Enhanced Narrowband Signal Detection and Estimation with a

Enhanced Narrowband Signal Detection and Estimation with a Synthetic Antenna

Array for Location Applications

Ali BroumandanPosition, Location And Navigation (PLAN) Group

University of Calgary2 October, 2009

ION Alberta Lunch Meeting

Enhanced Narrowband Signal Detection and Estimation with a Synthetic Antenna2 / 18

Outline• Motivation and applications of antenna array

systems• Signal detection challenges in fading environments• Signal detection performance by using a static

antenna • Signal detection performance by utilizing a moving

antenna• Analysis of processing gain in uncorrelated Channel• Analysis of processing gain in correlated Channel

• Data collection and performance verification• Conclusions


21

54

(10 ) 9.5 10

(10 ) 8.2 10

P dB

P dB

−

−

= ×

= × 1 2 3 4 5 6 7 8 9 10-20

-10

0

10

20

30

40

time (s)

SNR

(dB

)

Antenna 1Antenna 2

• In a dense multipath scattering environment, fading appears to be a random function of antenna location

• To reduce the fading margin required, the receiver can use multiple spatially separated antennas

• Spatial diversity• Assume 4 branch diversity system

Applications of Antenna Array in Multipath Environment

20 dB


Synthetic Array Concept • The size of the antenna

array is incompatible with the small size of the handheld receiver

• The only means of realizing the potential spatial processing is to move antenna

• For a direct comparison with the stationary antenna, the constraint

antenna

( )mt T+ Δp ( )mtp

( )1mt +p ( )1mt T+ + Δp

antenna trajectory

spatial coordinates

T M T= Δ


Problem Definition in Dense Fading• Consider a single channel handheld receiver detecting

narrowband signals in Rayleigh fading

• Static Antenna• Signal coherency is maintained• Signal will be subject to statistically large fading losses

• Moving Antenna• The coherency of the signal is decreasing leading to processing

losses• The spatial diversity contained in the snapshot data more than

compensates for this loss• Evaluating processing gain by moving an antenna instead of

keeping it stationary in Rayleigh fading• Comparing the detection performance of Estimator-Correlator

(EC) with that of Equal Gain (EG) combiner

Signal Detection Performance in Fading Environments with a Moving Antenna 6 / 18

Detection Problem• The optimal detection

processing based on the log likelihood Ratio Test (LRT) chooses H1 if

• The LRT reduces to the Estimator-Correlator (EC) formulation resulting in a sufficient statistic given as

( )( )

|( )

|1

0

p HL

p Hγ= >

xx

x

( ) 1H 2ECz σ

−= +s sx C C I x

-4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Variable

Prob

abili

ty o

f var

iabl

e

PDF under H1

Pd

PDF under H0

Pfa

-4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Variable

Prob

abili

ty o

f var

iabl

e

PDF under H1

Pd

PDF under H0

Pfa


Signal Detection by a Static Antenna

• The optimal NP detection processing is

• The average signal to noise ratio, ρ given as

• The Detection performance can be represented by

( )os t ( )0

1 T

dtT ∫ i

w(t)

z0

Channel

( )A p

Receiver

( )r t( ) 2i

xT

( )2 *0

0

, ( )T

T T oz x x r t s t dt= = ∫

( ) ( )detexp , expfaP P1γγρ

⎛ ⎞−= − = ⎜ ⎟⎜ ⎟+⎝ ⎠

2A

o

TNσρ ≡

( )( )det

lnln

faP1

Pρ = −


Signal Detection by a Moving Antenna

• In correlated Rayleigh fading the optimum detector in Gaussian signal and noise model is Estimator-Correlator(EC)

• The test statistics is

x1

x2

xM

Channel

( )os t

w 1(t)

( )( )1A tp

( )os t ∗

w 2(t)

( )( )2A tp

( )os t ∗

wM (t)

( )( )MA tp

( )os t ∗

( )1

1

1 t T

t

dtT

+ Δ

Δ ∫ i

( )2

2

1 t T

t

dtT

+ Δ

Δ ∫ i

( )1 M

M

t T

t

dtT

+Δ

Δ ∫ i

Rece iver processing

Decorrelator 11 1Hy x= v

Decorrela tor H

M M My x= v

Decorrela tor 22 2Hy x= v

y1

y2

yM

2⋅

2⋅

2⋅

1w

( )2

1i i

E Ci s s

E Gi

w

w

λ λ σ= +

=

2w

Mw

z

m

m

M2s

EC m2m 1 s

z yλ

λ σ=

=+∑

M2

EG mm 1

z y=

=∑

T M T= Δ

Signal Detection and Estimation Utilizing the Synthetic Array9 / 18

Synthetic Array Detection Performance in Uncorrelated Rayleigh fading

• In uncorrelated Rayleigh the optimum detector in Gaussian model is EG combiner

• The test statistics is

• Detection performance

where

21

1

M

mm

z x=

= ∑

( )22M

fa xP Q γ= det 22MxP Q

1M

γρ

⎛ ⎞⎜ ⎟

= ⎜ ⎟⎜ ⎟+⎝ ⎠

( . )( ) exp!

