1

Some Signal Processing Perspectives for Calibrating Phased Arrays

Prof Douglas Gray

Director UoA Radar Research Centre

Phased arrays examples

Why calibrate ?

Internal calibration

Self calibration

Calibration using sources of opportunity

Calibration using other sensors

2

Some phased arrays

Array Errors

transducerReceiver

unit

transducerReceiver

unit

transducerReceiver

unit

Calibration - anechoic chambers not always feasible

Gain and phase mismatch between receivers

constant across the receiver bandwidth

varying across the receiver bandwidth

Positional errors

Unknown mutual coupling

4

Why Calibrate ?

Key issue for

Maintaining beam pattern and hence SNR

Low sidelobe phased arrays

Advanced beamforming algorithms

DoA estimation

Maintaining dynamic range

Off-axis polarimetric biases

Not always necessary

Some advanced interference rejection techniques don’t require it !!

5

Sidelobe sensitivity

Increase in SLL due to phase errors

Sensitivity of MUSIC to Modelling Errors

The MUSIC ―spectrum‖ in the presence of phase errors, 0.005<β<0.05, the two sources are at directions

γ=13.5 and 16.5 degrees (4 uniformly spaced circular sensors, half a wavelength apart)

Performance of MUSIC degrades when the array manifold is not accurate

=> Calibration will improve sensitivity

7

Application Areas

Radar : gain and phase errors, mutual coupling

GPS : gain and phase errors, mutual coupling

Sonar : gain and phase errors across a range of frequencies

positional errors for towed arrays

Radio Astronomy : : gain and phase errors VLBI arrays

Communications : gain and phase errors

Synthetic aperture radar, ISAR : motion compensation, autofocus

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Some Key Issues

Parameters to calibrated

Amplitude and phase errors

Mutual coupling

Receiver positioning errors

Cross polarimetric terms

Quality of calibration

Dependent on type of processing

eg phase errors typically to within

10—30 degrees for conventional beamformers

5-10 degrees for optimum beamformers

<5 degrees for high accuracy DoA estimators

eg MUSIC

9

Calibration Techniques

Various signal processing techniques for calibration

Using internal calibration signals

Self-calibration – known as equalisation

Using sources of opportunity

Single sources

Multiple sources

Using other sensor information

OTHR – “Internal” calibration

transducerReceiver

unit

(b) calibrate by injecting signals at receiver

Array ~2.78 Kms in length

(a) calibrate using a nearfieldsource

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Some Necessary Theory

Signal Processing Techniques for Calibration

but first

Vary the steering direction

)()(1

.exp)(1),(1

fXkvK

ukjfXK

fkY

H

K

k

kk

Kukj

ukj

ukj

e

e

e

kv

.

.

.

2

1

)(

where

k

Ku

2u1u

Phase Shift Beamforming

Frequency domain

General Array

Geometry

Steering vector

Set of all steering vectors forms the

“array manifold”

Block Diagram

)(1

fX )(2

fX )(3

fX )( fXK

),( fkY

Kv*

1

Kv*

2 Kv*

3 KvK

*

Mean Output Beam Power

Cross-spectral Matrix

),,()(),,(1

),,(,,2

2fvfRfv

KfYEfP x

H

cbf

)()()( fXfXEfRH

x

Output Beam Power

2

2 ),,()()(),,(),,(

KfvfXfXfv

fYHH

Squaring

Cross-spectral Matrix

)()()()()()(

)()()()(

)()()()()()(

)()()(

)(

)(

)(

)()()(

21

2212

12111

21

2

1

fXfXEfXfXEfXfXE

fXfXEfXfXE

fXfXEfXfXEfXfXE

fXfXfX

fX

fX

fX

EfXfXEfR

KKKK

K

K

K

H

x

Elements are the spatial covariances

CSM Examples I

Uncorrelated receiver noise

IffR

ffNfNE

nx

ijnji

)()(

)()(*)(

2

2

CSM Examples II

Plane wave signal

)()()(

)()()(*)(

)()()(

2

s

H

ss

s

H

s

H

x

kvkvf

kvkvfSfSE

fXfXEfR

)()()(

)()()(

s

s

kvfSfX

kvtstx

CSM Examples III

)()()(

)()()(

fRfRfR

fnfsfx

nsx

Plane wave signal in white noise

)()()()()( 22

s

H

sinxkvkvfIffR

Example of an important principle

For uncorrelated processes

Defines signal subspace Defines noise subspace

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Finally

The CSM plays a pivotal role in

(a) Optimum beamforming

(b) High resolution DOA algorithms

(c) Calibration techniques

(d) Defines signal and noise subspaces

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Self-referencing

Adaptive equalisation or spatial prediction

(M Trinkle : GPS, G Frazer OTHR, J Horridge Acoustic Radar)

