Spectrum Sharing Between MIMO-MC Radars and Communication Systems

Transcription

1 Spectrum Sharing Between MIMO-MC Radars and Communication Systems Bo Li ands Athina Petropulus ECE Department, Rutgers, The State University of New Jersey Work supported by NSF under Grant ECCS and Raytheon. Asilomar November 07, 2016 Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 1 / 24

2 Outline 1 Introduction and Motivations 2 MIMO Radars with Matrix Completion 3 Spectrum Sharing Between MIMO-MC Radars and a MIMO Communication System 4 Conclusions & Future Work Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 2 / 24

3 Outline 1 Introduction and Motivations 2 MIMO Radars with Matrix Completion 3 Spectrum Sharing Between MIMO-MC Radars and a MIMO Communication System 4 Conclusions & Future Work

4 Bottleneck of Modern Radar Systems Reliable surveillance requires collection, communication and fusion of vast amounts of data from various antennas. Spectral Bandwidth consuming!! Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 3 / 24

5 Bottleneck of Modern Radar Systems Reliable surveillance requires collection, communication and fusion of vast amounts of data from various antennas. Spectral Bandwidth consuming!! Radar and communication systems may coexist and overlap in the spectrum. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 3 / 24

6 The Spectrum Crunch The widespread adoption of smartphones, laptops, and tablets has dramatically increased the need for more spectrum. Figure 1: Global mobile traffic per month(src: Cisco, 2013) Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 4 / 24

7 Motivation of Radar and Comm. Spectrum Sharing [Lackpour et al, 11], [DARPA SSPARC program, 12], [FCC & NTIA reports, 12-15] Radar and communications jointly consume most of the spectrum below 6 GHz. Spectrum held by commercial operators have heavy utilization in urban areas; Spectrum held by federal agencies have low average utilization. Figure 2: Spectrum utilization (0-6 GHz) in downtown Berkeley(src: UC Berkeley, 2007) Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 5 / 24

8 Motivation of Radar and Comm. Spectrum Sharing [Lackpour et al, 11], [DARPA SSPARC program, 12], [FCC & NTIA reports, 12-15] Radar and communications jointly consume most of the spectrum below 6 GHz. Spectrum held by commercial operators have heavy utilization in urban areas; Spectrum held by federal agencies have low average utilization. Figure 2: Spectrum utilization (0-6 GHz) in downtown Berkeley(src: UC Berkeley, 2007) FCC and NTIA have published the rule and order to share spectrum band MHz, previously held by federal agencies, with commercial wireless operators. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 5 / 24

9 Existing Approaches Existing spectrum sharing approaches basically include three categories. Avoiding interference by large spatial separation. Figure 3: Shipborne radar exclusion zones (src: NTIA 15) Dynamic spectrum access based on spectrum sensing. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 6 / 24

10 Existing Approaches Existing spectrum sharing approaches basically include three categories. Avoiding interference by large spatial separation. Figure 3: Shipborne radar exclusion zones (src: NTIA 15) Dynamic spectrum access based on spectrum sensing. Spatial multiplexing enabled by the multiple antennas at both the radar and communication systems Radar waveforms were projected to the interference channel null space [Sodagari et al, 12]. Spatial filtering at the radar receiver was used to reduce the communication interference [Deng et al, 13]. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 6 / 24

11 Joint Transmit Designs for Co-existence of MIMO Wireless Communications and Sparse Sensing Radars in Clutter Limitations on Existing Spatial Multiplexing Approach Prior work only addressed the interference in one direction and ignored the other. Clutter-free environments were assumed. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 7 / 24

12 Joint Transmit Designs for Co-existence of MIMO Wireless Communications and Sparse Sensing Radars in Clutter Limitations on Existing Spatial Multiplexing Approach Prior work only addressed the interference in one direction and ignored the other. Clutter-free environments were assumed. Our Contributions We propose cooperative spectrum sharing approaches which address the mutual interference simultaneously. The proposed spectrum sharing framework can handle strong clutter. The proposed spectrum sharing framework support MIMO radars with matrix completion. We further show that MC can even benefit spectrum sharing. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 7 / 24

