Low Resolution Adaptive Compressed Sensing for mmwave MIMO receivers

Transcription

1 Low Resolution Adaptive Compressed Sensing for mmwave MIMO receivers Cristian Rusu, Nuria González-Prelcic and Robert W. Heath

2 Motivation 2

3 Power consumption at mmwave in a MIMO receiver Power at 60 GHz 1GHz BW LNA RF Chain ADC 20mW 40mW 200 mw mw Baseband processing Baseband Precoding LNA RF Chain ADC u Infeasible to dedicate a separate RF chain and ADC for each antenna Alternative mmwave MIMO architectures are needed* * R. Heath, N. Gonzalez-Prelcic, S. Rangan, A. Sayeed, W. Roh Overview of signal processing techniques for millimeter wave MIMO systems, submitted IEEEE JSTSP

4 Low resolution mmwave MIMO receiver ` Transmit Processing H RF Chain u Output is heavily quantized 1-bit b-bit ADC <<10mW 1 bit, 240 Gs/s much less at 4 Gs/s RF 1-bit b-bit Chai ADC n ª Treating as Gaussian noise may not be a good approximation u Reasonable capacity is possible at moderate SNR values* u May have higher baseband complexity Baseband Processing u Channel estimates are useful at TX/RX for configuring the precoders/combiner " " ` With few bits 2b bits per complex dimension y Q phs ` vq With one sign bit p p ` `q q y sign phs ` vq threshold in real / imaginary *Jianhua Mo and R. W. Heath, Jr., Capacity Analysis of MIMO Systems with One-Bit Quantization', IEEE Trans. on Signal Processing, October 2015 See also extensive work by research groups led by U. Madhow, J. Nossek, G. Fettweis, C. Studer, G. Kramer, and O. Dabeer and others y 4

5 Low resolution channel estimation at mmwave Channel estimation at mmwave Channel seen through analog precoding Low SNR before beamforming and with few-bits ADCs Coarsely quantized received signal Large channel matrices u Estimation error with few-bit ADCs decreases at best quadratically per bit* u It also decreases with the sparsity of the channel* Channel estimation is challenging at mmwave, more with a few-bits RX *L. Jacques, J. Laska, P. Boufounos, and R. Baraniuk, Robust 1-bit compressive sensing via binary stable embeddings of sparse vectors, EEE Trans. Inf. Theory, vol. 59, no. 4, pp , April

6 Problem formulation 6

7 MmWave narrowband channel model θ Τ,1 θ R,1 θ Τ,2 θ R,2 N r xn p Physical spatial model N t xn p Array steering/response vectors evaluated at the AoAs/AoDs u Few scattering clusters at mmwave sparsity u We assume N r =N t =G path gains Virtual model Spatial resolution θ T,1 θ T,2 θ T,3 θ T,4 θ T,5 θ T,6 θ R,1 θ R,2 θ R,3 Virtual angles fixed a priori k-sparse matrix of size GxG N r xg N r xg Dictionaries of TX/RX array steering/response vectors with quantized angles Dictionaries are the DFT matrices A. M. Sayeed, "Deconstructing multiantenna fading channels," in IEEE Trans SP, vol.50, no.10, pp , Oct

8 Narrowband received signal model Training sequence Transmit Processing N t T P C N tˆp H Received signal N r RF Chain RF Chai n 1-bit b-bit ADC 1-bit b-bit ADC Y Q pht ` Nq Y Q pa r H v A t T ` Nq Baseband Processing Choose as DFT matrices (G=N t =N r ) 8

9 Channel estimation with low resolution ADCs as a sparse recovery problem u For a training sequence T P C N tˆp the vectorizedreceivedsignal is vecpyq Q `pt T Ā t b A r qvecph v q`vecpnq Quantized measurement Measurement matrix y Q pax ` nq Sparse vector iid Gaussian noise CS problem with quantized and noisy measurements 9

10 Prior work on low resolution mmwave channel estimation One-bit CS Adaptive CS CS CS with quantized measurements u 1-bit mmwave channel estimation reformulated via conventional one-bit CS* u Adaptive 1-bit CS based mmwave channel estimation** u No work on adaptive CS w/ low resolution (>1bit) noisy measurements u Adaptive estimation could help to reduce training overhead *J. Mo, P. Schniter, N. G. Prelcic and R. W. Heath, Jr. Channel Estimation in Millimeter Wave MIMO Systems with One-Bit Quantization, Asilomar 2014 ** C. Rusu, R. Méndez-Rial, N. González-Prelcic and R. W. Heath Jr., Adaptive One-bit Compressive Sensing with Application to Low-Precision Receivers at mmwave, Globecom

