Blind MIMO MC-CDMA parameters estimation over fading channels for cognitive radio

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Blind MIMO MC-CDMA parameters estimation over fading channels for cognitive radio

C. Nsiala Nzéza, K. Hassan and M. Berbineau Université Lille Nord de France FR59000, Lille, France

INRETS/LEOST, 20, Rue Elisée Reclus 59666, Villeneuve D’Ascq, France

crepin.nsiala@inrets.fr http://www.inrets.fr

R. Gautier and G. Burel

Université Européenne de Bretagne, France Université de Brest CNRS, UMR 3192 Lab-STICC ISSTB 6 Avenue Le Gorgeu CS 93837, 29238 Brest Cedex 3, France

roland.gautier@univ-brest.fr http://www.lab-sticc.fr

Abstract—Wireless communication systems must answer dras- tic requirements, such as reliability, availability and interop- erability in vehicular applications. Cognitive radio system is suitable technology able to answer these needs. It allows dynamic spectrum allocation and very often spectrum reuse especially in rural areas. In consequence, it require an accurate knowledge of the radio spectrum in which it operates while the vehicle is moving. Focusing on MIMO-MC-DS-CDMA systems, we present a blind technique that aims at recovering both spectral and temporal signal parameters. The proposed approach leads to a fast efficient parameters estimation, insensitive to receiver phase and frequency offsets, which is suitable for vehicular applications.

Index Terms—OFDM, MIMO, CDMA, spectral monitoring, railway domain,second order statistics, cyclo-stationarity.

I. INTRODUCTION

The reconfigurability of radio devices as a function of the environment where they operate constitutes a major issue in the railway domain to cope with the heterogeneity of wireless systems deployed for train to ground communications along railway lines in the world to satisfy control-command and passenger services. This functionality relies on cognitive or opportunistic radios [1], able to learn their environment and to adapt their emitting/transmitting parameters versus different criteria (coverage, interferences, availability of spectral resources, systems, services and even connection cost ...).

This kind of systems will be in a near future of considerable economic interest for railway industry and railway operators in order to make railway transports safer and greener.

To allow radio modems to reconfigure themselves as a function of the environment where they operate, the radio spectrum must be monitored. The reconfigurations of the communication system should be set up as a function of the available signals, while the train is moving. Unlike in coherent detection, the receiver does not have any prior knowledge on the time and frequency distribution of the transmitted signals along the railway lines among the possible existing wireless standards.

Orthogonal Frequency Division Multiplexing (OFDM), higher order MIMO (multiple Input Multiple Output) and Spread Spectrum are the key enabler technologies for very high throughputs and robustness in the emerging standards

that support vehicular mobility (IEEE 802.11p, IEEE 802.11n, Wimax, LTE) and that will be deployed along the railway lines all over the world to satisfy increasing demand on railway communications.

OFDM main well known advantages are its robustness in frequency-selective fading channels, its capability of portable and mobile reception, and its flexibility. The advantages and success of multi-carrier modulation spread spectrum technique [2], [3] motivated many researchers to investigate the suitability combination of both techniques, known as Multi- Carrier Spread Spectrum (MC-SS) which benefits from the main advantages of both schemes [4]. Among all combination of both techniques, the so called MIMO Multicarrier Di- rect Spread Spectrum CDMA (MC-DS-CDMA) is considered throughout this paper in order to propose a fast efficient method for spectral monitoring. The MIMO-MC-CDMA case can be easily inferred.

Recently, blind procedures have been proposed using autocorrelation and cyclic autocorrelation based features to extract parameters for OFDM signals using a Cyclic Prefix time guard interval (CP-OFDM) and for frequency-selective channels [5].

Some others have been proposed for OFDM signals using a Zero Padding time guard interval (ZP-OFDM) based on power autocorrelation feature. Among the recent studies of MIMO and MIMO-OFDM, blind approaches using High Order statistics [6], [7] or subspace-based [8], [9] have been addressed.

The algorithm described in [5] has caught our attention.

However, for vehicular applications, it may suffer from slow convergence speed due to an exhaustive search in solving their proposed optimization problem. Therefore, this paper presents a novel approach which lies in the joint use of Second Order Statistics scheme associated to spectral components and temporal parameters methods in order to respectively recover the number of transmit antennas, band-edges and central frequencies, symbol duration and OFDM periods and also the number of subcarriers. The resulting algorithm is then suitable for wireless telecommunications in the railway context (high mobility, interferences) due to fast processing.

The paper is organized as follows: Section II presents system model and hypothesis. Section III describes the proposed blind scheme while simulation results are given in Section IV.

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At last, Section V gives some comments about the proposed scheme, while Section VI concludes the paper.

