文獻2.窄帶相關器Theory and Performance of Narrow Correlator Spacing

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In the early days of GPS, there were a number of reasons why the 1.0 chip

spacing was ud. Some of the reasons are as follows:

1) This was a normal analog implementation in the early days to minimize

hardware. The advantages in noi are not prent if a T-dither DLL is

ud. This is becau the early and late noi components are uncorrelated

as a result of time-independence. A r-dither DLL is one that time-shares

a correlator between the early and late signals. However, most receivers

today, if not all, perform early and late (or early-minus-late) correlation

simultaneously .

2) The early receivers were usually of the P-code variety, Since the

P-code chip is already relatively short, narrower spacing makes the DLL

discriminator quite narrow. It was feared that Doppler and other distur-

bances would cau loss of code lock. Carrier aiding of the DLL minimizes

this problem. Carrier aiding was implemented in the early receivers, but

the designers of tho receivers feared the effects of jamming at initial

DLL acquisition. However, the linear range of a C/A-code DLL discrimi-

nator is 10 times greater than that of a P-code discriminator. Thus, this

reason does not have a foundation in carrier-aided C/A-code applications,

except possibly during DLL acquisition. However, variable spacing elimi-

nates that problem.

3) Narrower spacing requires faster clocking of the early/late gating. This

prented a technology problem, especially using the P-code. In fact, it

is still somewhat of a problem in P-code applications. However, in the

ca of a C/A-code receiver, the chipping rate is 1/10 that of a P-code

receiver. Today’s technology easily accommodates the clocking for nar-

rower spacing in a C/A-code receiver.

4) It is speculated that most designers failed to realize the potential of

narrower spacing. The results prented in this paper eliminate this

problem.

BASEBAND

AND AD/

CONVERSION

PUNCTUAL OR LATE

Q SAMPLES

,

-PUNCTUAL OR LATE

Q SAMPLES

DISCRIMINATOR SELECT

PUNCTUAL OR

LATE CODE

20.46 MHz

CODE CLOCK

Correlator Spacing

The key concept illustrated in Figure is the lection of the shift register

1

clock that generates the early, punctual, and late codes. N can take on values

from 1 to 20 to provide a variable spacing between tho codes, resulting in

early-late code spacing from 0.05 to 1.0 C/A-code chip, depending upon operat-

ing mode. This mechanization allows for a minimum E/L correlator spacing of

0.05 C/A-chip in the early-minus-late power mode and 0.1 C/A-chip in the

dot-

product or coherent modes.

SIMULATION AND TEST RESULTS

Band Limiting Effects

The performance of a DLL with narrow correlator spacing is very much

influenced by the precorrelation bandwidth. This is becau band limiting

tends to round the autocorrelation peak; thus the discrimination between early

and late correlation is limited when using very narrow correlator spacing.

To verify this connection, a simulation was developed using filtered

cross-

Vol. 39, No. 3 Van Dierendonck, et al.: Narrow Correlator Spacing

269

Discriminator Simulation

Before closing the DLL tracking loop, simulations were run to map discrimi-

nator versus correlator spacing d and code offt

150

160

CODE OFFSET - CYCLES

Fig. S-Early-Minus-Late Power Discriminator Simulated Versus Test Results (noi only)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1

EARLY/LATE CORRELATOR SPACING - CHIPS

Fig. 5-Dot-Product DLL (I Hz bandwidth) Tracking Performance Versus Spacing (noi only)

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To determine the performance for other loop noi bandwidths, one simply

multiplies the ordinate axis of Figure 5 by the square root of the loop bandwidth.

We have balined a bandwidth of 0.05 Hz (multiply by 0.2236) to achieve the

better than 10 cm accuracy at nominal

after the prentation

of [6] emphasized this problem. However, it was suspected that the narrow

spacing would also reduce the effects of multipath. This is becau distortion

of the cross-correlation function near its peak due to multipath is less vere

than that at regions away from the peak. Thus, it was reasoned that, if one

could track near the peak, the effects of multipath would be reduced. This

reasoning had to be verified. The results of that verification are prented here.

In

[2] it was indicated that narrow spacing did not reduce the maximum

ranging error due to multipath. However, that statement was made with

respect to coherent DLLs, which are more susceptible to carrier pha tracking

error. This is true becau the pha-lock-loop

(PLL), which is required when

using a coherent DLL, is tracking the composite true

+ multipath signal.

