radar precoding for spectrum sharing between matrix
TRANSCRIPT
Radar Precoding for Spectrum Sharing Between Matrix Completion Based MIMO
Radars and A MIMO Communication System
Bo Li and Athina Petropulu
Dec. 15, 2015
ECE Department, Rutgers, The State University of New Jersey, USA
Work supported by NSF under Grant ECCS-1408437
Outline
β’ Motivation
β’ Existing spectrum sharing approaches
β’ Background on Matrix Completion based MIMO (MIMO-MC) Radars
β’ The coexistence signal model
β’ The proposed spectrum sharing scheme
β’ Simulation results
β’ Conclusions and future directions
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Motivation
β’ Radar and communication systems overlap in the spectrum domain thus causing interference to each other.
β’ Spectrum sharing can increase spectrum efficiency.
3 Figure from DARPA Shared Spectrum Access for Radar and Communications (SSPARC)
Background
β’ Matrix completion based MIMO radars (MIMO-MC) [1] is a good candidate for reducing interference at the radar receiver [2].
Traditional MIMO radars transmit orthogonal waveforms from their transmit (TX) antennas, and their receive (RX) antennas forward their measurements to a fusion center to populate a βdata matrixβ for further processing.
Based on the low-rankness of the data matrix, MIMO-MC radar RX antennas forward to the fusion center a small number of pseudo-randomly obtained samples. Subsequently, the full data matrix is recovered using MC techniques.
MIMO-MC radars maintain the high resolution of MIMO radars, while requiring significantly fewer data to be communicated to the fusion center, thus enabling savings in communication power and bandwidth.
The sub-sampling at the RX antennas introduces new degrees of freedom for system design enabling additional interference power reduction at the radar receiver [2].
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[1] S. Sun, W. U. Bajwa, and A. P. Petropulu, IEEE TAES 2015.
[2] B. Li and A. Petropulu, IEEE ICASSP 2015.
β’ Existing Spectrum Sharing Approaches Avoiding interference by large spatial separation;
Dynamic spectrum access based on spectrum sensing;
Spatial multiplexing: MIMO radar waveforms designed to eliminate the interference
at the communication receiver [1].
β’ Our previous work [2] Spectrum sharing between a MIMO-MC radar and a MIMO communication system is achieved by
Sharing the radar waveforms with the communication system, and
Jointly designing the communication system signals and the radar system sampling
scheme.
β’ In this work A new framework for spectrum sharing between a MIMO-MC radar and a MIMO communication system is proposed
Radar precoding is jointly designed with the communication codewords to maximized
the radar SINR while meeting certain rate and power constraints at the
communication system.
Only the radar precoder is shared with the communication system, rather than the
radar waveforms, which preserves the radar waveform confidentiality.
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[1] A. Khawar, A Abdel-Hadi, and T.C. Clancy, IEEE DySPAN 2014.
[2] B. Li and A. Petropulu, IEEE ICASSP 2015.
Introduction to Matrix Completion MIMO Radars (MIMO-MC)
β’ The received data at the radar receivers can be expressed as
ππ = ππΊππππ +ππ β πππ +ππ
π: βππ‘,π ΓπΎ, π: βππ,π ΓπΎ, transmit/receive manifold matrices;
πΊ: βπΎΓπΎ, diagonal matrix contains target reflection coefficients;
π: βππ‘,π ΓπΏ, coded MIMO radar waveforms, taken to be orthonormal;
π: βππ‘,π Γππ‘,π , the radar precoding matrix; π β ππΊππ;
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Notation
ππ‘,π # of radar TX antennas
ππ,π # of radar RX antennas
πΎ # of targets
πΏ Length of waveform
ππ Additive noise
β¦ β¦
TX antennas RX antennas
target
Fusion center
π 1 π‘ π ππ‘,π π‘ π¦1 π‘ π¦ππ,π
π‘
β’ Matrix πππ is low rank if ππ,π and πΏ >> πΎ.
β’ Random subsampling is applied to each receive antenna. The matrix formulated at the fusion center can be expressed as:
π β ππ = π β πππ + π βππ
where π is a matrix with binary entries, whose "1"s correspond to sampling times at the RX antennas, and β denotes Hadamard product.
β’ Matrix completion can be applied to recover πππ using partial entries of ππ if: πππ has low coherence;
π has large spectral gap.
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1 0 00 0 10 1 1
1 0 00 0 11 1 0
1 1 00 0 10 1 0
1 0 00 0 10 1 0
0 0 11 0 00 1 1
0 1 11 0 00 0 1
ππ,π Γ πΏ
π
Subsampling rate
π = π 0/πΏππ,π
β¦ β¦
TX antennas RX antennas
target
β¦
Random Sampling
Controller
100100110 001001001 011110010 β¦β¦β¦β¦β¦β¦.