22M

kM 1

xk 0

1 0 5xQ x x2 k

−

=

⎛ ⎞= −⎜ ⎟⎝ ⎠

∑


Required SNR and Processing Gain for Stationary and Moving Antenna

• Average required SNR for static and moving antenna is compared as a function of the target parameters with several values of M

• G is the processing gain which is defined by

5 10 15 20 25 30 35 4010

15

20

M

aver

age

SNR

(dB

)

(a) pfa = 0.01 , pdet = 0.95

5 10 15 20 25 30 35 400

2

4

6

M

G (d

B)

(b)

StaticMoving

Mopt

Mopt

10log( )s mG ρ ρ=

Processing Gain of the Synthetic Antenna Array

Enhanced Narrowband Signal Detection and Estimation with a Synthetic Antenna Array 11 / 18

• Synthetic array gain with respect to static antenna versus PDand M for PFA=0.01

• The black line represents the optimum M as a function of PD


Channel Correlation Coefficient

• The received signal

• The correlation coefficient can be defined by

• where θ is mean of Angle of Arrival and Φ is angle spread

( ) ( ) ( )os t A s tθ=

0 1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

1

ΔD/λ

Nor

mal

ized

cor

rela

tion

coef

ficie

nt

θ = 0, φ = 360

θ = 0, φ = 90

θ = 0, φ = 15

θ = 45, φ = 180θ = 45, φ = 15

θ = 45, φ = 10

α= ΨsC

( ) ( ) ( )os t A s tθ=


• Required SNR and equivalent gain of the dual antenna relative to the signal antenna processing schemes

• EC has better or identical performance as the static antenna

0 0.5 111

12

13

14

15

Correlation coefficient

ρ avg (d

B)

PFA=0.1, PD=0.9

0 0.5 1-2

-1

0

1

2

3

Correlation coefficient

Gai

n (d

B)

ρs

ρEG

ρEC

GEG

GEC

0 0.05 0.1 0.15 0.2 0.250.9

0.92

0.94

0.96

0.98

1

Pfa

P d

SNR= 16 (dB)

r = 0r = 0.5

r = 0.75

r = 0.99

EGEC

Performance of EC and EG in Correlated Rayleigh

• ROC curves versus r for given SNR=16 dB and M=2

• For moderate correlation coefficient performance of EG and EC are almost identical


Indoor Data Collections

• GPS L1 C/A code• Data collection in different indoor environments• Linear moving table is used to realize the synthetic array• The antenna was moved by 2 cm/s

Antenna

Receiver Linear Moving Table


• Detection variables used for estimating PDF under H0 and H1

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

Code offset

Nor

mal

ized

cor

rela

tion

func

tion

Detection variables under H0



0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

ΔD/λN

orm

aliz

ed c

orre

latio

n co

effic

ient

MeasuredTheoritical model

• Theoretical model is

• ΔD is antenna spacing and λ is the wavelength

( )0 2 /J Dπ λΔ

Correlation Functions and Indoor Channel Correlation Coefficient


Indoor GNSS Channel Measurements and Verification

0 10 20 30 40 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

variablePr

obab

ility

of v

aria

ble

Chi-Squared distribution with 10 DOF

Estimated PDF under H1

Composit PDF σ2 =1.5, σ2 =3Estimated PDF under H0

Composit PDF σ2 =0.15, σ2 =0.32

• PDF of the receiver signal amplitude and theoretical Rayleigh PDF fit

• Theoretical and measured PDF under H0 and H1states

Enhanced Narrowband Signal Detection and Estimation with a Synthetic Antenna17 /18

Detection Performance

SNR measurements

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

PFa

P D

Synthetic Array, M = 4, r = 1Synthetic Array, M = 2, r = 1StaticSynthetic Array, M = 2, r = 0Synthetic Array, M = 4, r = 0

ROC Curves

CDF of SNR0 20 40 60 80 100-20

-10

0

10

20

30

t (s)

ISN

R (d

B)

Stationary AntennaSynthetic array, M=2Synthetic array, M=4

• Synthetic array reduces the fading effect

• ROC performance of the synthetic array is higher than the static antenna


Conclusions• The detection performance of a narrow bandwidth wireless

signal subjected to Rayleigh fading has been considered for a single antenna handheld receiver

• Of specific interest was to determine the merits of moving the antenna while capturing the signal that provides diversity gain

• It was shown that substantial processing gains are possible by moving the antenna

• Substantial processing gains are possible by moving the antenna

• Experimental measurements were done to verify the assumption of the Rayleigh fading and also to verify the calculated processing gain

• Good agreement between the experimental and theoretical results was obtained

Documents

Enhanced Narrowband Signal Detection and Estimation with a