Slide 1

ION GNSS, Fort Worth, Texas, September 28, 2006

Maximum Likelihood Multipath Estimationin Comparison with Conventional Delay Lock Loops

Michael Lentmaier, Bernhard KrachGerman Aerospace Center (DLR)

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 2

Outline

Multipath signal model and maximum-likelihood delay estimator

Methods for efficient implementation- data size reduction- interpolation techniques - Newton optimization

Integration of the ML estimator into receiver tracking loop:- comparison with conventional DLL (narrow correlator)- ML estimator in absence of multipath- ML estimator in presence of multipath

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 3

The Effect of Multipath

Multipath: superposition of received signals with different amplitude and delay

Offset in Loop-S curve: DLL estimate gets biased → error due to multipathExtended signal processing model: multipath can be taken into account:

Estimation problem with parameters ak and τk and additive noise n.

delayed path offset

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 4

For two paths:Cost function for single path:

Maximum Likelihood Estimation:

τ1τ1

τ2

Difficulties: multi-dimensional nonlinear optimization problem

complexity reduction techniques by Jesùs Selva (PhD Thesis, 2004)

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 5

Data Size ReductionConcept: projection of received y into smaller subspaceNo loss of information if estimation is based on the reduced model

provided that and

Then is a sufficient statistic for estimating τ and a according to the Neyman-Fisher factorization:

The estimation problem can then be reformulated within thereduced space as

with

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 6

Two-Fold Reduction

Principal Components:

where Qp is formed by eigenvectors of the dominant eigenvalues of

Canonical Components: use convolutional factorization

Allows using the output of a bank of simple code-matched correlators

Principal ComponentsCanonical Components

Bank of Correlators

Further Reduction

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 7

Canonical Components

Signal matched correlation

Code matched correlation

Correlator Outputs:

operates at chip rate

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 8

Efficient Cost Function MinimizationNewton Optimiser with explicitformulas for gradient and Hessian:

where

The optimization can be performed in the subspace of small dimension.

MVPmethod

Vandermonde vectorInverse DFT matrix DFT of pulse

with

Interpolation techniques:

continuous time-shift possible

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time [samples]

Am

plitu

de

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 9

Integration into Tracking Loop

LoopFilter

NCO

Code Generator

from PLL

CarrierWipe-Off

Bank of Correlators ML

Estimator

to PLL

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 10

30 35 40 45 50 55 60 65 70

10-10

10-9

10-8

10-7

C/N0 [dB-Hz]

RM

S e

rror [

s]

Performance without Multipath

30 35 40 45 50 55 60 65 70

10-10

10-9

10-8

10-7

C/N0 [dB-Hz]

RM

S e

rror [

s]1 chip

0.5 chips

0.3 chips

0.1 chips

Cramer Rao bound

MLE, 1ms

MLE, 10ms

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 11

Multipath Error Envelopes

Multipath error resulting from a singlereflection without noiseShown as function of relative delayRelative amplitude: 1/10 Phase: 0°10 MHz bandlimited rectangular pulse

The curves show:1. Standard correlator, 1 chip spacing2. Narrow correlator, 0.1 chip spacing3. ML estimator, assuming N=1 path4. ML estimator, assuming N=2 paths

narrow correlator

MLE N=1

MLE N=2

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 12

Simulation of Multipath Tracking Error

0 10 20 30 40 50 60-2

0

2

4

6

8

10

Time [s]

Trac

king

erro

r [m

]

Relative amplitude: 0.5

Relative phase: 180°

C/N0: 50 dB-Hz

Integration time: 10ms

DLL, 0.1 chip

1 path MLE2 path MLE

Multipath delay: 10m

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 13

Conclusions

Signal compression and interpolationa small data set contains all necessary information on parametersNewton optimizationforms an efficient way to find the ML solution

Further topics:Path number problemsoft information better than fixed selection/decision?Sequential estimation (Bayesian filtering)using history, not only independent ML snapshotsIncorporation of prior knowledge about channelmovement models, echo statistics

ION GNSS, Fort Worth, Texas, September 28, 2006Slide 14

Thanks for your attention!