Statistical MIMO Radar

Collaborative Research with:Eran Fishler/NJITAlex Haimovich/NJITDmitry Chizhik/Bell LabsLen Cimini/U. Del. Reinaldo Valenzuela/Bell Labs

Rick S. BlumECE DepartmentLehigh University

Overview

MIMO radar with angular diversity

MIMO radar channel

Spatial Resolution Gain

Diversity gain

Numerical results

Extensions – space-time code

Radar Information Theory (time permitting)

Conclusions

Motivation Radar targets provide a rich

scattering environment.

Conventional radars experience target fluctuations of 5-25 dB.

Slow RCS fluctuations

(Swerling I model) cause long fades in target RCS, degrading radar performance.

MIMO radar exploits the angular spread of the target backscatter in a variety of ways to extend the radar’s performance envelope.

Backscatter as a function of azimuth angle, 10-cm wavelength [Skolnik 2003].

Traditional Beamformer• A typical array radar is composed of many closely spaced

antennas.

1 je 2je 3je

( )u t ( )u t ( )u t ( )u t

Beamformer mode:• Copy waveform

• The performance of radar systems is limited by target RCS fluctuations, which are considered as unavoidable loss.

• The common belief is that radar systems should maximize the received signal energy. This is made possible by the high correlation between the signals at the array elements.

MIMO Radar Terminology Recently, the radar community has been discussing

“MIMO” radars that utilize multiple transmitters to transmit independent waveforms (Dorey et. al. 1989, Pillai et. al. 2000, Fletcher and Robey 2003, Rabideau and Parker 2003).

Main characteristics of MIMO radars:● Each antenna transmits a different

waveform● Isotropic radiation – LPI advantage

1( )u t 2 ( )u t3( )u t 4 ( )u t

MIMO mode:• Orthogonal waveforms

MIMO with Angular Diversity

MIMO mode:• Exploit angular spread• Orthogonal waveforms

Angular spread - a measure of the variability of the signals received across the array. A high angular spread implies low correlation between target backscatter

MIMO with Angular Spread

The Radar MIMO Concept With MIMO radar, many "independent" radars collaborate to

average out target fluctuations, while maintaining the ability to detect target (measure range, AOA).

MIMO radar offers the potential for significant gains:● Detection/estimation performance through diversity gain● Resolution performance through spatial resolution gain

A first step to a cooperative radar network.

Signal Model Point source assumption dominates current models used in

radar theory. This model is not adequate for an array of sensors with large

spacing between array elements. Distributed target model

Many random scatterers

Signal Model (Cont.)

( ) ( ) ( )

( )

Signal energy:

Transmit antennas:

Vector of transmitted wavef orms:

Observation:

Et t t

M

E

M

t

t

·

·

·

·

= - +r Hs n

s

( ) ( ) ( )[ ]

[ ]

( ) ( ) ( )[ ] ( )

1

21

, ,

Channel matrix: target gain between transmitter k

and receiver :

Noise ,...,

~ 0,

N

kk

TN n

rt t r t

h

n t n t CNt s

=

·

=

· =

r

H

n

ll

L

l

T

See: E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, R. Valenzuela, "Spatial Diversity in Radars - Models and Detection Performance", to appear in IEEE Trans on Signal Processing, 2005.

Distributed Target

1 2

1 2

I t can be shown that ~ 0,1 .

Take and . We can show that if either /

or ' ' / ' , then 0, and otherwise

1

k

i cjk

Hc ijk

Hijk

h CN

h h d d d

d d d E h h

E h h

l

l

l

l

;

;

r1 r2

d

d1

d2

d’1

t1 t2

d’2

d’

Target beamwidth

Channel Matrix

( ) ( )

( )

Q

1

The channel matrix can be expressed

as a sum of dyads

Target refl ection coeffi cients are

iid, CN 0,RCS

From the channel matrix defi nition,

it is clear

that

T T Tq q q

q

a q f=

·

·

·

= = åH GΣK g k

:

( )rank min Q,rank ,rank £H G K

Spatial Resolution Gain

The rank of the channel matrix can be used to determine the number of dominant scatterers or the number of targets in the range resolution cell.