II. SYSTEM MODEL AND ASSUMPTIONS

We consider a Spatial Multiplexing MIMO-MC-DS-CDMA system withNt transmit andNr receive antennas, employing M subcarriers and the spreading code c= [c0,· · ·, cL−1] of length Lwherecℓstands for theℓ^thsequence chip, as shown on Fig. 1. The channel is assumed to be a slowly-varying frequency-selective Rayleigh fading channel. The principal motivation for using MC-DS-CDMA is to allow a frequency- selective channel to appear as flat fading on each subcarrier, assuming thatM is sufficiently large. The required bandwidth (main lobe) over each transmit antenna is equal to W =

(M+1)L

M Ts = ^M+1_{M T}_c,Tc=^T_L^s standing for the chip duration. The complex flat fading channel experienced by each subcarrier is described as:

Hm=hnr,nt,mexp(jπǫnt,m)δ(t) (1) wherehnr,nt,m represents path gain between then^thr receive antenna and then^tht transmit antenna over them^thsubcarrier, assumed i.i.d. circular complex Gaussian random varaibles with zero mean and varianceσ² [10]. Also,ǫnt,m is the i.i.d.

uniformly-distribution phase over[0 2π]. This system can then be viewed asM independant(Nt, Nr)-MIMO systems, which channel matrix is denoted as H_m. Let the transmitted signal from then^th_t antenna over them^th subcarrier be denoted as:

snt,m(t)=

+∞X

k=−∞

am(k)

L−1X

ℓ=0

cℓψ(t−kTs) exp (2πjfmt)

=

+∞X

k=−∞

am(k)g(t−kTs) exp (2πjfmt)

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whereg(t−kTs)=PL−1

ℓ=0 cℓψ(t−lTs), andam(k) ∈a^T = [a0,· · ·, aM−1]the complex symbol transmitted over them^th subcarrier at the symbol period Ts, m = 0,· · ·, M −1, are assumed to be independent, centered, noise-free and of variance σ_a². Therefore, the complex transmitted signalsnt(t) from over the n^th_t antenna is given by:

snt(t)=

M−1X

m=0

snt,m(t)

=

M−1X

m=0 +∞X

k=−∞

am(k)g^′(t−kT_u^′) exp (2πjfmt) (3)

wherefm=f0+_{M T}^mL_s =f0+_{M T}^m_c,f0 stands for the center band frequency,Tu= ^{M T}_L^s =M Tc represents the an OFDM symbol period. Thus, Tu^′ =Tu+Tcp = (1 + ∆Tu)Tu, with standing for Tcp the cyclic prefix (CP) duration,0 ≤∆Ts <

1. Hence, g^′(t) can be viewed as the convolution between sequence waveform and the OFDM modulator filter. Without loss of generality,g^′(t)is supposed to be a rectangular window defined in [0T_u^′].

Therefore, the complex received signal ynr,m(t)at then^th_r

antenna over them^thsubcarrier can be expressed as:

ynr,m(t)=

Nt

X

nt=1

hnrnt,msnt,m(t) exp(jπǫnt,m)+bnr,m(t) (4) where bnr,m(t) is an AWGN noise vector at the n^thr

received antenna n^thr antenna, supposed to be uncorrelated with symbols and of varianceσ²_b. From (4), the recieved signal at then^thr antenna is done by:

ynr(t) =

MX−1 m=0

ynr,m(t) (5) For a better understanding of the subsection III-A in the sequel, let us set to as N the complet set of transmit spread OFDM symbols, and write the (LN M Nr × 1)-vector of received signal at all Nr antennas in a matrix form at time t as:

y=Hs+b (6)

where,H=I_{N L}⊗L PM−1 m=0 H_m

, which diagonal blocks are(Nr×Nt)-matricesH_m,⊗andL

stand for the Kroneker product and the direct subspace sum, respectively. Vector s = [s¹^T· · ·s^N^T]^T of size (LN M Nt × 1) contains N spread OFDM symbols sⁿ over all M subcarriers, defined as sⁿ = [sⁿ₀^T · · ·sⁿ_M^T₋₁]^T, n = 1,· · ·, N. Vector sⁿ_m stands for the n^th spread OFDM symbol transmitted over the m^th subcarrier, and is defined assⁿ_m^T = [sⁿ_1,m^T · · ·sⁿ_N^T_t_,m]^T, where sⁿ_nt,m is the symbol transmitted at the n^th_t transmit antenna over the m^th subcarrier as illustrated on Fig 1. Also, the (LN M Nr×1)-vector b, defined like for vector s, contains AWGN at the receiver side. One can notice that in the case of MIMO-MC-CDMA transmission, it should set to as:L=M, since each sequence chip spread the corresponding subcarrier.

