1. Beamforming Issues in Modern MIMO Radars with Doppler
Chun-Yang Chen and P. P. Vaidyanathan
California Institute of Technology
Ladies and gentlemen, welcome to my talk. The title of this talk is beamforming issues in modern MIMO radars with doppler.
DSP Group, EE, Caltech, Pasadena CA

2. DSP Group, EE, Caltech, Pasadena CA
Outline
Review of the MIMO radar
Spatial resolution. [D. W. Bliss and K. W. Forsythe, 03]
MIMO space-time adaptive processing (STAP)
Problem formulation
Clutter rank in MIMO STAP
Clutter subspace in MIMO STAP
Numerical example
This is the outline of the talk. I will first review the MIMO radar and the review will be focus on the concept of using MIMO radar to improve the spatial resolution. Then I will talk about space-time adaptive processing in MIMO radar. The STAP problem will first be formulated. Then the clutter rank and the clutter subspace will be explored. This paper is focused on the derivation of the clutter subspace in MIMO radar. Finally the numerical example will be presented.
DSP Group, EE, Caltech, Pasadena CA

3. SIMO Radar Transmitter: M elements Receiver: N elements ej2p(ft-x/l)dR dT
w2f(t)
w1f(t)
w0f(t)
Number of received signals: N
Transmitter emits coherent waveforms.
DSP Group, EE, Caltech, Pasadena CA

4. MIMO Radar Transmitter: M elements Receiver: N elements ej2p(ft-x/l)dR dT MF … MF
f2(t)
f1(t)
f0(t) …
The transmitter is a uniform linear array with M elements and the receiver is also a uniform linear array with N elements. The transmitter emits M orthogonal waveforms. In the receiver, each antenna uses a set of matched filters to extract the signals sent from M orthogonal waveforms. So there will be totally NM sufficient statistics be extracted.
Matched filters extract the M orthogonal waveforms.
Overall number of signals: NM
Transmitter emits orthogonal waveforms.
DSP Group, EE, Caltech, Pasadena CA

5. DSP Group, EE, Caltech, Pasadena CA
MIMO Radar (2)
ej2p(ft-x/l)
ej2p(ft-x/l) q q dR
dT=NdR MF … MF
f2(t)
f1(t)
f0(t) …
Transmitter: M elements
Receiver: N elements q
The spacing dT is chosen as NdR, such that the virtual array is uniformly spaced.
The orthogonal waveform sent by the first transmitter is extracted by the matched filter in each receiver antenna. The phase difference of the antennas are the same as this array. The second orthogonal waveform is then extracted by the corresponding matched filter. If the spacing of the transmitting antennas dT is equal to NdR then the phase difference of these extracted element are the same as the virtual array elements because the phase difference of these blue elements and the green elements is created by the transmitter. And here goes the third orthogonal waveform. The phase difference of these red elements and blue elements are created by the locations of this red antenna and this blue antenna. If dT is less than NdR we will not have a critically sampled virtual array like this and the virtual array will be smaller because the phase difference created in the transmitter is smaller.
Virtual array: NM elements
DSP Group, EE, Caltech, Pasadena CA

6. DSP Group, EE, Caltech, Pasadena CA
MIMO Radar (3)
[D. W. Bliss and K. W. Forsythe, 03] + =
Virtual array: NM elements
Transmitter : M elements
Receiver: N elements
The clutter resolution is the same as a receiving array with NM physical array elements.
A degree-of-freedom NM can be created using only N+M physical array elements.
Therefore, with M transmitting antennas and N receiving antennas and with dT=N times dR we are able to create NM signals. The clutter resolution of this system is the same as a receiving array with NM physical array elements. A degree-of-freedom NM can be created using only N+M physical array elements.
DSP Group, EE, Caltech, Pasadena CA