13 Outline 1 Introduction and Motivations 2 MIMO Radars with Matrix Completion 3 Spectrum Sharing Between MIMO-MC Radars and a MIMO Communication System 4 Conclusions & Future Work

14 MIMO Radar Systems Multiple-Input-Multiple-Output (MIMO) Radars: MIMO radar system is a novel radar system of multiple antennas. Each transmit antenna radiates an independent waveform. Each receive antenna can apply a matched filter bank to extract the target information. Target TX antennas RX antennas Fusion Center MIMO Radars gain superiority over traditional radars High spatial resolution. M tm r spatial degrees of freedom can be achieved with M t TX and M r RX antennas. Omnidirectional illumination. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 8 / 24

15 Matrix Completion What is Matrix Completion? The task of filling in the missing entries of a partially observed matrix. For a low-rank matrix M, matrix completion can be done by solving relaxed nuclear norm optimization problem [Candès & Recht, 2009], [Candès & Tao, 2010] min s.t. X X ij = M ij, (i, j) Ω where Ω denotes the set of observed entries. M can be perfectly recovered with high probability, given that M is incoherent, The observed entries are sampled uniformly at random and its size is large enough. It is also shown that matrix completion is stable under noisy observations [Candès & Plan, 2010]. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 9 / 24 (1)

16 MIMO-MC: MIMO Radars with Matrix Completion [Kalogerias & Petropulu, 14] [Sun & Petropulu, 15] [Li & Petropulu] For a traditional MIMO radar, the noisy data matrix at the radar fusion center equals Y R = V r ΣV T t S + W R DS + W R, (2) Target TX antennas RX antennas Fusion Center Figure 4: Traditional MIMO Radar Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 10 / 24

17 MIMO-MC: MIMO Radars with Matrix Completion [Kalogerias & Petropulu, 14] [Sun & Petropulu, 15] [Li & Petropulu] For a traditional MIMO radar, the noisy data matrix at the radar fusion center equals Y R = V r ΣV T t S + W R DS + W R, (2) Target TX antennas RX antennas Fusion Center Figure 4: Traditional MIMO Radar Figure 5: MIMO-MC Radar A new MIMO radar with matrix completion (MIMO-MC) was recently proposed based on the fact that the data matrix DS C M r,r L is low rank. Random sub-sampling is applied to each receive antenna. Only the sub-sampled data are sent to the fusion center, where the full data matrix is recovered with high accuracy. Save power and spectral bandwidth! Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 10 / 24

18 MIMO-MC Radars Using Random Unitary Waveforms Our Contributions We propose a MIMO-MC radar framework which supports waveform agility, i.e., waveforms are secure and easy to generate, and can operate under low SINR conditions, e.g., strong clutter and interference. To achieve this, we propose to use random unitary waveform matrix S and introduce radar precoding P Ω Y R = Ω (V r ΣV T t PS + W R ) Ω (M + W R ), (3) where Ω denotes the sub-sampling matrix and denotes Hadamard product. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 11 / 24

19 MIMO-MC Radars Using Random Unitary Waveforms Our Contributions We propose a MIMO-MC radar framework which supports waveform agility, i.e., waveforms are secure and easy to generate, and can operate under low SINR conditions, e.g., strong clutter and interference. To achieve this, we propose to use random unitary waveform matrix S and introduce radar precoding P Ω Y R = Ω (V r ΣV T t PS + W R ) Ω (M + W R ), (3) where Ω denotes the sub-sampling matrix and denotes Hadamard product. Remarks We show that the coherence of M is bounded by a small constant. A random unitary matrix S can be easily obtained from any random Gaussian matrix by performing the Gram-Schmidt orthogonalization. The coherence of M is independent of P. This means that we can design P, without affecting the incoherence property of M, for the purpose of interference suppression. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 11 / 24