Choose weights to

minimise output

backward/forward

prediction error

power for each

array channel+

1z

1z

1z

][kxref

x

x

x

x

x

][kebf

]2/[ Nkx j

1w

2w

3w

1w1z

x

2w

3w

+ ][kx j

GPS – adaptive digital equalisers

Channel Mismatch < 0.05 dB Difficult to achieve good analogue mismatch with 50 dB out of band rejection

Measured Interference cancellation ratios

2 4 6 8 10 12 14 16 18 20 22 24-0.5

-0.2-0.1

00.10.2

0.5

1

Frequency (MHz)

Mis

matc

h (

dB

)

Channel Mismatch Over Signal Bandwidth

Before EqualisationAfter Equalisation (7 taps)

Mismatch < 0.05 dB

0 5 10 15 20 25-10

0

10

20

30

40

50

60

70

80

Frequency (MHz)

Pow

er

(dB

)

Input SignalCancellation After EqualisationNon-Interference Signal

Noise floor

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Self-referencing ; Radio Astronomy

Phase errors only for illustration (easily extended to include amplitude)

Assume a initial model for the exact covariance matrix,

e.g., )()( H

x vvR

KKKK jj

jj

jj

j

j

j

j

j

j

ee

ee

ee

e

e

e

v

e

e

e

v

22

11

2

1

2

1

~

~

~

~

Compute the estimated covariance matrix

Phase errors to be

estimated

Refine the model and iterate

for a coherent point source

xR

2

,,,

ˆ

lk

lkx

j

lkx ReR lk

Estimate the phase errors, as those that minimise k

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VLBI Example

P. J. Napier, R. T. Thompson, R. D. Ekers,

Proc. IEEE 71, 1295 (1983).

Originally formulated for optical astronomy (Muller and Buffington 1974)

Applied to sonar (Bucker 1978, Ferguson 1989)

One Definition (several variants)

Sharpness

)sin,,max(argˆ0

2

dxfPSx

set of unknown parameters, eg receiver positions, phasesx

xfP ,, Output of conventional or optimum beamformer

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Sources of Opportunity

Signal Subspace Methods

A simple eigen-based phase calibration technique

Key Idea

A single strong far field source of opportunity

Compare the measured phases with those predicted

Estimating the phases

Estimate the CSM

Use of eigenvector corresponding to largest eigenvalue CSM

Some maths

H

s

H

sinx EkvkvEfIffR )()()()()( 22

ekVkvEu ss )()()( max

[D. A. Gray, W. O. Wolfe and J. L. Riley, “An eigenvector method for estimating the positions of the elements of an array of receivers”, Australian Symposium on Signal Processing Applications, 1989, Gold Coast, Australia, Vol. 2, pp 391-393.

If the noise is directional then need to prewhiten

)()( ss kvEkv E is diagonal matrix of amplitude

and phase errors

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Maintaining SNR

Sonar arrays depart significantly from linearity during manoeuvres

Use of targets of opportunity to measure and correct for positional

errors

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Sources of Opportunity

Noise Subspace Methods

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Friedlander’s approach I

Extension

Multiple signals of opportunity

Unkown DOAs

Incorporates mutual coupling

)()()()()( faCEVfXkvfafX

)(:)(:)( 21 NkvkvkvV Matrix of steering vectors

E is diagonal matrix of amplitude and phase errors

C is matrix of mutual coupling coefficients

B. Friedlander and A. J. Weiss, “Direction finding in the presence of mutual coupling”, IEEE Transactions on Antennas and Propagation, March

1991, Vol. AP-39, pp. 273-284.