7. Space-Time Adaptive Processing (STAP)
The adaptive techniques for processing the data from airborne antenna arrays are called space-time adaptive processing (STAP).
airborne radar v
vsinqi qi
The clutter Doppler frequencies depend on looking directions.
The problem is non-separable in space-time.
jammer
Now, let’s review some space-time array processing. The space-time array processing is the technique for processing data from the airborne antenna array. This figure illustrates the airborne antenna array. The key point of this kind of radar is that the radar station itself is moving. Therefore the Doppler frequency of the clutter will not be zero. The Doppler frequency will depend on the looking direction. The Doppler frequency for this target is this. vt is the target speed toward the radar station and v is the speed of radar station. \theta t is the looking direction of the target. This is the Doppler frequency for the ith clutter and \theta I is the looking direction of the i-th clutter. The clutter Doppler frequencies are chaning at different looking directions. Therefore the problem is non-separable in space and time. Therefore we need to processing space and time at the same time.
target vt
i-th clutter
DSP Group, EE, Caltech, Pasadena CA

8. Formulation of MIMO STAP
target
target
ej2p(ft-x/l)
ej2p(ft-x/l) vt q vt q
vsinq
vsinq dR
dT=NdR MF … MF
f2(t)
f1(t)
f0(t) …
Transmitter : M elements
Receiver: N elements
target
The space-time MIMO signals can be expressed as ynml. n is the index of receiving antenna and m is the index of the m-th orthogonal waveform and l is the index of radar pulse. We can view n,m as index in space domain and l as index in time domain. The signal consists of four parts: target, clutter jammer and noise. We can also stack them into an NML vector and the corresponding covariance matrix is NML times NML.
NML
noise
clutter
jammer
NML x NML
DSP Group, EE, Caltech, Pasadena CA

9. DSP Group, EE, Caltech, Pasadena CA
Clutter in MIMO Radar
size: NML
Now we focus on the clutter. We assume the clutter signal is contributed by Nc clutters. This rho i is the magnitude of the signal reflected by the i-th clutter and the phase difference are also caused by the transmitter, receiver and the Doppler. This term is created by the transmitter. This term is created by the receiver. This term is created by the Doppler. We define the parameters. Normalized spatial frequency for the i-th clutter fsi. \gamma is the ratio of the spacing in transmitting radar and receiving radar. \beta is related to the speed of the radar station. The clutter signal can be simplified as this. One can see that this gamma and beta are very important parameter for the clutter signal.
size: NMLxNML
DSP Group, EE, Caltech, Pasadena CA

10. Clutter Rank in MIMO STAP: Integer Case
Integer case: g and b are both integers.
The set {n+gm+bl} has at most N+g(M-1)+b(L-1) distinct elements.
Theorem: If g and b are integers,
If gamma and beta are both integers, then this set is a subset of this. Because the largest integer we can get is by plug in capital N-1, captial M-1, , and captal L-1 in this term. And all these elements must be integers. Therefore they must be one of these integers. This means the vector ci have at most this many distinct elements. This also means the matrix C has at most this many distinct rows. Therefore we can conclude the following. The rank of Rc must less than these three numbers. Usually the first one tend to be the smallest one. If M equals one, it reduces to the brennan’s rule. This result can be viewed as the MIMO extension of the brennan’s rule. This derivation is only for the integer case. For non-integer case we will need a tool called prolate spheroidal wave function.
This result can be viewed as the MIMO extension of Brennan’s rule.
DSP Group, EE, Caltech, Pasadena CA

11. Clutter Signals and Truncated Sinusoidal Functions
ci is NML vector which consists of
It can be viewed as a non-uniformly sampled version of truncated sinusoidal signals. X
We can view this clutter vector ci as a non-uniformly sampled version of truncated sinusoidal signals. It can be viewed as a non-uniformly sampled version of this function. Note that x is less than this number because it is the largest element we can have in here. Also the normalized frequency is usually confined in -.5 to .5 to avoid aliasing. Therefore, this signal is time-limit and most of its energy is concentrate in the band [-.5, .5] as shown in this figure. The top shows the real part of this function. It is just a truncated cosine. The bottom shows the magnitude response of the function. It is just a shifted sinc. We can see that it is somehow concentrated in a time-frequency region. 2W
The “time-and-band limited” signals can be approximated by linear combination of prolate spheroidal wave functions.
DSP Group, EE, Caltech, Pasadena CA