20 Outline 1 Introduction and Motivations 2 MIMO Radars with Matrix Completion 3 Spectrum Sharing Between MIMO-MC Radars and a MIMO Communication System 4 Conclusions & Future Work

21 The Coexistence Signal Model Consider a MIMO communication system which coexists with a MIMO-MC radar: The radar searches in particular directions of interest for targets. There are multiple point scatters (clutter or interfering objects) with known angles and RCS. One pair of communication transmitter and receiver communicate through a wireless MIMO channel. Collocated MIMO radar Communication TX Communication RX Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 12 / 24

22 The Coexistence Signal Model Consider a MIMO communication system which coexists with a MIMO-MC radar: The radar searches in particular directions of interest for targets. There are multiple point scatters (clutter or interfering objects) with known angles and RCS. One pair of communication transmitter and receiver communicate through a wireless MIMO channel. Assumptions: Flat fading channel, narrow band radar and comm. signals; Block fading: the channels remain constant for L symbols; The two systems are time-synchronized and have the same symbol rate; The two systems cooperate on channel estimation and feedback. Collocated MIMO radar Communication TX Communication RX Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 12 / 24

23 The Coexistence Signal Model The received signals at the MIMO-MC radar and communication RX are Radar fusion center: Ω Y R = Ω ( ) DPS }{{} + CPS + G }{{ 2XΛ } 2 + W R, (4a) signal interference Communication receiver: }{{} noise Y C = }{{} HX + G 1PSΛ 1 + W }{{} C, (4b) signal interference }{{} noise where C K c k=1 βc k vr (θc k )vt t (θc k ) is the clutter response matrix. X [x(1),..., x(l)]: the code-book of the communication system. We consider the circularly symmetric complex Gaussian codewords x(l) CN (0, R xl ). H C M r,c M t,c : the communication channel; G 1 C M r,c M t,r : the interference channel radar communication RX antennas; G 2 C M r,r M t,c : the interference channel communication TX antennas radar. Λ 1 and Λ 2 are diagonal matrices with entries e jα il, denoting the random phase offset between the MIMO-MC radar and the communication system. {α 1l } L l=1 are distributed as N (0, σα 2 ), where σ2 α is small [Razavi, 96]. W C and W R denote the additive noise. Channels H, G 1 and G 2 are estimated and feedback to the radar and communication transmitter. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 13 / 24

24 Design Overview The radar and communication systems cooperate with each other in order to Optimize certain figure of merit of the MIMO-MC radar While maintaining certain communication constraints. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 14 / 24

25 The Mutual Interference At the communication receiver, the interference plus noise covariance is given as R Cin = E{G 1Ps l s H l P H G H 1 } + σ 2 C I G 1ΦG H 1 + σ 2 C I. (According to [Jiang, 06], the entries of S can be approximated by distribution CN (0, 1/L). Φ PP H /L is positive semidefinite.) Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 15 / 24

26 The Mutual Interference At the communication receiver, the interference plus noise covariance is given as R Cin = E{G 1Ps l s H l P H G H 1 } + σ 2 C I G 1ΦG H 1 + σ 2 C I. (According to [Jiang, 06], the entries of S can be approximated by distribution CN (0, 1/L). Φ PP H /L is positive semidefinite.) MIMO-MC radar sub-sampling actually modulates the interference channel G 2. The effective channel during the l-th symbol duration is G 2l l G 2 with l = diag(ω l ). Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 15 / 24

27 The Mutual Interference At the communication receiver, the interference plus noise covariance is given as R Cin = E{G 1Ps l s H l P H G H 1 } + σ 2 C I G 1ΦG H 1 + σ 2 C I. (According to [Jiang, 06], the entries of S can be approximated by distribution CN (0, 1/L). Φ PP H /L is positive semidefinite.) MIMO-MC radar sub-sampling actually modulates the interference channel G 2. The effective channel during the l-th symbol duration is G 2l l G 2 with l = diag(ω l ). The interference channel is time-varying! Dynamic/adaptive communication transmission is optimal, i.e., distinct R xl for i = 1,..., L. The effective interference power from communication transmitter to MIMO-MC radar: L l=1 ( ) Tr G 2l R xl G H 2l Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 15 / 24

