Adaptive sparse algorithms for estimating sparse channels in broadband wireless communications systems

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1 Wireless Signal Processing & Networking Workshop: Emerging Wireless Technologies, Sendai, Japan, 28 Oct Adaptive sparse algorithms for estimating sparse channels in broadband wireless communications systems Guan Gui co-works with Prof. Fumiyuki Adachi PhD & JSPS postdoctoral researcher fellow Department of Communications Engineering Graduate School of Engineering Tohoku University, Sendai, Japan 1

2 Outline 1. Background and motivation - Fast evolution of mobile communication systems - Conventional adaptive channel estimation methods - Sparse channel models in various broadband systems - Objective 2. Invariable-step-size based adaptive sparse filtering algorithms - Proposed sparse ISS-NLMS filtering algorithms - Simulations results - Summary of the sparse ISS-NLMS algorithms 3. Variable-step-size-based adaptive sparse filtering algorithms - Proposed sparse VSS-NLMS filtering algorithms - Simulations results - Summary of the sparse VSS-NLMS algorithms 4. Related works 5. Concluding remarks and future works 2

3 Fast evolution of mobile comm. systems 3 Ultimate goal of mobile wireless communication systems [Adachi2005/2007] - extremely high data rate - as low energy as possible - as high spectral efficiency as possible Narrowband Era [Adchi2005] F. Adachi, D. Grag, S. Takaoka, and K. Takeda, IEEE Wireless Communications, vol. 12, no. 2, pp. 8 18, [Adchi2007] F. Adachi and E. Kudoh, Wireless Communications and Mobile Computing, vol. 7, no. 8, pp , Broadband signal transmission technique is becoming an indispensable to realize high-speed date rate. WCDMA: wideband code division multiple access; LTE-A: long-term evolution advanced; MIMO: multiple-input multiple-output OFDM: orthogonal frequency division multiplexing; DAN: distributed antenna network Wideband Era Broadband Era November 1, G 2G 3G 4G 5G 1978~ years 1991~ years 2001~ years 2011~ years? FDMA TDMA CDMA/WCDMA LTE/LTE-A (MIMO+OFDM) Analog ~2.4kbps (very low rate) Digital ~64kbps (low rate) Digital ~2Mbps/~14Mbps (high rate) Digital 100Mbps~1Gbps (higher rate) 2021~2031? 10years???? (e.g., massive MIMO or DAN) Digital 1Gbps~10Gbps(?) (super-high rate)

ISS-NLMS filtering algorithm (2-1) Update equation of ISS-NLMS filtering algorithm[widrow1985] input signal vector x(n) unknown channel h additive noise z(n) h T x(n) Σ h n + 1 = h n + μ e n x n x n

4 ISS-NLMS filtering algorithm (2-1) Update equation of ISS-NLMS filtering algorithm[widrow1985] input signal vector x(n) unknown channel h additive noise z(n) h T x(n) Σ h n + 1 = h n + μ e n x n x n 2 2 where μ is a ISS of gradient descend and n-th updating error as e n = d n h T n x n d n = h T x n + z n d(n) estimate channel h(n) adaptive algorithm h T (n)x(n) Adaptive channel estimation e(n) Σ [widrow1985] B. Widrow and D. Stearns, New Jersey: Prentice Hall,

5 ISS-NLMS filtering algorithm (2-2) Gradient descent for finding minimum linear adaptive channel estimator. h(0)=0 step-size of gradient descend Linear minimum plane (many solutions) of adaptive channel estimator (steady-state solution) sparse minimum point of adaptive channel estimator Solution plane Solution point 5

6 Dense channel and sparse channel (2-1) Generations [Adachi2001] 2G Cellular (IS-95) 3G Cellular (WCDMA) 4G/5G Cellular (LTE-Advanced~) Bandwidth BW=1.23MHz BW=10MHz BW=20MHz ~100MHz Time delay (if) 0.4μs 0.4μs 0.4μs Sampling channel length [Gui2013] 1-taps case 4-taps case 6-taps case ~80 Channel model dense appro. sparse sparse - dominant channel taps of sampling channel are very limited - even time delay spread of three cases is same, relative sampling channel length is different since different bandwidth is adopted [Adchi2001] F. Adachi, IEICE Trans. Fundamentals, vol. E84-A. no. 1, pp , Jan [Gui2013] G. Gui et. Al., Wireless Comm. and Mobile Comput, pp.1-10, Aug

Magnitude Magnitude Dense and sparse channel (2-1) [Czink2007] Same delay-spread was considered in the following two channel models but at the different trans. bandwidth [Gui2013]. 0.8 0.6 0.4 0.