With suitable processing, this property of MIMO radar can be applied to enhance radar resolution by allowing the measurement of one scatterer at a time.

Radar Detection Problem

0

1

The radar detection problem: : Target does not exist at delay : Target exists at delay

Assume that all the parameters are known. The optimal detector is the LRT det

HypH

ecto

yp

r, and

0

1

1

0

1 1

it is given by,

The channel matrix is unknown, but its distribution is assumed known

| Hyplog

| Hyp

| Hyp | ,Hyp

H

H

p tT

p t

p t p t p d

r

r

r r H H

H

H

Detection with MIMO Radar

I t can be shown that the suffi cient statistic f or the

MI MO radar problem with unknown channel is the

vector the output of a bank of matched fi lters

sampled at .

Orthogonality of wavef orm

x

1

22

0

22 1

s is assumed

The optimal NP detector is

where

Non-cohe

,

rent processing in a network of MN rada

rs.

12 MN

Hn

FAH

T F P

x

2 22 2

21

2

I t is possible to compute the probability of detection as a f unction of the probability of f alse alarm, and it equals

The test statist

1

i

1

MN MN

nFAD

n

P F F PEM

c and threshold are independent of energy E. improves with E: test is UMPDP

UMP Test

Invariance Detector

1

0

2

H2

2H

Assume access to a vector that contains samples of the noise process.

Note that is the ML estimate of the noise level.

The optimal det

/

ector

This test sta

yp

yp

L

L

T

y

y

x

yd

>

·

<

·

·

·

=

tistic is very intuitive. I t normalizes the UMP test by the best estimate of the noise level.

Example: ROC SNR=10dB, N=4 M=2

10-10

10-8

10-6

10-4

10-2

100

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pfa

PD

S-MIMO Phased Array I-S-MIMO L=64Phased Array L=64

Miss Probability

• Probability of missed detections, M=2.

• MIMO Diversity (red): MIMO is used to enhance SNR.

• Beamforming (blue): isotropic transmission (MIMO mode), beamforming on receive.

0 5 10 15 20 25 3010

-6

10-5

10-4

10-3

10-2

10-1

100

SNR

PM

D

PMD

vs. SNR PFA

= 1e-006

N= 2N= 2N= 8N= 8N= 12N= 12

Concluding Remarks Introduced a concept of a radar with widely separated antennas

– a network of collaborating radars

MIMO radar exploits the angular diversity of the target to provide diversity gains.

Developed the channel model (illustrate spatial resolution gain).

Non-coherent processing of the sensor outputs is optimal, which challenges the conventional wisdom that array radars are best utilized by maximizing the coherent processing gain.

Introduced an Alamouti type space-time code for radar, which provides diversity gains without requiring orthogonal waveforms

Introduced radar information theory.

Alamouti Space-Time Code for Radar

1 2

1 2

Space-time code f or M = 2 transmitters, N = 1 receiver

Channel matrix given by

Two wavef orms and are trans

mitted f r

Th h SNR

s s

ré ùû· ë

·

=

·

=

·

H

Diversity gain without transmitting orthogonal waveforms

( ) ( )( ) ( )

( )1 1 1 2 2 1

2 2

1

1

2

2 1

PRI 1 PRI 2

Antenna 1 s

Antenna 2

om two antennas

over two time intervals as f ollows

Received signals at PRI s 1,2

2

s

2

t s t

t

r s h s

h

t

h

r s

s

nr

r

*

*

*

=

= -

-

+ +

·

( )1 2 2

2

s h n

r

*

= +

+ +

r SH n

Alamouti Space-Time Code for Radar

0 1

Consider the f ollowing detection problem:

: Target does not exist; : Target exists

The optimal detector f or unknown and Swerling case 1

target (Rayleigh) is the energy de

Hyp Hyp

tector

r

·

·

( )

2

† † †

2 2 †1 2

2

22 2

2

2

where 2

2

noise terms

The signal compon ent in has a dist

t

s s

y

y S r S SH S n

H S n

y H

y

r

r

c

r

´

>

<

= = +

= + +

=

·

+

ribution hence a

. diversity gain of 2

Radar Information Theory A fundamental approach to radar design and analysis

After initial breakthroughs by Davies and Woodward at the time when Shannon theory was published, the field has been laying dormant.