28 The Mutual Interference At the communication receiver, the interference plus noise covariance is given as R Cin = E{G 1Ps l s H l P H G H 1 } + σ 2 C I G 1ΦG H 1 + σ 2 C I. (According to [Jiang, 06], the entries of S can be approximated by distribution CN (0, 1/L). Φ PP H /L is positive semidefinite.) MIMO-MC radar sub-sampling actually modulates the interference channel G 2. The effective channel during the l-th symbol duration is G 2l l G 2 with l = diag(ω l ). The interference channel is time-varying! Dynamic/adaptive communication transmission is optimal, i.e., distinct R xl for i = 1,..., L. The effective interference power from communication transmitter to MIMO-MC radar: L l=1 ( ) Tr G 2l R xl G H 2l The interference depends on both {R xl } and Ω. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 15 / 24

29 The Design Variables and Constraints Design Variables The radar transmit precoder P, in terms of Φ; The communication transmit covariance matrices R xl for i = 1,..., L; Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 16 / 24

30 The Design Variables and Constraints Design Variables The radar transmit precoder P, in terms of Φ; The communication transmit covariance matrices R xl for i = 1,..., L; The MIMO-MC radar sub-sampling scheme Ω (Original result [Candès, et al, 10]) Ω corresponds to sampling uniformly at random. (Recent result [Srinadh, et al, 14]) Ω is required to have a large spectral gap. The spectral gap of Ω remains unaltered under row and column permutations but the interference changes. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 16 / 24

31 The Design Variables and Constraints Design Variables The radar transmit precoder P, in terms of Φ; The communication transmit covariance matrices R xl for i = 1,..., L; The MIMO-MC radar sub-sampling scheme Ω (Original result [Candès, et al, 10]) Ω corresponds to sampling uniformly at random. (Recent result [Srinadh, et al, 14]) Ω is required to have a large spectral gap. The spectral gap of Ω remains unaltered under row and column permutations but the interference changes. Design Constraints The power budget at the communication transmitter: L l=1 Tr(R xl) P t, The requirement on the average communication rate achieved during the L symbol periods C avg({r xl }) 1 L L log 2 I + R 1 CinHR xl H H C (5) l=1 Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 16 / 24

32 The Design Objective The signal-to-interference-plus-noise ratio (SINR) is a widely used radar performance metric. MIMO-MC radar not only sub-samples the communication interference but also the echoes from targets and clutter. Only the sampled target signal and sampled interference determine the matrix completion performance. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 17 / 24

33 The Design Objective The signal-to-interference-plus-noise ratio (SINR) is a widely used radar performance metric. MIMO-MC radar not only sub-samples the communication interference but also the echoes from targets and clutter. Only the sampled target signal and sampled interference determine the matrix completion performance. Based on this observation, we define the radar effective SINR (ESINR) as follows effective signal power ESINR effective interf. plus noise power mtr (ΦD t) = mtr (ΦC t) + L l=1 Tr ( ). G 2l R xl G H 2l + mσ 2 R m is the number of entries sampled by the MIMO-MC receiver; D t = K k=1 σ2 β 0 vt (θ k)vt T (θ k), C t = K c k=1 σ2 β k c vt (θc k )vt t (θc k ); σβ 2 0, σβ 2 k c, and σr 2 denote the variances of target, clutter and noise. (6) Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 17 / 24