2 0 0 10 20 30 40 50 60 70 80 Taps index Adaptive sparse channel estimation (ASCE) methods can exploit the sparse structure information and it can achieve better mean square error (MSE)

7 Magnitude Magnitude Dense and sparse channel (2-1) [Czink2007] Same delay-spread was considered in the following two channel models but at the different trans. bandwidth [Gui2013] Dense channel Sparse channel Traditional ACE methods were based on the assumption of dense channel. Hence, the inherent channel sparsity is neglected Taps index Adaptive sparse channel estimation (ASCE) methods can exploit the sparse structure information and it can achieve better mean square error (MSE) performance. [Czink2007] N. Czink, et. al., IEEE Transactions on Wireless Communication, vol. 6, no. 4, pp , Apr [Gui2013] G. Gui et. Al., Wireless Comm. and Mobile Comput, pp.1-10, Aug

8 Objective [C-E] Traditional ISS-NLMS filtering algorithm - invariable step-size based NLMS filtering algorithm cannot trade off between convergence speed and instantaneous MSE on the process of error updating - neglecting the inherence channel structure information Objective we propose variable step-size based sparse NLMS filtering algorithms to achieve two merits: - more flexible and more efficient: variable step-size based algorithms can trade off well between convergence speed and instantaneous MSE on the process of error updating - better estimation performance: sparse constraint functions can exploit channel sparsity, more accurate channel estimators can be obtained 8

9 Outline 1. Background and motivation - Fast evolution of mobile communication systems - Conventional adaptive channel estimation methods - Sparse channel models in various broadband systems - Review of compressive sensing - Research objective 2. Invariable-step-size based adaptive sparse filtering algorithms - Proposed sparse ISS-NLMS filtering algorithms - Simulations results 3. Variable-step-size-based adaptive sparse filtering algorithms - Proposed sparse VSS-NLMS filtering algorithms - Simulations results 4. Related works 5. Conclusion and future works 9

10 ISS-ZA-NLMS filtering algorithm (2-1) updating equation of ISS-ZA-NLMS filtering algorithm [Gui2013WCNC] h n + 1 = h n + μ e n x n x H n x n ρ ZAsgn h n where ρ ZA = μλ ZA, both μ and λ ZA denote step-size of gradient descend and sparse regularization parameter, respectively. sgn(.) is defined as sgn h(n = h(n) 1 h(n) = 1, h i n > 0 0, h(n) = 0 1, h n < 0 Even thought the ISS zero-attracting normalized least mean square (ISS-ZA-NLMS) filtering algorithm can obtain sparse solution, the efficiency of exploiting sparse ability is very limited. The detailed performance analysis was given in [Gui2013EURASIP]. [Gui2013WCNC] G. Gui, et. al., in IEEE WCNC, Shanghai, China, April 4-7, [Gui2013EURASIP] G. Gui et. al, EURASIP J. on Wireless Comm. and Networking, vol. 2013, no.1, pp. 1-18,

11 ISS-ZA-NLMS filtering algorithm (2-2) Linear minimum plane (many solution) of adaptive channel estimator (steady-state) h(0)=0 zero-attracting to exploit channel sparity sparse minimum point of adaptive channel estimator ISS Solution plane 11

12 zero-attracting ability ISS-RZA-NLMS filtering algorithm (2-1) Updating equation of ISS-RZA-NLMS filtering algorithm [Gui2013WCNC] h n + 1 = h n + μ e n x n x T n x n ρ RZA sgn h n 1 + ε RZA h n where ρ RZA = μλ RZA ε RZA, λ RZA and ε RZA denote sparse regularization parameter and reweighted factor, respectively value of channel taps ε RZA = 0 (ISS-ZA-NLMS) ε RZA = 1 (ISS-RZA-NLMS) ε RZA = 2 (ISS-RZA-NLMS) ε RZA = 5 (ISS-RZA-NLMS) ε RZA = 10 (ISS-RZA-NLMS) ε RZA = 20 (ISS-RZA-NLMS) ε RZA = 100 (ISS-RZA-NLMS) ε RZA = 500 (ISS-RZA-NLMS) ε RZA = 1000 (ISS-RZA-NLMS) [Gui2013WCNC] G. Gui, et. al., in IEEE WCNC, Shanghai, China, April 4-7,

13 ISS-RZA-NLMS filtering algorithm (2-2) Linear minimum plane (many solution) of adaptive channel estimator (steady-state) h(0)=0 reweighted zero-attracting to exploit channel sparity sparse minimum point of adaptive channel estimator ISS Solution plane 13

Sparse constraint [Donoho2006] [Donoho2006] D. L. Donoho, IEEE Trans. Inf. Theory, vol. 52, no.

sparse constraint solution plane solution plane solution plane non-sparse constraint e.g.