Thus, coherent is not really coherent to the true signal, and extreme cross-

correlation function distortion can occur, even with narrow spacing. The results

prented in

[2] were verified to be true for strong multipath returns, which

will be discusd later.

It was found, however, that this statement did not apply to noncoherent

DLLs. This is becau tho types of discriminators cancel most reliance on

carrier pha, and work even when the carrier pha is not being tracked.

Thus, our investigations were directed to determining the effects of multipath

in noncoherent

DLLs. Since we have balined the dot-product DLL, the in-

vestigations are limited to that discriminator. However, dot-product DLL

performance in a multipath environment is esntially the same as the

early-

minus-late power DLL for what is called Region I tracking [7]. For C/A-code

applications, this is the only region of interest, since carrier aiding should

never allow a transfer to Region II. Generally speaking, in Region

II the

multipath signal itlf is being tracked, while in Region I, the multipath is

simply causing a signal tracking error by distorting the discriminator

[2,7].

Multipa th Error Analysis

The composite true + multipath signal is given as

AC&t) crAC& 61 $1

+ is the carrier pha,

C,Jt) is the filtered PN code,

Vol. 39, No. 3 Van Dierendonck, et Spacing

al.: Narrow Correlator

273

and 8 is the relative time delay of the multipath signal with respect to the

receipt of true signal. Theoretically,

w,&

Equation (8) esntially reflects the signal model ud in [2] and [7], with

the exception that, here, we are filtering the code. This is important when we

narrow the correlator spacing, becau the bandwidth affects the multipath

distortion. If we process equation (8) in terms of I and Q samples similar to

tho in equations (A-l) and (A-2) of the appendix, we obtain the correlated

noi-free samples

where

I$

E[dT,] =

+ - R+JT~ 6 + cos(+, +

d/Z) R,JT~ + cos+,

&., of 180 deg and an

Tk, depending upon the multipath delay and correlator spacing

when the spacing is less than 1.0 chip. Thus, for the period of time the

conditions occur, the DLL would not be tracking at all. This was the phenome-

non that prompted the dishonorable mention for narrow spacing in

[2]. How-

ever, that phenomenon would be rare and does not occur in the ca ofnoncoher-

ent DLLs. The analysis to follow will be only for the noncoherent ca.

Noncoherent DLL

By then applying equation (2) and normalizing with

EL&l =

+aIR&Tk d/2)

d/2) R&k +

d/2) R&k 6 + +(r’[Rf(Tk 6

fi&k + co.%&

-

6 +

Cr2Rf2(Tk 6) 17&l

(13)

Plots of the normalized discriminator equation versus

4,

are given in Figures 6 and 7 for a t

to 0.5 and two values of d (1.0 and 0.1 chip, respectively). The filter bandwidth

in the 0.1 chip ca is that of the GPSCardTM

(8 MHz), while the bandwidth

TRACKING ERROR -CHIPS

Fig.

6-Discriminator Multipath Distortion for 1 .O Chip Spacing

TRACKING ERROR -CHIPS

Fig. i’-Discriminator Multipath Distortion for 0.1 Chip Spacing

an envelope of the multipath error versus

multipath delay. Varying relative pha caus the tracking error to take on

all values inside the envelope for a given multipath delay.

To evaluate the tracking error envelope versus multipath delay, equa-

tion (12) is t to zero and solved in an iterative manner for

+, t to 0 and 180 deg and

,J.O chip spacing, but not as small as the P-code

error envelope, for two reasons. First, the CIA-code correlates with the multi-

path signal with up to 10 times the delay than is the ca with the P-code.

Second, the 8 MHz bandwidth limits the reduction of the multipath effect.

To evaluate what would happen if the bandwidth were opened up to 20 MHz

with a spacing of 0.05 chip, the same process is repeated for that ca. The

results are shown in Figure 9. Note that for the region of 0.15 chip multipath

delay or less, the small C/A-code correlator spacing slightly outperforms the

conventional P-code performance. Although the existing GPSCardTM does not

have this 20 MHz capability, it is certainly a consideration for future develop-

ment.

1 CHIP SPACING c/A CODE, BW

II

1

W--6.-.60

d

h-

1.20/l

I

- C/A CHIPS

Fig.