34 Put Everything Together: Cooperative Spectrum Sharing Cooperation & Knowledge Shared Cooperate on estimation of G 1, G 2. Share H, G 1, and G 2 with the central processor. Jointly design of Φ, Ω and {R xl }. The cooperative approach for spectrum sharing problem can be formulated as (P 1) max ESINR ({Rx}, Ω, Φ), {R xl } 0,Φ 0,Ω s.t. C avg({r xl }, Φ) C, L Tr (R xl ) P C, LTr (Φ) P R, l=1 Tr (ΦV k ) ξtr(φ), k N + K, ξ > 1 Ω is Boolean and has large spectral gap Problem (P 1) is non-convex w.r.t. optimization variable triple ({R x}, Ω, Φ). A local solution can be found by using alternating optimization in a loop. Solving {R n xl } with fixed Φn 1 and Ω n 1 Solving Φ n with fixed {R n xl } and Ωn 1 Solving Ω n with fixed {R n xl } and Φn (7a) (7b) (7c) (7d) Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 18 / 24

35 Simplifications and Comparisons Constant-Rate Communication Transmission A sub-optimal transmission approach of constant rate, i.e., R xl R x, l N + L, has a lower implementation complexity. (P 1 ) max R x 0,Φ 0 ESINR s.t. C(R x, Φ) C, Tr (ΦD t) Tr (ΦC t) + Tr ( l G 2 R x G H ), 2 /m + σ 2 R LTr (R x ) P C, LTr (Φ) P R, Tr (ΦV k ) 0, k N + K, Pros: Only one matrix variable lower complexity Cons: Cannot adapt to the variation of the effective interference channel R xl performance degradation Traditional MIMO Radar: a special case when Ω = 1 Cons: There is no more modulation of the interference channel G 2 due to sub-sampling. Smaller interference channel null space for multiplexing performance degradation Pros: Constant-rate communication transmission is sufficient lower complexity Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 19 / 24

36 Numerical Results: Radar TX Beampattern and MUSIC Spectrum Radar TX Beampattern (db) Proposed Precoding Scheme Uniform Precoding Scheme Null Space Projection Scheme Spatial Spectrum in db Azimuth Angle Azimuth Angle Figure 6: The radar transmit beampattern and the MUSIC spatial pseudo-spectrum. M t,r = M r,r = 16, M t,c = M r,c = 4. The true positions of the targets and clutters are labeled using solid and dashed vertical lines, respectively. CNR=30 db. MC Relative Relative RCS Precoding schemes ESINR Recovery Errors Est. RMSE Joint-design precoding 31.3dB Uniform precoding -44.3dB NSP based precoding -46.3dB Table 1: The radar ESINR, MC relative recovery errors, and the relative target RCS estimation RMSE. The simulation setting is the same as that for Fig. 6. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 20 / 24

37 Numerical Results: Constant-Rate vs Adaptive ESINR in db Constant Rate Comm. TX Adaptive Rate Comm. TX The variance of interference channel G2 MC Relative Recovery Error Constant Rate Comm. TX Adaptive Rate Comm. TX The variance of interference channel G2 Relative RCS Estimation RMSE Constant Rate Comm. TX Adaptive Rate Comm. TX The variance of interference channel G2 Figure 7: Comparison of spectrum sharing with adaptive and constant-rate communication transmissions under different levels of interference channel G 2 from the communication transmitter to the radar receiver. M t,r = 16, M r,r = M t,c = 8, M r,c = 2. Observations Adaptive method outperforms the Constant-Rate counterpart. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 21 / 24

38 Numerical Results: MIMO-MC Radars and Traditional MIMO Radars ESINR in db Traditional MIMO Radar MIMO-MC Radar (p varies) Radar Subsampling Rate p MC Relative Recovery Error Traditional MIMO Radar MIMO-MC Radar (p varies) Radar Subsampling Rate p Relative RCS Estimation RMSE 0.6 Traditional MIMO Radar MIMO-MC Radar (p varies) Radar Subsampling Rate p Figure 8: Comparison of spectrum sharing with traditional MIMO radars and the MIMO-MC radars with different subsampling rates p. M t,r = 16, M r,r = M t,c = 8, M r,c = 2. Observations The MIMO-MC radar achieves better target RCS estimation accuracy than the traditional radar if p is between 0.4 and 0.7. p < 0.4: The number of entries sampled is too small; 0.4 p 0.7: Best compromise achieved! p > 0.7: Small ESINRs for p > 0.7 introduce high distortion in MC. MIMO-MC radars can co-exist with comm. systems and achieve better target RCS estimation while saving up to 60% data samples. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 22 / 24