12. Prolate Spheroidal Wave Function (PSWF)
Prolate spheroidal wave functions (PSWF) are the solutions to the integral equation [van tree, 2001]. X -W W
in [0,X]
Time window
Frequency window
Only the first 2WX+1 eigenvalues are significant [D. Slepian, 1962].
The “time-and-band limited” signals can be well approximated by the linear combination of the first 2WX+1 basis elements.
The PSWF are defined as the follows. It is the eigenfunction of the integral equation. This integral equation can be viewed as first truncated the signal in the time domain. This is because of the interval in the integration. And then there is another truncation in the frequency domain. This is because of convoluting with sinc.
DSP Group, EE, Caltech, Pasadena CA

13. PSWF Representation for Clutter Signals
The “time-and-band limited” signals can be approximated by 2WX+1 PSWF basis elements.
This truncated sinusoidal function is concentrated in a time-frequency region. Therefore, it can be well approximated by the linear combination of 2WX+1 basis elements. Also W is 0.5 and X is this value. Therefore, the number of elements required is N plus gamma times M minus one plus beta times L minus one. This number is exactly the clutter rank when gamma and beta are both integers.
clutter rank in integer case
DSP Group, EE, Caltech, Pasadena CA

14. PSWF Representation for Clutter Signals (2)
non-uniformly sample
U: NML x rc A: rc x rc
The clutter signals are non-uniformly sampled version of the truncated sinusoidal signals. Since the truncated sinusoidal functions can be approximated by the linear combination of the PSWF basis elements. The clutter signal can also be approximated by the non-uniformly sampled version of the PSWF basis elements. Therefore, this clutter vector can be written as a linear combination of these vectors uk. This uk is a NML vector consists of non-uniformly sampled PSWF. The PSWF can be computed off-line and stored in the computer. The uk can be obtained by sampling the PSWF. When the parameter gamma and beta changed the new subspace can be obtained by taking different samples.
The PSWF yk(x) can be computed off-line
The vector uk can be obtained by sampling the PSWF.
DSP Group, EE, Caltech, Pasadena CA

15. DSP Group, EE, Caltech, Pasadena CA
truncated sinusoidal
Linear
combination
PSWF
Non-uniformly
sample
Non-uniformly
sample
i-th clutter signal
Linear
combination
Sampled PSWF
Stack
Stack
Linear
combination
Sampled PSWF
i-th clutter signal
Clutter covariance matrix
U: NML x rc
A: rc x rc
DSP Group, EE, Caltech, Pasadena CA

16. DSP Group, EE, Caltech, Pasadena CA
Numerical Example
qkH Rcqk
The figure shows the clutter power in the orthonormalized basis elements.
The proposed method captures almost all the clutter power.
Parameters:
Proposed method
N=10
M=5
L=16
g=N=10
b=1.5
NML=800
N+g(M-1)+b(L-1)=72.5
This figure shows the clutter power in the orthonormalized basis. Here are the parameters. The total dimension is 800. We have only plot the result for the first 200 basis. The red line is the basis created by doing eigenvalue decomposition. Therefore it is the optimal basis for this. The blue line is the basis computed by the proposed method. We can see that this method is very accurate. It captures almost all the clutter power. The subspace estimated by the proposed method is larger than the eigenvectors. This is because for some range bin, the clutter looking direction is limited.
Eigenvalues k
DSP Group, EE, Caltech, Pasadena CA

