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Transcript of Statistical MIMO Radar Collaborative Research with: Eran Fishler/NJIT Alex Haimovich/NJIT Dmitry...
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.