In contrast, communication information theory has been an active and productive field leading to many breakthroughs in communications including MIMO.

We will show that radar IT:● Unifies seemingly disparate concepts● Provides insight into trade-offs among radar parameters● Guides radar design● May help us draw from the rich experience gained with comm

IT.

The Essence of Radar Operation Four performance measures capture the essence of

radar operation [Siebert 1956]:

1. Reliability of detection2. Accuracy of target parameter estimation3. Resolution of multiple targets4. Ambiguity of estimates. The ambiguity problem

has two aspects: Noise ambiguity, where the noise is high and may

interpreted as target returns Ambiguity in the ability to measure a target

parameter in the absence of noise.

t

t

t

0

Range-Delay Estimation

The range-delay for which the output of the correlator is maximized provides an estimate of the target range

Threshold

APD

Noise waveform

Transmitted waveform

Received waveform

Correlator output Estimate of target range

A Posteriori Distribution

The output of the correlator provides information for the calculation of the a posteriori distribution (APD) of the range-delay given the observation

p(τ|y)

The APD describes the probability that a target is at a specified range-delay. It takes into account any prior information on the location of the target and it is the most information that can be extracted from the observation.

What Can We Learn from the APD? Woodward (1952) has shown that the variance

of the APD at the target’s range-delay is given by

στ2 = 1/(β2ρ)

β = bandwidth, ρ = SNR. This variance is a measure of the accuracy of

the range-delay estimation. The expression reflects a power-bandwidth

exchange: a deficit in each is compensated by increasing the other.

Relation to Resolution

Relation to resolution: The product Θβ is approximately the number of targets that can be resolved over the range-delay interval Θ.

APD for low bandwidth

APD for high bandwidth

Relation to Noise Ambiguity Relation to ambiguity: At low-SNR and high bandwidth

β, there are higher chances that noise might "pop up." As the SNR increases, the noise peaks diminish in size

compared to the target peak and the system transitions to a regime where true power-bandwidth exchange can take place.

A Unified Framework Traditionally, radar researchers have treated the

detection and estimation problems separately (let alone resolution, noise ambiguity and waveform ambiguity).

The APD provides a unified framework to detection and estimation.

The APD also captures the noise ambiguity problem and the resolution factor Θβ.

Is there an even better representation of the radar information that unifies all of the above factors?

Entropy

In IT, entropy is a measure of the amount of uncertainty of a random variable x (similar to the thermodynamics concept). Given the pdf p(x), the entropy is defined

h(x) = -∫ p(x) log p(x) dx

Radar Information Gain

The radar information gain is defined as the reduction in entropy (uncertainty) due to the processing of the radar observation.

How is this concept useful? How does it relate to APD (or to detection and estimation)? How does it motivate MIMO radar?

Consider the radar information gain for a single target

2 21log high SNR

2 2 1 1 2

- log low SNR2 2

eI

e

Power-bandwidth exchangeResolution factor

Noise ambiguity

Example of Information Gain

• The information gain increases linearly with the actual number of resolvable targets.

Ambiguity Region

Power-BandwidthExchange Region

Does not capturewaveform ambiguity

What We Learn from Radar IT? At sufficiently high SNR, the radar IG increases

logarithmically with the SNR and the bandwidth (power-bandwidth exchange).

The IG increases linearly with the actual number of resolvable targets.

With conventional radar, the IG increases only logarithmically with the number of targets if the targets cannot be resolved in range.

Based on preliminary analysis, with statistical MIMO, the IG increases linearly with the number of targets even when the targets are not resolvable in range, but are resolvable by the MIMO radar.