39 Outline 1 Introduction and Motivations 2 MIMO Radars with Matrix Completion 3 Spectrum Sharing Between MIMO-MC Radars and a MIMO Communication System 4 Conclusions & Future Work

40 Conclusions & Future Work We have proposed: A new MIMO-MC radar using random unitary waveform and supporting precoding Waveform agility and stable performance under strong interference are supported. Cooperative spectrum sharing between a MIMO-MC radar and a MIMO communication system Proposed spectrum sharing methods based on the joint design of the radar transmit precoder, the radar sub-sampling scheme, and the communication transmit covariance matrix; Provided efficient algorithms and extensive comparisons; Higher level cooperation means better performance, at the cost of complexity. MIMO radars and wireless MIMO communication coexistence is a new line of work with many interesting challenges. We propose to work on the following directions: Different design objective for radar and communication co-existence. Other design logistics would be more favorable depending on the priority. Muti-objective design is also an interesting topic. Spectrum sharing between MIMO radars and multiple MIMO communication systems. It would be very interesting to extend current work to the scenario under which MIMO radars coexist with multiple pairs of MIMO transmitters and receivers. Distributed implementation for the centralized spectrum sharing algorithm in such case is also an important topic. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 23 / 24

41 Thank You Thank You! Questions please Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 23 / 24

42 Future Work MIMO radars and wireless MIMO communication coexistence is a new line of work with many interesting challenges. We propose to work on the following directions: Different design objective for radar and communication co-existence. Other design logistics would be more favorable depending on the priority. Muti-objective design is also an interesting topic. Spectrum sharing between MIMO radars and multiple MIMO communication systems. It would be very interesting to extend current work to the scenario under which MIMO radars coexist with multiple pairs of MIMO transmitters and receivers. Distributed implementation for the centralized spectrum sharing algorithm in such case is also an important topic. The investigation of the coexistence of MIMO-OFDM radars and MIMO-OFDM communication systems. It is possible to formulate an optimization problem which optimally allocates the radar and communication resources to multiple antennas and multiple sub-carriers simultaneously. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 23 / 24

43 Matrix Incoherence and Performance Guarantee Subspace Coherence [Candès & Recht, 2009] Let U C n be an r-dimensional subspace spanned by a set of orthogonal vectors {u i C n } i=1,...,r, P U be the orthogonal projection onto U, and e i be the standard basis vector. The coherence of U is defined as µ(u) = n r max 1 i n P U e i 2 [1, n r ]. Subspace U has low coherence if the energy of u i, i = 1,..., r elements is spread out, i.e., uncorrelated with the standard basis. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 23 / 24

44 Matrix Incoherence and Performance Guarantee Subspace Coherence [Candès & Recht, 2009] Let U C n be an r-dimensional subspace spanned by a set of orthogonal vectors {u i C n } i=1,...,r, P U be the orthogonal projection onto U, and e i be the standard basis vector. The coherence of U is defined as µ(u) = n r max 1 i n P U e i 2 [1, n r ]. Subspace U has low coherence if the energy of u i, i = 1,..., r elements is spread out, i.e., uncorrelated with the standard basis. A rank-r matrix M C n 1 n 2 with SVD UΣV H is said to be incoherent with parameters µ 0 and µ 1 if (A0) max(µ(u), µ(v )) µ 0 for some µ 0 > 0 (A1) The entries of 1 i r u iv H i have magnitudes upper bounded by µ 1 r n 1 n 2 Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 23 / 24