17. DSP Group, EE, Caltech, Pasadena CA
Conclusion
The clutter subspace in MIMO radar is explored.
Clutter rank for integer/non-integer g and b.
Data-independent estimation of the clutter subspace.
Advantages of the proposed subspace estimation method.
It is data-independent.
It is accurate.
It can be computed off-line.
We have explored the clutter space in MIMO radar. We have derive the clutter rank for both integer and non-integer cases. We have also derive the clutter rank for arbitrary array. We have also introduced a data-independent estimation for the clutter subspace. The advantages of this method is the following. First, it is data-independent. Therefore we don’t have to cumulate data to estimate the subspace. Second, it is quite accurate. By accurate we meant it capture almost all clutter power. Third, the PSWF can be computed off-line. Therefore the complexity is low. We have a STAP method using this clutter subspace estimation technique. And the result has been submitted to ICASSP. In this paper, we consider only the idea case, in fact there are things like (internal clutter motion) ICM will increase the clutter rank. In that case, a better way is estimating the clutter subspace by using a combination of both geometry and the data. This idea will be explored in the future.
DSP Group, EE, Caltech, Pasadena CA

18. Further and Future Work
Further work
The STAP method applying the subspace estimation is submitted to ICASSP 07.
Future work
In practice, some effects such as internal clutter motion (ICM) will change the clutter space.
Estimating the clutter subspace by using a combination of both the geometry and the data will be explored in the future.
New method
DSP Group, EE, Caltech, Pasadena CA

19. DSP Group, EE, Caltech, Pasadena CA
References
[1] D. W. Bliss and K. W. Forsythe, “Multiple-input multiple-output (MIMO) radar and imaging: degrees of freedom and resolution,” Proc. 37th IEEE Asilomar Conf. on Signals, Systems, and Computers, pp , Nov
[2] D. Slepian, and H. O. Pollak, "Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-III: the dimension of the space of essentially time-and-band-limited signals," Bell Syst. Tech. J., pp , July 1962.
[3] D. J. Rabideau and P. Parker, "Ubiquitous MIMO Multifunction Digital Array Radar," Proc. 37th IEEE Asilomar Conf. on Signals, Systems, and Computers, pp , Nov
[4] N. A. Goodman and J.M. Stiles, "On Clutter Rank Observed by Arbitrary Arrays," accepted to IEEE Trans. on Signal Processing.
Here are my main references.
DSP Group, EE, Caltech, Pasadena CA

20. DSP Group, EE, Caltech, Pasadena CA
Thank you
Thank you for attending my talk.
DSP Group, EE, Caltech, Pasadena CA

21. Comparison of the Clutter Rank in MIMO and SIMO Radar
N+g(M-1)+b(L-1)
N+b(L-1)
Total dimension
NML NL
Ratio (g=N)
>
>
<
Now we compare the estimated rank in the MIMO and SIMO. The approximate clutter rank we obtained is this. And this is the result of the brennan’s rule. The total dimension is shown here. This row shows the ratio of clutter rank and the total dimension. We can see that the clutter rank in MIMO becomes a smaller portion of the total dimension. Therefore the MIMO radar receiver can null out the clutter subspace without affecting the SINR too much. Therefore the SINR performance can be improved.
The clutter rank is a smaller portion of the total dimension.
The MIMO radar receiver can null out the clutter subspace without affecting the SINR too much.
DSP Group, EE, Caltech, Pasadena CA

22. Formulation of MIMO STAP (2)
target
target
ej2p(ft-x/l)
ej2p(ft-x/l) vt q vt q
vsinq
vsinq dR dT MF … MF
f2(t)
f1(t)
f0(t) …
Transmitter : M elements
Receiver: N elements
T: Radar pulse period
Let’s first look at the target signal. The phase difference can be caused by the transmitting array and receiving array and Doppler frequency. This term corresponding to the phase difference caused by the receiving array. This term corresponding to the phase difference caused by the transmitting array. This term corresponding to the phase difference caused by the Doppler frequency. With this definition of normalized spatial frequency fst and normalized Doppler frequency fdt and we define the ratio of the spacing of transmitting antenna and receiving antenna gamma. The target signal can be simplified as this.
DSP Group, EE, Caltech, Pasadena CA

