for spread spectrum systems,” IEEE Trans. Commun., vol 44, pp. 1947-1956, Feb/Mar/Apr. 1994.

23

[25] J.-C. Lin, “A modified PN code tracking loop for direct-sequence spread-spectrum communication over arbitrarily correlated multipath fading channels,” IEEE Journ. Select. Area. Commun., vol. 19, no. 12, pp. 2381-2395, Dec. 2001. [26] Kamran Kiasaleh, “Channel-aided, decision-directed delay-locked loop for pilot symbol aided CDMA communications impaired by fast frequency-selective Rayleigh fading and user interference ,” IEEE Trans. Veh. Technol., vol. 49, no. 4, pp. 1392-1407, Jul. 2000. [27] W.-H. Sheen and C.-H. Tai, “A noncoherent tracking loop with diversity and multipath interference cancellation for directsequence spread-spectrum systems,” IEEE Trans. Commun., vol. 46, pp. 1516-1524, Nov. 1998. [28] G. Fock, J. Baltersee, P. Schulz-Rittich, and H. Meyr, “Channel tracking for RAKE receivers in closely spaced multipath environments,” IEEE Journ. Select. Area. Commun., vol. 19, no. 12, pp. 2420-2431, Dec. 2001. [29] J. J. Caffery and G. L. St¨uber, “Effects of multiple-access interference on the noncoherent delay lock loop,” IEEE Trans. commun., vol. 48, no. 12, pp. 2109-2119, Dec. 2000. [30] W. K. Wong, Y. T. Wu, S. H. Leung, and Y. S. Zhu, “A modified de-correlated delay lock loop for synchronous DS-CDMA systems,” IEEE Trans. Veh. Technol., pp. 3293-3299, Sep. 2008. [31] A. Papoulis, Probability, Random variables and Stochastic Processes, 3rd Edition. New York : McGraw-Hill, 1991.

24