45 Matrix Incoherence and Performance Guarantee Subspace Coherence [Candès & Recht, 2009] Let U C n be an r-dimensional subspace spanned by a set of orthogonal vectors {u i C n } i=1,...,r, P U be the orthogonal projection onto U, and e i be the standard basis vector. The coherence of U is defined as µ(u) = n r max 1 i n P U e i 2 [1, n r ]. Subspace U has low coherence if the energy of u i, i = 1,..., r elements is spread out, i.e., uncorrelated with the standard basis. A rank-r matrix M C n 1 n 2 with SVD UΣV H is said to be incoherent with parameters µ 0 and µ 1 if (A0) max(µ(u), µ(v )) µ 0 for some µ 0 > 0 (A1) The entries of 1 i r u iv H i Theorem [Candès & Recht, 2009] have magnitudes upper bounded by µ 1 r n 1 n 2 Suppose that we observe m entries of M, sampled uniformly at random. Let n = max(n 1, n 2 ). There exist constants C and c such that if m max{µ 2 1, µ1/2 0 µ 1, µ 0 n 1/4 }nrβ log n for some β > 2, the minimizer to the program of (1) is unique and equal to M with probability at least 1 cn β. It is also shown that matrix completion is stable under noisy observations [Candès & Plan, 2010]. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 23 / 24

46 Q: What is the effect of the particular type S and P on the incoherent property of M? The key step is to provide the upper bounds on µ(u) and µ(v ) for M V r ΣV T t PS: Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 24 / 24

47 Q: What is the effect of the particular type S and P on the incoherent property of M? The key step is to provide the upper bounds on µ(u) and µ(v ) for M V r ΣV T t PS: Theorem 1 (Bounds on µ(u) and µ(v )). Consider a MIMO-MC radar with S being random unitary. For any transmit precoder P such that the rank of M is K 0 K, and arbitrary transmit array geometry and target angles, the coherence of subspace V and U obeys the following with probability 1 L 2 and µ(u) K K 0 µ r 0. µ(v ) K K 0 ln L + 6 ln L K 0 µ t 0 Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 24 / 24

48 Q: What is the effect of the particular type S and P on the incoherent property of M? The key step is to provide the upper bounds on µ(u) and µ(v ) for M V r ΣV T t PS: Theorem 1 (Bounds on µ(u) and µ(v )). Consider a MIMO-MC radar with S being random unitary. For any transmit precoder P such that the rank of M is K 0 K, and arbitrary transmit array geometry and target angles, the coherence of subspace V and U obeys the following with probability 1 L 2 and µ(u) K K 0 µ r 0. µ(v ) K K 0 ln L + 6 ln L K 0 µ t 0 The incoherence parameters of M can be obtained as µ 0 max{µ(u), µ(v )} and µ 1 Kµ 0. For values of the rank K 0 greater than ln L, µ(v ) is O(1). For K = K 0 and M r,r goes to infinity, µ(u) gets close to 1. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 24 / 24

49 Q: What is the effect of the particular type S and P on the incoherent property of M? The key step is to provide the upper bounds on µ(u) and µ(v ) for M V r ΣV T t PS: Theorem 1 (Bounds on µ(u) and µ(v )). Consider a MIMO-MC radar with S being random unitary. For any transmit precoder P such that the rank of M is K 0 K, and arbitrary transmit array geometry and target angles, the coherence of subspace V and U obeys the following with probability 1 L 2 and µ(u) K K 0 µ r 0. µ(v ) K K 0 ln L + 6 ln L K 0 µ t 0 The incoherence parameters of M can be obtained as µ 0 max{µ(u), µ(v )} and µ 1 Kµ 0. Remarks For values of the rank K 0 greater than ln L, µ(v ) is O(1). For K = K 0 and M r,r goes to infinity, µ(u) gets close to 1. A random unitary matrix S can be easily obtained from any random Gaussian matrix by performing the Gram-Schmidt orthogonalization. We show that the bound on the coherence parameters of M are independent of P. This means that we can design P, without affecting the incoherence property of M, for the purpose of interference suppression. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 24 / 24