B. Friedlander and A. J. Weiss, “Eigenstructure methods for direction finding with sensor gain and phase uncertainties”, ICASSP, 1998, New

York,U.S.A., pp 2681-2684.

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Friedlander’s approach II

Key Observation

MUSIC DOA estimation algorithms extremely sensitive to manifold errors

Perturbed signal subspace orthogonal to noise subspace

Thus columns of the matrix CEV are orthogonal to the noise subspace

Freidlander’s metric2

1,,,,

)(ˆmin21

L

n

n

H

ECwrt vCEUL

U

Neat three stage iterative algorithm

Matrix of estimated noise subspace eigen-vectors

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Disjoint sources

Daniel Solomon’s PhD thesis

Use back scattered returns from ionised trails of meteors as they burn up on entering the atmosphere

Friedlander’s approach of using subspace methods to estimate DOAs, array positional errors and mutual coupling coefficients

Range and time gating used to separate sources (disjoint)

Cost function modified to 2

1,,,,

)()(ˆmin21

L

n

n

H

ECwrt vCEnUL

Matrix of estimated noise subspace eigen-vectors for each source)(ˆ nU

I. S. D. Solomon, D. A. Gray, Y. I. Abramovich, and S. J. Anderson, Overthe- horizon radar array calibration using echos from ionized me- teor trails, Proc. Inst. Elect. Eng. Radar, Sonar, Navigation, vol. 145, pp. 173180, June 1998.

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Calibration – HF array

Sidelobe

comparison

Variant developed by Zili Xu for calibration of GPS antenna using GPS satellites as sources of opportunity

Frequency dependent calibration

Gain and phase responses vary across

operating bandwidth

Limits dynamic range in presence

of strong interferences

Increases dimension of signal subspace

At each range bin find a matrix

that minimises a time averaged error

between the range bin in question

and a reference range bin.

Uses strong AM interferences

G A Fabrizio, D A Gray and M D Turley "Using Sources of Opportunity to Estimate Digital

Compensation for Receiver Mismatch in HF Arrays" IEEE Trans on Aerospace and

Electronic Systems. (AES), Vol. 37, No. 1, Jan 2001, pp310-316.

Oracular S. McMillan and Y Abramovich

Calibration of amplitude and phase for an OTH HF radar

Uses LOS active returns from ISS ―Zarya‖ space station.

Key point : Moving target of opportunity over a CIT

Technique

Estimate signal and noise CSM

Prewhiten using estimated noise CSM

Decompose signal CSM into the ―signal‖ and ―noise‖ eigen-spaces

(Use of Slepian kernel to determine dimension of signal subspace)

(Signal subspace is the space spanned by the steering vectors of

the moving target over the CIT)

Form an ―integrated projection operator‖ onto noise subspace

Amplitude and phase errors obtained from the eigenvector

corresponding to the minumum eigenvalue of this operator

ISAR Calibration

Coarse translational motion

compensation of target/range

realignment

Autofocus – fine motion

compensation/phase realignment

Four techniques for estimating

the phase correction across PRIs

Max CBF, Max MVDR,

Max projection onto signal subspace,

Min projection onto noise subspace

All subject to a unity norm constraint

Z. She, D A Gray and R E Bogner. "Autofocus for Inverse Synthetic Aperture

Radar (ISAR) Imaging" Signal Processing, Vol 81, 2001, pp275=291

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Using instrumentation

37

Calibration – towed sonar array

Sonar arrays depart significantly from linearity during manoeuvres

Use of compass data and model of array dynamics

Design Kalman filter to estimate the time varying array shape

Incorporate unknown biases in compass outputs into the KF estimates

Summary

Novel solutions, based on signal processing, for calibrating large

arrays of receivers

Wide variety of techniques across a number of different

application areas

Self-referencing or model based approaches

Sources of opportunity

Instrumented arrays

Closure Phase

Key Idea

Sum of phases around a closed loop of 3 receivers is independent of

the phase errors.

Provides info regarding source

eg if the sum is zero then the source is highly coherent.

Also for highly redundant arrays in homogeneous noise fields can be

used to constrain individual phases to lower dimensional subspaces.

Amplitude closure as well.