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-0.25

MULTIPATH DELAY -C/A CHIPS

Fig. 9-Multipath Error Envelopes with 20 MHz Bandwidth

Test Results in a Multipath Environment

To verify the theory derived above on the effects of multipath, we t up an

experiment on the roof of

NovAtel’s headquarters in Calgary, where multipath

was known to exist becau of many reflecting objects. In this experiment, four

GPS receivers were connected to a common antenna/preamplifier. One of the

purpos of the experiment was to compare the effects of multipath on C/A-

code and P-code operation. Thus, two

Ashtech P-12 receivers were loaned to

us by The University of Calgary. The other two receivers were

GPSCardTM's.

The common antenna/preamplifier was that of one of the Ashtech receivers so

as to provide data on both the Ll and L2 frequencies. Multiple satellites were

tracked in each receiver. The Ashtech receivers were t up to track the P-

code on both Ll and L2 and the C/A-code on Ll with an output data rate of

0.2 Hz. The NovAtel receivers were t up to track the same satellites using

different correlator spacings.

First, we analyzed the collected data to find ctions of the data that had

obvious multipath effects, and to ensure that data were available for each of the

variations described above. For a lected ction of data, the code pudorange

measurements (PR) and the carrier pha measurements (ADR) were differ-

enced to remove satellite motion and receiver and satellite clock effects. Thus,

the results had a constant bias, plus the code-carrier divergence due to the

ionosphere. L2 differences were compared with the

Ll differences to verify

that the suppod multipath effects were not ionosphere effects. Procesd raw

measurement data for the Ashtech Ll P-code, and the

GPSCard’“,

the tracking loop bandwidth was t

at 0.05 Hz. The ramping of all the data over the last hour is due to the iono-

spheric code-carrier divergence; over this time the elevation angle varies from

40 to 16 deg. Data collection ended at that time becau the Ashtech data

buffer was full.

Fig. IO-Raw Measurement Data in Multipath Environment (bias are arbitrary)

The multipath effects are most noticeable in the P-code and 0.1 chip spacing

data for which the noi does not dominate. For a better obrvation, the data

prented in Figure

10

were smoothed through a first-order digital filter with

a 100 s time constant. The resulting data are shown in Figure 11. Although

a small portion of the multipath effects may also have been filtered, the differ-

ence in the effects due to correlator spacing is indeed noticeable. The data also

show that the effects using the P-code and the C/A-code with 0.1 chip spacing

are almost identical. The general difference in the effects between the P-code

and C/A-code with 1.0 chip spacing agrees with data obrved by other sources

93.

To further identify the multipath effects, the C/A-code measurements were

differenced with the P-code measurements, thus removing the effects of the

‘P

CHI

I

SPfiCINGt-t-

f--

4

3

IIII

3.6 4

3.8

1.8

2

2.2 2.4 2.6

2.8 3 3.2

3.4

TIME IN SATELLITE PASS

Data in Multipath Environment

(bias are arbitrary)

CONCLUSIONS

The theory and performance of narrow early-late correlator spacing delay

lock loops

(DLLs) have been prented. Remarkable performance advantages

over

the conventional 1.0 chip spacing DLL have been shown. The same is

true

in the prence of multipath. The following obrvations can be made:

The noi performance is proportional to the square root of the spacing,

provided sufficient precorrelation bandwidth is available.

For noncoherent DLLs, the maximum multipath errors are directly pro-

portional to the spacing, but conditioned to the availability of precorrela-

tion bandwidth. For coherent

DLLs, this obrvation is not necessarily

true under conditions of strong multipath reception. The error envelope

is also shortened by the narrower spacing, decreasing the radius to the

objects that cau multipath.

5

4

3

a-

1

&loil*h*d'I110

a=&122 Max. -Min.

!

j

1.6 2 2.2 2.4 2.6 2.6 3 3.2 3.4 3.6 3.6 4 4.2 4.4 4.6 4.6

TIME

IN SATELLITE PASS- HOURS

Fig. 12-Smoothed Differences Between Ll P-Code and Ll CIA-Code

Measurements (bias are arbitrary)

In this appendix we derive in four steps the equations for the noi perform-

ance of a DLL with the correlator spacing as an input variable. First, we

prent early, late, and punctual signal and noi models in the form of

in-

pha and quadrapha components. Second, we derive the distortion of the

signal cross-correlation function due to filtering the input signal. Then, we

derive the correlation properties of the early, late, and punctual noi compo-

nents. Finally, using the models and derived quantities, we derive the DLL

noi performance.