50 Alternating Iterations The Alternating Iteration w.r.t. {R xl } convex; dual decomposition can be used to reduce the computation. The Alternating Iteration w.r.t. Ω Solve Ω by searching the best permutation of an initial Ω 0 We propose to reduce the search space as follows min Ω Tr(ΩT Q) s.t. Ω (Ω 0 ), (9) Proposition 2. min Ω Tr(ΩT Q) s.t. Ω r (Ω 0 ), (10) For any Ω 0, searching for an Ω in r (Ω 0 ) can achieve the same EIP as (Ω 0 ). Further, (10) can be formulated as a linear assignment problem and solved efficiently in polynomial time O(Mr,R 3 ) using the Hungarian algorithm. The Alternating Iteration w.r.t. Φ The sequential convex programming technique and Charnes-Cooper Transformation are applied to solve Φ. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 24 / 24

51 Discussions Convergence of the Alternating Optimization The objective ESINR is nondecreasing during the alternating iterations and is upper bounded. Convergence guaranteed according to the monotone convergence theorem [Yeh, et al, 06]. Feasibility Proposition 3. If C, ξ, P C > 0, P R > 0 are chosen such that C < C max(p C ) and ξ ξ max, then (P 1 ) is feasible. C max(p C ) and ξ max are given by 1 L I C max(p C ) max log {R xl } 0 L 2 + σ 2 C HR xl H H L, s.t. Tr (R xl ) P C l=1 l=1 ξ max max ξ, s.t. Tr(ΦV k) ξtr(φ), k N + Φ 0,ξ 0 K. Rank of Φ Proposition 4. Any optimal solution of Φ has rank at most K. All rank-k solutions Φ K have the same range space. Any solution Φ K with rank less than K has range space such that R(Φ K ) R(Φ K ). Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 24 / 24

52 Simplifications and Comparisons, contd. Traditional MIMO Radar: a special case when Ω = 1 Cons: There is no more modulation of the interference channel G 2 due to sub-sampling. Smaller interference channel null space for multiplexing performance degradation Pros: Constant-rate communication transmission is sufficient lower complexity N (G 2) is empty All transmit directions would introduce non-zero interference. N (G 2l ) is nonempty There are directions that would introduce zero interference. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 24 / 24

53 Complexity Analysis Performance vs Complexity (computation and cooperation) Joint design Adaptive Rate Const. Rate Traditional spectrum sharing (P 1) (P 1) MIMO Radar Variables {R xl }, P, Ω R x, P R x, P Computation O(LMt,C 7 + Mt,R 7 + Mr,R) 3 O(Mt,C 7 + Mt,R) 7 O(Mt,C 7 + Mt,R) 7 Table 2: Different joint design based methods proposed in the dissertation. Trivial/None Null space projection Proposed Precoder P I P VV H joint-design Performance Computation Low Low High Cooperation Low level Medium level High level Ignore each other Ignore comm. interf. No clutter mitigation No clutter mitigation Table 3: Different radar precoding schemes. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 24 / 24

54 Numerical Results: Comparison of Different Levels of Cooperation ESINR in db Uniform Precoding + Selfish Comm. NSP Precoding + Selfish Comm. Uniform Precoding + Design R xl &Ω Design P + Selfish Comm. Proposed Joint Design MC Relative Recovery Error Uniform Precoding + Selfish Comm. NSP Precoding + Selfish Comm. Uniform Precoding + Design R xl &Ω Design P + Selfish Comm. Proposed Joint Design Relative RCS Estimation RMSE Uniform Precoding + Selfish Comm. NSP Precoding + Selfish Comm. Uniform Precoding + Design R xl &Ω Design P + Selfish Comm. Proposed Joint Design Radar TX Power Budget PR Radar TX Power Budget PR Radar TX Power Budget PR 10 5 Figure 9: Comparison of spectrum sharing with different levels of cooperation between the MIMO-MC radar and the communication system under different P R. M t,r = M r,r = 16, M t,c = M r,c = 4. Observations The propose joint design method significantly outperforms other non-cooperative and partial cooperative methods. Li & Petropulu (Rutgers University) Spectrum Sharing Between MIMO-MC Radars and Communication Systems 24 / 24