Design of Multi-Carrier MIMO Radar Array for DOA ...

Design of Multi-Carrier MIMO Radar Array for DOA Estimation Michael Ulrich, Yinglai Yang and Bin Yang Institute of Signal Processing and System Theory, University of Stuttgart, Germany

Example for N = 3 Rx and C = 2 carriers: △ △ carrier 1:  carrier 2: 

△ 

Formulation: arg min η · CRBux + (1 − η) · CRBuy PTx,PRx,f

s.t.

px λ

This work: 0 1 2  Design approach for • antenna positions • carrier frequencies  Cramer-Rao bound as design criterion, sidelobes as contraint

f i ∈ Rf ,

pTx,i ∈ RTx,

pRx,i ∈ RRx,

constraint for maximum sidelobe height PSLLmax



additional constraints for frequency bands (Rf ), array size (RTx, RRx) and antenna spacing δmin

2. Signal Model and Cramer-Rao Bound

2π B = f ⊗ Q, c Q = [P, −2 1M N ] , P = PTx ⊗ 1N + 1M ⊗ PRx, θ = [ux, uy , r]T , ∗ ρ=α ⊙α

Vector containing (pulse compressed) measurements of all C × M × N channels Parameter vector θ u Electrical angles (DOA) r Range α Complex amplitude f Vector of C carrier frequencies PT x Matrix of M Tx antenna physical positions PRx Matrix of N Rx antenna physical positions additive white Gaussian noise with n covariance σ 2I

The CRB of the two electrical angles ux and uy for a planar MC-MIMO
−1 array is κ 2 γxx − γxy /γyy CRBux = γf
−1 κ 2 γyy − γxy /γxx CRBuy = γf     2 1 σ c 2 γxx γxy 2 with κ= , γf = ||f || , = PT P γxy γyy 2 ||ρ||2 2π ⇒ The CRB is  independent of range and DOA  inversely proportional to the SNR

is non-convex



has a low dimensionality of C + 2 · (M + N ) parameters for a planar array of C × M × N different channels

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r in ramb

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1

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0 ux

β(θ0 , θ)

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can be solved using the Augmented Lagrangian Genetic Algorithm (ALGA)

The multidimensional (range-DOA) ambiguity function must be calculated



use a grid to locate high sidelobe and refine using local optimization



reduce computational cost by considering unambiguous range ramb and DOA

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0.1 0 ux

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—-– —-– —-–

optimized MC-MIMO optimized SC-MIMO state-of-the-art design

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0 ux

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The optimized MC-MIMO has a narrow main beam and a definable PSLL. u0x = −1

Multi-carrier array optimized with ALGA

To evaluate the constraint on the sidelobes: 

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Comparison of optimization metrics γf · γxx and PSLL: γf · γxx PSLL design rule of [1] 0.5848 · 1021 0.55 optimized SC-MIMO 0.3352 · 1021 0.20 optimized MC-MIMO 1.6690 · 1021 0.20 RMSE over SNR. The optimization of MC-MIMO is able to improve the performance of DOA estimation: 100 —-– —-– —-– —– ---

10−1 CRB, RMSE

x

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4. Examples and Evaluations

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optimized MC-MIMO optimized SC-MIMO state-of-the-art design RMSE √ CRB

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√

x = [x111, ..., x11N , ..., x1M N , ..., xCM N ] = α ⊙ exp(jBθ) + n

T

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This optimization problem Assumptions:  A single stationary target in far field (colocated MIMO)  Orthogonal waveforms Signal model after sampling and pulse compression:

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Multi-carrier (MC) array with state-of-the-art design Single-carrier (SC) array optimized with ALGA rule [1]

minimize CRB (η for weighting ux and uy )



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0 −1

min(δ Tx) ≥ δmin, min(δ Rx) ≥ δmin, PSLL ≤ PSLLmax.



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β(u0x , ux )



3. Array Design Approach

1

β(θ0 , θ)



Multi-Carrier (MC) MIMO radar uses C carrier frequencies, M transmitters and N receivers to obtain a virtual array of size C × M × N An MC-MIMO array design for high-accuracy direction of arrival (DOA) estimation is possible with low hardware effort Problem: The DOA and range are coupled in the estimation, hence an array design considering only DOA is not sufficient

β(θ0, θ) = ||a(θ0) a(θ)||

1

2

r in ramb



H

r in ramb

1. Motivation

Optimization results:

β(θ0, θ)

Noise-free ambiguity function for range and DOA:

Example design:  

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C = 2, M = 4, N = 4 linear array (ux only) frequencies: f = [f0, f1] with f1 ∈ [0.8, 0.95]f0



aperture 100 fc0 for Tx and Rx



PSLLmax = 0.2



−10

T

32 different channels, but only 7 optimization parameters

This work was supported by the German Federal Ministry of Education and Research, Grant No. 13N13480.

25th European Signal Processing Conference Eusipco 2017, August 28 - September 2, 2017, Kos, Greece





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10 15 SNR in dB

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Multiple carriers can provide a lower CRB at the same sidelobe level (PSLL) compared to traditional (single-carrier) MIMO radar This leads to a lower mean-squared error above the threshold in the DOA estimation!

[1] M. Ulrich and B. Yang, “Multicarrier MIMO Radar: A Concept of Sparse Array for Improved DOA Estimation,” IEEE Radar Conference, pp. 1-5, May 2016