5) l 0 -norm (a) w/ sparse penalty: l 1 -norm (b) w/ sparse penalty: l p -norm, 0 < p < 1 (c) w/

14 Sparse constraint [Donoho2006] [Donoho2006] D. L. Donoho, IEEE Trans. Inf. Theory, vol. 52, no. 4, pp , sparse constraint solution plane solution plane solution plane non-sparse constraint e.g., l p -norm (p = 0.5) l 0 -norm (a) w/ sparse penalty: l 1 -norm (b) w/ sparse penalty: l p -norm, 0 < p < 1 (c) w/ sparse penalty: l 0 -norm L 1 -norm sparse penalty strength < L p -norm sparse penalty strength < L 0 -norm sparse penalty strength w/ stronger sparse penalty on cost function can obtain more accurate sparse channel estimator using corresponding update equation. 14

15 ISS-LP-NLMS filtering algorithm (2-1) Using the p-norm sparse function, updating equation of ISS-LP-NLMS filt ering algorithm [Gui2013WCNC] h n + 1 = h n + μ e n x n x T n x n ρ LP h n 1 p p sgn h n ε LP + h n 1 p where 0 < p < 1 is the p-norm sparse constraint ε LP is a small positive parameter ρ LP = μλ LP, λ LP is a sparse regularization parameter to trade off the estimation error and channel sparsity [Gui2013WCNC] G. Gui, et. al., in IEEE WCNC, Shanghai, China, April 4-7,

16 ISS-LP-NLMS filtering algorithm (2-2) Linear minimum plane (many solution) of adaptive channel estimator (steady-state) h(0)=0 l p -norm zero-attracting to exploit channel sparity sparse minimum point of adaptive channel estimator ISS Solution plane 16

17 ISS-L0-NLMS filtering algorithm (2-1) updating equation of ISS-L0-NLMS filtering algorithm [Gui2013WCNC] h n + 1 = h n + μ e n x n x T n x n ρ L0g L0 h n where 0 < p < 1 is the p-norm sparse constraint ρ L0 = μλ LP, λ L0 is a sparse regularization parameter to trade off the estimation error and channel sparsity g L0 h n is a approximate sparse constraint function g L0 h 2β2 h 2βsgn h, when h 1 β 0, others. β is a threshold parameter [Gui2013WCNC] G. Gui, et. al., in IEEE WCNC, Shanghai, China, April 4-7,

18 ISS-L0-NLMS filtering algorithm (2-2) Linear minimum plane (many solution) of adaptive channel estimator (steady-state) h(0)=0 l 0 -norm zero-attracting to exploit channel sparity sparse minimum point of adaptive channel estimator ISS Solution plane 18

19 Simulation parameters Type of parameters Value channel length N = 16 nonzero channel tap K = 1 & 4 signal-to-noise ratio (SNR) 10log(E 0 σ 2 n ) transmit power E 0 =1 step-size of sparse LMS μ = 5e 1 step-size of sparse NLMS μ N = 5e 2 regularization parameter of ZA-(N)LMS 2 2 λ ZA = 0.02σ n & λ ZAN = 0.002σ n regularization parameter of RZA-(N)LMS 2 2 λ RZA = 0.002σ n & λ RZA = σ n threshold parameter of RZA-(N)LMS ε RZA =1/20 regularization parameter of LP-(N)LMS 2 2 λ LP = 0.05σ n & λ LPN = 0.005σ n regularization parameter of L0-(N)LMS 2 2 λ L0 = 0.02σ n & λ L0N = 0.002σ n threshold parameter of L0-(N)LMS Q = 1 & β = 10 The estimation performance between actual and estimated channel is evaluated by average 2 mean square error (MSE) standard which is defined : Average MSE h n = E h h n 2 19

20 Avergae MSE Average MSE MSE performance comparison of ISS-LP-NLMS with p - ASCE using smaller l p -norm NLMS algorithm, more accurate estimator can be obtained; but it also too small to unstable in low SNR region. - estimate sparser channel, more accurate sparse channel estimator achieved SNR=10dB LMS LP-LMS(p=0.3) LP-LMS(p=0.5) LP-LMS(p=0.7) LP-LMS(p=0.9) NLMS LP-NLMS(p=0.3) LP-NLMS(p=0.5) LP-NLMS(p=0.7) LP-NLMS(p=0.9) SNR=15dB LMS LP-LMS(p=0.3) LP-LMS(p=0.5) LP-LMS(p=0.7) LP-LMS(p=0.9) NLMS LP-NLMS(p=0.3) LP-NLMS(p=0.5) LP-NLMS(p=0.7) LP-NLMS(p=0.9) 10-2 K=1 K= K=1 K= Iterations Iterations 20