11 ` What is the potential of adaptive CS? ` Classical, non adaptive CS* Sparsity level Et}x ˆx } 2 2u C s 2 Measurement marix BOUNDS ON THE RECONSTRUCTION ERROR Noise variance }A} 2 F N log N. Size of the sparse vector Adaptive CS** Et}x ˆx } 2 2u C s 2 }A} 2 F u Adaptive sensing schemes may not lead to dramatic improvements in MSE u Is adpatation still worthwhile in a mmwave one-bit/few-bits receiver? N * M. Davenport, M. Duarte, Y. Eldar and G. Kutyniok, Introduction to compressed sensing, Compressed Sensing: Theory and Applications, Cambridge University Press, ** E. Arias-Castro, E. Candes and M. Davenport, On the fundamental limits of adaptive sensing, IEEE Trans. Inform. Theory,

12 Adative CS with quantized and noisy measurements 12

13 Contributions u An adaptive CS algorithm design from quantized noisy measurements u Application to channel estimation at mmwave with a low resolution RX u Training sequence adaptation to reduce MSE in the estimation u Performance analysis of the algorithm in terms of ª Resolution of the ADC ª Training length ª SNR ª Sparisty level in the channel u Comparison to conventional CS approaches 13

14 Adaptive CS based on quantized and noisy measurements y Q pax ` nq u Sparse recovery from quantized and noisy measurements ª General Expectation Maximization (GEM) algorithm based on fixed A* Considers noise and quantization together Estimates the noise variance from the data Considers the joint distribution between the observed (quantized) data and missing (unquantized) data Convergence to ML estimator under mild assumptions ª Non adaptive approaches in prior work u We propose a new adaptive technique based on GEM *K. Qiu and A. Dogandzic, Sparse signal reconstruction from quantized noisy measurements via GEM hard thresholding, IEEE Trans. Signal Process., vol. 60, no. 5, pp ,

15 Main ideas for AGEM Assumptions on A o o Restrict the rows of A to a given set A of size N Assume A is made up of orthonormal vectors Main idea for solving o o Solve using GEM at successive steps Increase the number of measurements at every step Adaptation of A o o After solving at each step, Update A to maximize correlation with current estimate 15

16 The new adaptive CS algorithm - AGEM Initialization, i=1 1. Set the total measurement budget to m 2. Randomly select A 1 from A 3. Compute m/2 measurements in y 1 4. Solve x 1 =GEM(A 1,y 1,s) Iterate, i=i+1 1. Take m/2 i new measurement vectors a T j maximized T such that a j ˆx i 1 is 2. Add them to A i-1 to construct A i 3. Compute the new measurements y i 4. Solve x i =GEM(A i,y i,s) 5. Return to step 1 until reaching the total measurement budget 16

17 Adaptive channel estimation using AGEM - 1 Measurement matrix vecpyq Q `pt T Ā t b A r qvecph v q`vecpnq Training sequence p: number of training pilots Sparse channel vector u A t and A r are fixed, depend on the antenna array geometry u An adaptive measurement matrix is obtained adaptingthe training pilots ª Each training pilot contributes N r new measurements at receiver ª More training pilots p more measurements m=pn r 17

18 Adaptive channel estimation using AGEM - 2 RF Chain 1-bit b-bit ADC H Transmit y Processing Baseband Processing p x log N t Feedback channel RF Chain 1-bit b-bit ADC Selected index in A u Adaptation requires limited feedback to the transmitter ª RX computes the channel estimate at every step ª At every step, feedbacks the new measurement vectors from A If A is of size N t, we need log N t bits for the feedback ª For the last step, feedback the channel estimate 18

19 Simulation results 19

20 Performance analysis of AGEM in terms of SNR Adapting to noise s=8 p=8 N t =N r =16 Better performance w/ adaptive u It is not worthwhile to use more than 3 bits in the ADCs u Adaptive CS reduces MSE if SNR is not very low Adaptive CS overcomes conventional CS when SNR> -3 db 20

21 Performance analysis of AGEM in terms of training length Too few measurements SNR=5dB s=8 N t =N r =16 Plenty of measurements u For AGEM there is a working regime in terms of the number of pilots u If there are too many measurements AGEM and GEM perform equivalently Adaptive CS outperforms conventional CS when using a few pilots 21

22 Performance analysis of AGEM in terms of sparsity SNR=5dB p=8 N t =N r =16 Adaptive CS is worthwhile for any sparsity level 22

23 How the MSE reduction in the channel estimateimpacts on the achievable rate? 23

24 Achievables rates in terms of SNR u Improvement on achievable rate by adaptive CS does not depend on q u Significant improvement on achievable rate if SNR is not very low 24

25 Conclusions u Adaptive CS overcomes conventional CS when SNR is not too low ª Works various sparsity levels ª Helps reduces training length and thus overheads CS u If many measurements are possible, adaptive does not make sense Quantization does not dramatically affect performance for channel estimation 25