π 1 π‘ π ππ‘,π π‘ π¦1 π‘ π¦ππ,π
π‘
Parameters Radar System Communication System
Carrier Freq. (ππ) 3550 MHz 3550 MHz
Baseband Bandwidth (π€) 0.5 MHz 0.5 MHz [3]
Sub-pulse/Symbol duration (ππ) 2 us 2 us
Transmit power 750kW [1] 790 W [1]
Range resolution π/(2*π€) = 300m [2]
Pulse repetition freq. (PRF) 1 kHz
Unambiguous range π/(2*PRF)= 150 km
Symbols per pulse (πΏ) 128
Duty cycle 25%
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β¦ β¦
β¦ β¦
Collocated MIMO radar
Comm. TX Comm. RX
π
π2 π1
[1] F. H. Sanders, et al, NTIA Tech. Rep TR-13-490, 2012.
[2] βRadar performance,β Radtec Engineering Inc., [online] (Accessed:July 2015).
[3] Telesystem Innovations, LTE White paper, 2010.
β’ Consider a MIMO communication system which coexists with a MIMO-MC radar system using the same carrier frequency.
β’ Assumptions: Flat fading, narrow band radar and comm. signals;
Block fading: the channels remain constant for πΏ symbols;
Both systems have the same symbol rate;
The Coexistence Signal Model
β’ The received signals at the MIMO-MC radar and communication RX are
ππ β π²π π = ππ β πππ¬ π + πππΌ2ππ2π± π + π°π π ,
π²πΆ π = ππ± π + πππΌ1π π1ππ¬ π + π°πΆ π , βπ β βπΏ+,
where
π is the sampling time instance, ππ is the π-th column of π ;
π: βππ‘,πΆΓππ,πΆ, the communication channel;
ππ: βππ‘,π Γππ,πΆ, the interference channel from the radar TX to comm. RX;
ππ: βππ‘,πΆΓππ,π , the interference channel from the comm. TX to radar RX;
π¬(π) and π±(π): transmit vector by radar and communication system;
πππΌ1π and πππΌ2π: random phase jitters
We do not make any assumption on πΌππ.
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β¦ β¦
β¦ β¦
Collocated MIMO radar
Communication TX Communication RX
π
π2 π1
β’ Grouping πΏ samples together, we have
π β ππ = π β πππ + π2ππ²2 +ππ ,
ππΆ = ππ + π1πππ²1 +ππΆ ,
where π²π = diag πππΌπ1 , β¦ , πππΌππΏ , π β {0,1}.
β’ We consider radar precoding; the precoding matrix π is employed and shared with the communication system We take π to be a random orthonormal matrix;
Communication codewords are circularly symmetric complex Gaussian
π± π ~ππ© 0,ππ₯ , βπ;
β’ Matrices π and ππ₯ are jointly designed to maximize the SINR at the MIMO-MC radar receiver, while maintaining
certain communication system rate and power constraints.
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The Proposed Spectrum Sharing Method
β’ For the MIMO communication system: The total TX power of the communication TX antennas equals
πΌ Tr πππ» = πΏTr(ππ₯).
The interference plus noise covariance is given as
ππ€ β π1ππΌ π¬ π π¬π» π ππ»π1π» + ππΆ
2π
= π1π½π1π» + ππΆ
2π,
where π½ = πππ»/πΏ. The second equality holds because the entries of π can be
approximated by i.i.d. Gaussian random variables with distribution π© 0, 1 πΏ ,
if ππ‘,π = πͺ(πΏ lnπΏ ) [1].
The communication rate achieved, also a lower bound of the channel capacity,
is given by
C ππ₯, π½ β log2|π + ππ€β1πππ₯π
π»|.
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[1] T. Jiang, Annals Prob. 2006.
β’ For the MIMO-MC radar: The total interference power (TIP) exerted at the RX antennas equals
TIP β πΌ Tr π2ππ²2π²2π»ππ»π2
π» = πΏTr(π2ππ₯π2π»).
Recall that only partial entries of ππ are forwarded to the fusion center, which
implies that only a portion of TIP affects the MIMO-MC radar.
The effective interference power (EIP) to MIMO-MC radar is given as:
EIP β πΌ Tr π β π2ππ²2 π β π2ππ²2π»
= Tr(βπ2ππ₯π2π»)
where β β βππΏπ=1 and βπ= diag ππ . We note that the EIP is a re-weighted
version of the TIP.
Similarly, the effective signal power (ESP) of the target echo signal forwarded to
the fusion center equals
ESP β Tr(βππ½ππ»)
We assume that π is given a priori. In practice, such information could be
obtained in various ways, e.g., in tracking applications, the parameter estimates
obtained from previous tracking cycles.