21 Average MSE Steady-state MSE comparison Average MSE - ASCE in higher SNR region, more accurate channel estimator can be obtained - estimate sparser channel, more performance gain can be achieved SNR=5dB K=1 K=4 LMS ZA-LMS RZA-LMS LP-LMS L0-LMS NLMS ZA-NLMS RZA-NLMS LP-NLMS L0-NLMS SNR=15dB LMS ZA-LMS RZA-LMS LP-LMS L0-LMS NLMS ZA-NLMS RZA-NLMS LP-NLMS L0-NLMS 10-3 K=1 K= Iterations Iterations 21

22 Outline 1. Background and motivation - Fast evolution of mobile communication systems - Conventional adaptive channel estimation methods - Sparse channel models in various broadband systems - Review of compressive sensing - Research objective 2. Invariable-step-size based adaptive sparse filtering algorithms - Proposed sparse ISS-NLMS filtering algorithms - Simulations results 3. Variable-step-size-based adaptive sparse filtering algorithms - Proposed sparse VSS-NLMS filtering algorithms - Simulations results 4. Related works 5. Conclusion and future works 22

23 VSS-ZA-NLMS filtering algorithm (2-1) Updating equation of VSS-ZA-NLMS filtering algorithm [Gui2013WCSP] e n x t h n + 1 = h n + μ n + 1 x T ρsgn h n t x t where ρ = μ(n + 1)λ depends on regularization parameter and variable stepsize (VSS) μ(n + 1) which controls the convergence speed and steady/transient state MSE. - the VSS is defined by μ n + 1 = μ max p T n + 1 p n + 1 p T n + 1 p n C - threshold parameter C is a positive threshold parameter which is related to received signal-to-noise ratio (SNR) and μ max is the maximal step-size. - p n is defined by p n + 1 = βp n + 1 β x t e n x T t x t where β [0,1) is the smoothing factor for controlling the VSS and estimation error. [Gui2013WCSP] G. Gui, et. al., in WCSP, Hangzhou, China, Oct

24 VSS-ZA-NLMS filtering algorithm (2-2) Gradient descent for finding minimum linear adaptive channel estimator. Linear minimum plane (many solution) of adaptive channel estimator (steady-state) sparse minimum point of adaptive channel estimator h(0)=0 VSS zero-attracting (ZA) to exploit channel sparity Solution plane 24

25 Computer simulation parameters values channel length N = 16 no. of nonzero coefficients K = 1 and 4 distribution of nonzero coefficient random Gaussian CN(0,1) threshold parameter for VSS-NLMS C {10 3, 10 4 } step-size μ = 0.2 and μ max = 2 regularization parameter ρ = 0.002σ n 2 modulation schemes QPSK, 8PSK, 16PSK and 32PSK 16QAM, 64QAM and 256QAM 25

26 Average MSE Average MSE v.s. iteration times (SNR=5dB) Average MSE VSS-ZA-NLMS algorithm can achieves a better estimation performance than ISS-ZA-NLMS. VSS-ZA-NLMS obtains better estimation performance than VSS-NLMS in the extreme sparse channel case, e.g., K = 1 VSS-ZA-NLMS may not work well in two cases: low SNR and low sparse 10-1 ISS-NLMS VSS-NLMS ISS-ZA-NLMS VSS-ZA-VNLMS 10-1 ISS-NLMS VSS-NLMS ISS-ZA-NLMS VSS-ZA-VNLMS SNR=5dB Sparsity: K=1 Parameter: C= Iterative times (n) SNR=5dB Sparsity: K=4 Parameter: C= Iterative times (n) 26

27 Average MSE Average MSE v.s. iteration times (SNR=15dB) Average MSE VSS-ZA-NLMS algorithm can achieves a better estimation performance than ISS-ZA-NLMS. VSS-ZA-NLMS obtains better estimation performance than VSS-NLMS in the different sparse channel cases No. of nonzero taps increase, performance gag between VSS-ZA-NLMS and VSS- NLMS reduce SNR=15dB Sparsity: K=1 Parameter: C= ISS-NLMS VSS-NLMS ISS-ZA-NLMS VSS-ZA-VNLMS SNR=15dB Sparsity: K=4 Parameter: C= ISS-NLMS VSS-NLMS ISS-ZA-NLMS VSS-ZA-VNLMS Iterative times (n) Iterative times (n) 27