Signal and Noi Models

We assume here that the receiver is tracking the carrier and removes Doppler

frequency uncertainty effects. This is true for the

Skt), sampled at time

(A-1)

dm

sin& +

S/N,T is the signal-to-noi ratio in a predetection bandwidth of l/T Hz

(usually

50 Hz), and tk.

WTk) is the cross-correlation between the

incoming filtered signal PRN code and the unfiltered reference code; and

are the in-pha and quadrapha noi samples, respectively.

Equations (A-l) and (A-2) reprent a normalized signal and noi model,

so that the noi samples are Gaussian random variables with unity variance

and zero mean. Furthermore, the I and Q noi samples

are

statistically independent. However, it will be shown later that the early and

late or early-minus-late and punctual samples are, in general, correlated.

The cross-correlation function describing the correlation between the refer-

ence code and the incoming signal code of the same PRN number is, in general,

qQk

qQk)

wTk) =

0

R(u)h(Tk u)du

(A-3)

where

1-1~1,

luJ>l

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Fall 1992

is the unfiltered PRN code autocorrelation function, and

I

S,WlH(jfl(“df

(A-5)

s

=

sin*nfL

Tc

l/Tc

equal to 1.023 MHz,

I 1. This allows the variance of the noi samples in equations A-l

and A-2 to remain at unity.

The I and Q samples of equations A-l and A-2 reprent the punctual samples

Qpk). The equations also apply to the early and late samples

QEk,

QLk) and early-minus-late samples QE_L,k) with

the following substitutions:

R&T~ d/2),

(early)

(A-7)

R&7k +

R.&l = d/2) M(A-9)

d/2), (early-minus-late)

for early-late correlator spacing d in chips for early, late, and

7)IEk7 = tearly)

= TQLk, TQk

7)QELk,

and

(A-10)

q]k T)IELkY

PEL = R(d) = -

PE-L.P =

Vol. 39, No. 3 Van Dierendonck, et al.: Narrow Correlator Spacing

281

are correlated by the same d/2, which cancels. The variances of the

1

E[T-&J =

where

-d) = 2d

(1) and noncoher-

ent DLL discriminators, we have their expected values

B;K % @,c

=

2

- &+ic%k

=

2

R1(Ck +

F@r, d/21

(A-16)

Eb,l f,-,,,

=

[7,l%l/2 to

+

S/N,,T(2 d)Tk, (early-minus-late power)

S/N,T$l )Q)), (dot-product)

2v!~~, (coherent)

The evaluation of the variances of the noncoherent discriminators requires a

fair amount of algebra, especially for the early-minus-Iate power discriminator.

The key relationship in tho evaluations is the fact that, for correlated Gauss-

ian random variables 1111,

(A-19)

(A-20)

(A-21)

E[x,x21E[x,x,l + E[x,x,lE[x2x,l (A-22)

Also, to evaluate tho variances with filtered cross-correlation functions,

as given in equation (A-3), would be extremely messy. That is left for computer

simulation, results of which are prented in the paper. The evaluation using

infinite precorrelation bandwidths, although tedious, yields the following:

4d(2 d)[(2

&_O =

(z&

=

d)S/&,T +

Of,k+l =

-

K:u:& D

However, in steady state,

7

= =

+

1.k

so that

=

8B,T(l 2BLT)

for

,2-BLd

-

2 SIN,

- S/N,T

2

1

a: = +&I, (dot-product)

=

(A-29)

(A-30)

(A-31)

Vol.

39, No. 3 Van Dierendonck, et al.: Narrow Correlator Spacing

283

ACKNOWLEDGMENT

The authors express their appreciation to Dr.

D.,

Comparison of the Noncoherent Delay-Lock Loop and the Data

Estimating Delay-Lack Loop in the Prence of Specular

Multipath,

Stanford

Telecommunications, Inc., Report No. STI-TR-33048,5 April 1983.

8. Hatch, R., Magnavox, private communication.

9. Lachapelle, G., Falkenberg, W., Neufeldt, D., and Kielland, P.,

Marine DGPS

Using Code and Carrier in a Multipath Environment,

Proceedings of ION

GPS-89, Second International Technical Meeting of the Satellite Division of The

Institute of Navigation, Colorado Springs, CO, 27-29 September 1989,

pp. 343-47.

10. Cannon, M. E. and Lachapelle, G.,

Analysis of a High-Performance C/A-Code

GPS Receiver in Land Kinematic Mode,

this issue of NAVIGATION.

11. Brown, R. G. and Hwang, I? Y. C.,

Introduction to Random Signals and Applied

Kalman Filtering, Second Edition,

John Wiley and Sons, Inc., 1991.