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β’ If the MIMO-MC radar shares its random sampling scheme with the communication system, the spectrum sharing problem can be formulated as:
P1 maxππ₯,π½ ESINR β‘Tr βππ½ππ»
Tr βπ2ππ₯π2π» +πππ€π
s.t. LTr(ππ₯) β€ ππΆ , LTr(π½) β€ ππ ,
C ππ₯, π½ β₯ πΆ, ππ₯ β½ 0,π½ β½ 0.
where ππ€π = πΏππ,π ππ 2.
β’ Problem (P1) is non-convex w.r.t. both ππ₯ and π½. A solution can be obtained via alternating optimization.
β’ Fixing π½, the ππ₯ sub-problem is given as
PR minππ₯β½0Tr π2π»βπ2ππ₯
s.t. C ππ₯, π½ β₯ πΆ, LTr(ππ₯) β€ ππΆ
The above problem is convex w.r.t. ππ₯ and can be solved efficiently using the
interior point method.
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β’ Fixing ππ₯, the π½ sub-problem is given as
PΞ¦ minπ½β½0Tr ππ»βππ½
s.t. C ππ₯, π½ β₯ πΆ, LTr(π½) β€ ππ
We can express πΆ(ππ₯, π½) as follows:
πΆ(ππ₯, π½) = log2|π1π½π1π» + π π₯| β log2|π1π½π1
π» + ππΆ2π|,
where π π₯ β ππΆ2π + πππ₯π
π». C ππ₯, π½ β₯ πΆ is a non-convex constraint.
To overcome the non-convexity, an auxiliary πΏ is introduced by transforming
PΦ into the following problem:
πΦΨ maxπ½βͺ°0
Tr π«ππ½ππ» , s. t. πΏππ(π½) β€ ππ ,
log2 π1π½π1π» + π π₯ +max
πΏβͺ°0log2 πΏ
βTr((π1π½π1π» + ππΆ
2π)πΏ) + ππ,πΆ β₯ πΆ
Again, alternating optimization is applied as an inner iteration. During the π-
th outer alternating iteration, let (π½ππ , πΏππ) be the variables at the π-th
inner iteration.
One inner iteration is given as follows
πΏππ = (π1π½π(πβ1)π1
π» + ππΆ2π)β1
πΞ¦
β² π½ππ = argπππ₯π½βͺ°0
Tr π«ππ½ππ» , s. t. πΏTr(π½) β€ ππ ,
log2|π + π1π»(π π₯)
β1π1π½| β Tr π1π»πΏπππ1π½ β₯ πΆβ²,
where πΆβ² is a constant w.r.t. π½. (πΞ¦β² ) is convex and can be solved efficiently.
β’ The proposed spectrum sharing algorithm can be summarized as
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Simulations β’ MIMO-MC radar with half-wavelength uniform linear TX&RX arrays transmit random
orthonormal waveforms. Two far-field targets at angles Β±60β.
β’ Entries in π are i.i.d. and πππ βΌ ππ© 0,1 ; Entries in π1 and π2 are i.i.d. ππ©(0,0.01).
β’ πΏ = 32, Οπ 2 = ΟC
2 = .01, πΆ = 20bits/symbol, ππΆ = πΏππ‘,πΆ (the power is normalized by the power of radar waveform).
β’ The obtained ππ₯ is used to generate π₯(π) = ππ₯1/2
randn(ππ‘,πΆ , 1).
β’ The TFOCUS package is used for matrix completion at the radar fusion center.
β’ ESINR and MC relative recovery error ( πππ β πππ πΉ
πππ πΉ ) are used as the
performance metrics.
β’ Comparing methods include
Method #1: no radar precoding, i.e., π = πΏππ /ππ‘,π π, plus βselfish communicationβ, where
the communication system minimizes the TX power to achieve certain rate without any
concern about the interferences it exerts to the radar system.
Method #2: no radar precoding, but ππ₯ being designed to minimize the interferences it exerts
to the radar system while achieving certain communication rate.
Method #3: our previous approach where π is shared with the communication receiver, and
there is no radar precoding. 16
Simulations
Figure 1. Spectrum sharing under different sub-sampling rates. ππ‘,π = 4,ππ,π = 8,ππ‘,πΆ = 8,ππ,πΆ = 4,
ππ = 10πΏππ‘,π , πΆ = 20bits/symbol.
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Simulations
Figure 2. Spectrum sharing under different radar transmit power budget. ππ‘,π = 4,ππ,π = 8,ππ‘,πΆ = 8,ππ,πΆ = 4,
π = 0.8, πΆ = 20bits/symbol.
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Conclusions
β’ We have investigated a new framework for spectrum sharing between a MIMO communication system and a MIMO-MC radar system.
β’ Spectrum sharing is achieved by joint design of the radar precoding matrix and the communication codeword covariance matrix.
β’ Simulation results show that, the radar precoder plays a key role in improving the radar SINR and matrix completion recovery accuracy over previous approaches.
β’ Potential future directions include
spectrum sharing problem with targets distributed across different range bins
spectrum sharing in an environment with clutter.
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Thank you!