1. Statistical MIMO Radar
Rick S. Blum
ECE Department
Lehigh University
Collaborative Research with:
Eran Fishler/NJIT
Alex Haimovich/NJIT
Dmitry Chizhik/Bell Labs
Len Cimini/U. Del.
Reinaldo Valenzuela/Bell Labs

2. Overview Radar Information Theory (time permitting) Conclusions
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
We that S-MIMO will serve as fertile ground for radar R&D for years to come.

3. 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.
Swerling I model (slow fluctuations) provides a better fit than other models for airborne targets.
In this presentation, we will argue that statistical MIMO builds on concepts developed by the radar community, but at the same time it is a radical departure.
Backscatter as a function of azimuth angle, 10-cm wavelength [Skolnik 2003].

4. Traditional Beamformer
A typical array radar is composed of many closely spaced antennas.
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.
We argue
Beamformer mode:
Copy waveform

5. 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
MIMO mode:
Orthogonal waveforms

6. MIMO with Angular Diversity
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 mode:
Exploit angular spread
Orthogonal waveforms

7. MIMO with Angular Spread

8. 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.

9. 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

10. Signal Model (Cont.)
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. T

11. Distributed Target Target beamwidth d’2 d2
Channel coefficients are uncorrelated if an imaginary beam generated by the target resolves the radar antennas. d d’ d1
d’1 r1 r2 t1 t2

12. Channel Matrix

13. 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.

14. Radar Detection Problem

15. Detection with MIMO Radar

16. UMP Test

17. Invariance Detector

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

19. 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.

20. 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.

21. Alamouti Space-Time Code for Radar

22. Alamouti Space-Time Code for Radar

23. 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.

24. The Essence of Radar Operation
Four performance measures capture the essence of radar operation [Siebert 1956]:
Reliability of detection
Accuracy of target parameter estimation
Resolution of multiple targets
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.

25. Range-Delay Estimation
Noise waveform
The range-delay for which the output of the correlator is maximized provides an estimate of the target range
Transmitted waveform
Received waveform
Correlator output Estimate of target range
Threshold
APD

26. 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.

27. 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.

28. 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
This shows the well known result that range-delay resolution is inverse proportional to bandwidth.
APD for high bandwidth

29. 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.

30. 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?

31. h(x) = -∫ p(x) log p(x) dx
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

32. 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
Resolution factor
Power-bandwidth exchange
Noise ambiguity

33. Example of Information Gain
Power-Bandwidth
Exchange Region
Ambiguity Region
Does not capture
waveform ambiguity
The information gain increases linearly with the actual number of resolvable targets.

34. 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.