23. Fully Adaptive STAP for MIMO Radar
Solution: ^
Difficulty: The size of Ry is NML which is often large.
The convergence of the fully adaptive STAP is slow.
The complexity is high.
After formulated the target signal, we are able to formulated the fully adaptive space-time adaptive processing. The problem can be formulated like this. The target gain is constrained to be unity while minimizing the total variance. The solutions is well-known and can be expressed as this. In the fully adaptive method. Ry is estimated using this direct method and the target signal can be expressed as this. As we discussed in the last slide. But the difficulty for this method is that the size of Ry is N times M times L. It is very large. Therefore the convergence of this method is slow and the complexity is high.
DSP Group, EE, Caltech, Pasadena CA

24. Clutter Subspace in MIMO STAP: Non-integer Case
Non-integer case: g and b not integers.
Basis need for representation of clutter steering vector ci.
Data independent basis is preferred.
Less computation
Faster convergence of STAP
We study the use of prolate spheroidal wave function (PSWF) for this.
DSP Group, EE, Caltech, Pasadena CA

25. Extension to Arbitrary Array
This result can be extended to arbitrary array.
XR,n is the location of the n-th receiving antenna.
XT,m is the location of the m-th transmitting antenna.
ui is the location of the i-th clutter.
v is the speed of the radar station.
The result can be extended to arbitrary array case. For an arbitrary array, the clutter signals can be expressed as this. This ui is the location of the i-th clutter and this xrn is the location of the n-th receiving element. This v is the speed of the radar station. All these parameters are three dimensional vectors. We can also view this signal as a non-uniformly sampled data from a truncated exponential function. Therefore the same argument can be applied here. This is the result we have, the rank of Rc can be approximated by this number. If we change this xrn, xtm and v into scalar, then it reduced to the ULA case.
DSP Group, EE, Caltech, Pasadena CA

26. Review of MIMO radar: Diversity approachdR
ej2p(ft-x/l) MF …
If the transmitting antennas are far enough, the received signals of each orthogonal waveforms becomes independent. [E. Fishler et al. 04]
This diversity can be used to improve target detection.
Receiver:
DSP Group, EE, Caltech, Pasadena CA

27. Prolate Spheroidal Wave Function (PSWF) (2)
By the maximum principle, this basis concentrates most of its energy on the band [-W, W] while maintaining the orthogonality.
Only the first 2WX+1 eigenvalues are significant [D. Slepian, 1962].
The “time-and-band limited” signals can be well approximated by the linear combination of the first 2WX+1 basis elements.
By the maximum principle. The basis functions are the solution to the following optimization problem. The first basis element concentrated most of its energy on thie band from –W to W. The i-th function orthogonal to the previous basis while maximize the energy in the band. Therefore the first few basis functions concentrates most of their energies in the band from –W to W. Only the first 2WX+1 eigenvalues are significant. Therefore, it requires only 2WX+1 elements to approximate the signal concentrate in a the time-frequency region using linear combination.
DSP Group, EE, Caltech, Pasadena CA

28. Review of MIMO Radar: Degree-of-Freedom Approach
Transmitter : M elements
dT=NdR
ej2p(ft-x/l)
f2(t)
f1(t)
f0(t) q
ej2p(ft-x/l) q + = dR MF … MF …
Receiver: N elements
The clutter resolution is the same as a receiving array with NM physical array elements.
A degree-of-freedom NM can be created using only N+M physical array elements. q
Therefore, with M transmitting antennas and N receiving antennas and with dT=N times dR we are able to create NM signals. The clutter resolution of this system is the same as a receiving array with NM physical array elements. A degree-of-freedom NM can be created using only N+M physical array elements.
Virtual array: NM elements
[D. W. Bliss and K. W. Forsythe, 03]
DSP Group, EE, Caltech, Pasadena CA