28 Outline 1. Background and motivation - Fast evolution of mobile communication systems - Conventional adaptive channel estimation methods - Sparse channel models in various broadband systems - Review of compressive sensing - Research objective 2. Invariable-step-size based adaptive sparse filtering algorithms - Proposed sparse ISS-NLMS filtering algorithms - Simulations results 3. Variable-step-size-based adaptive sparse filtering algorithms - Proposed sparse VSS-NLMS filtering algorithms - Simulations results 4. Related works 5. Conclusion and future works 28

29 Recent works relates to this presentation (3-1) Step-size Introducing the threshold parameter to adaptive control the step-size, we also proposed sparse VSS-LMS/F filtering algorithm in [Gui2013PIRMC] and analyzed in [Gui2013WCMC] LMS e 2 (n)=0.1 - threshold parameter (λ) controls the step-size, i.e., bigger threshold incurs smaller step size and verse versa. - suitable choosing threshold parameter is necessary for controlling the variable step size LMS/F Threshold parameter ( ) [Gui2013PIRMC] G. Gui, et. al., in PIRMC, London, UK, 5-8 Sept [Gui2013WCMC] G. Gui et. al., accepted by Wireless Comm. and Mobile Comput. 29

30 Recent works relates to this presentation (3-2) Step size of gradient descend Considering least mean fourth (LMF) rather than LMS, we also proposed sparse VSS-LMF filtering algorithm in [Gui2013VTC-fall] and performance analyzed in [Gui2013IET] NLMS-type and LMF-type: s and f NLMF-type: f (n) step-size (ISS): μ f step-size(vss): μ f n = μ fe 2 (n) x n 2 2 +e 2 (n) e 2 n, μ f (n) μ f e 2 n 0, μ f (n) e 2 (n) [Gui2013PIRMC] G. Gui, et. al., in VTCfall, Las Vegas, USA, 2-5 Sept [Gui2013IET] G. Gui et. al., submitted for IET Communications. 30

Recent works relates to this presentation (3-3) Considering two independent x(t) w

1 (n) Σ NLMS filtering algorithm in [Gui2014VTC] and analyzed in [Gui2013IEEE]

, submitted for VTC-Spring, Seoul, Korea, 18-21 May 2014. [Gui2013IEEE] G. Gui, et.

31 Recent works relates to this presentation (3-3) Considering two independent x(t) w o d(t) NLMS filters, we proposed sparse affine combination VSS- Adaptive filter I e 1 (n) Σ NLMS filtering algorithm in [Gui2014VTC] and analyzed in [Gui2013IEEE] [Gui2014VTC] G. Gui, et. al., submitted for VTC-Spring, Seoul, Korea, May [Gui2013IEEE] G. Gui, et. al., in preparation for IEEE Trans. on Circuits and Systems. w 1 (n) w 2 (n) Adaptive filter II e 2 (n) y 1 (n) λ(n) y 2 (n) 1 λ(n) Σ Σ y(n) Σ 31

32 Outline 1. Background and motivation - Fast evolution of mobile communication systems - Conventional adaptive channel estimation methods - Sparse channel models in various broadband systems - Review of compressive sensing - Research objective 2. Invariable-step-size based adaptive sparse filtering algorithms - Proposed sparse ISS-NLMS filtering algorithms - Simulations results 3. Variable-step-size-based adaptive sparse filtering algorithms - Proposed sparse VSS-NLMS filtering algorithms - Simulations results 4. Related works 5. Conclusion and future works 32

33 Conclusion and future works In this presentation, we proposed - sparse ISS-NLMS algorithms to exploit channel sparsity so that the they can improve channel estimation performance - sparse VSS-NLMS algorithms to exploit channel sparsity and to adaptive change the step-size simultaneously so that they can improve channel estimation performance and convergence speed In the next work, some related works should be developed - optimal regularization parameter (λ) selection problem for sparse ISS-NLMS and VSS-NLMS filtering algorithms. Optimal λ should depend on the channel sparseness - optimal parameter selection (C) problem of sparse VSS-NLMS filtering algorithms 33

34 Wireless Signal Processing & Networking Workshop: Emerging Wireless Technologies, Sendai, Japan, 28 Oct Thank you very much for your kind attention! Comments or suggestions? 34

Research Article Variable Is Better Than Invariable: Sparse VSS-NLMS Algorithms with Application to Adaptive MIMO Channel Estimation

e Scientific World Journal, Article ID 74897, 10 pages http://dxdoiorg/101155/014/74897 Research Article Variable Is Better Than Invariable: Sparse Algorithms with Application to Adaptive MIMO Channel