III. PROPOSED APPROACH

Fig. 2 shows different steps of the proposed algorithm for spectral monitoring and parameter estimation of MIMO spread-spectrum multi-carrier modulations. The novelty in this paper is threefold. First, spectral components (e.g., central frequency, bandwidth) are estimated by the averaged periodogram non-parametric approach using a Fast Fourier Transform (FFT) combined with a detection threshold automatically computed from the signal received power estimation. One should note that in [5], a targed filter is used instead of a threshold.

Second, within each detected bande-edge and together with the estimation process of Nt, the covariance matrix second- order statistics analysis allows to estimate the number Nt

of transmit antennas. Although methods using high order statistics have been adressed in the literature, we only focus on second-order statistics in order to reduce the computational cost. Indeed, vehiculars high mobility must be taken into account.

Third, we extend the blind temporal parameters method previously proposed in [11] to the case of transmissions

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P/S CP

IFFT

P/S CP

IFFT

Proposed Blind Parameters Estimation Method

Estimated Parameters sⁿ_N_t_,0

Nt

c= [c0,· · ·, c_L−1]^T

sⁿ_N_t_,M−1

sⁿ1,0

c= [c0,· · ·, c_L−1]^T 1

sⁿ1,M−1

1

Nr

Figure 1. MIMO-MC-DS-CDMA modulator scheme with Spatial Multiplexing, andNr≥Nt.

considered in this paper. Then, with the knowledge of Nt

and focussing only on the recieved signal over one receive antenna, we perform the extend apporach in order to assess temporal paremeters such as CP duration, symbol period,..., with no ohter prior knowledge.

A. Determination of the number of antennasNt

Among all approaches aim at detecting the number of transmit antennas, the method addressed in [12], based on the covariance matrix predicted eigenvalues threshold (PET), appears more suitable for our application. In fact, it is very simple to implement, robust in most of propagating environments, and presents a very low computational cost. The method consists in adaptively modelize noise eigenvalues decrease. Then, the number of transmit antennas correspond to the index of the largest eigenvalue which does not reach eigenvalues decrease model. First, from (6), the correlation matrix computation, set to as R=E

yy^H , gives

R=σs²HH^H+σb²I_NrP (7) whereP =LM N,(∗)^H represents the Hermitian transpose, and bothLandN are assumed to be known. By applying the EVD on R, (7) can be expressed by

R=U LU^H (8) where L=I_P⊗Λ with Λ = diag{λ1,· · · , λNr}, such that λ1 ≥ · · · ≥ λNr, and U=I_P⊗U are (Nr ×Nt)-matrices unitary (i.e.,UU^H=I_Nr). Therefoe, matrixRcontainsP Nr

diagonal blocks(Nr×Nr)-matrices, conrresponding to a sub- MIMO system experienced by each subcarrier. Let us set to as F, the diagonal block matrix containing matrixINrand zeros.

Then we can extract from (8) the correlation channel matrix R which analysis allows to estimate the number of transmit antennas. Matrix Ris defined as

R= [vec{F R}] =UΛU^H (9) where ”vec” stands forF R columns whithout zeros. Then, we have the following theorem:

analysis

Input parameters

Covariance matrix Averaged periodogram

Fluctuations measurement

Threshold

Estimated parameters Second

band−edge estimation Esimation

of

y= [y⁰,· · · ,y^K−1]^T

Y

e Y

Φ(θ) b

R η

Tcp,M, f0ⁱ

fWi



 Wi

... Wn



 Nt

K, TF, Ns, Te, TF =NsTe, Nr

Nt, Ts,Tu

Figure 2. Overview of the proposed approach.

Theorem 1: Assuming a full rank(Nr×Nr)-matrixR, the lowest eigenvalue of R is equal to σ_b² and its multiplicity order is equal toNr−Nt, i.e.,λNt+1=· · ·=λNr =σ²_b [7], [12].

More details about the PET method can be found in [12].

Therefore, with the knowledge of both Nr and the lowest eigenvalue multiplicity order, we can deduce the number of transmit antennas. Furthermore, this multiplicity order gives an estimate of the noise subspace dimension, from which we get the following power thresholdη defined as in [13]:

η= 1 Nr−Nt

Nr

X

nt=Nt+1

λnt+3σ²_b (10)

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In practice, the covariance matrix R is unknown at the receiver side. Hence, using K windows of duration TF such that TF = NsTe, where Ns and Te stand for the number of samples in each window and the sampled period, respectively. Ns can not be defined as TF/Tc, since T c is still unknown at this point. For simplicity, the sampled channel- corrupted received sequence over a received antenna set to as y= [y⁰,· · ·, y^K−1]^T is computed. Therefore, an estimate of the covariance matrix of the received sequence is computed as in [11]:

b R= 1

K

K−1X

k=0

y^k(y^k)^∗ (11) B. Spectral components analysis

Various spectral estimation approaches can be found in [14]. While parametric methods are suited for short data records, non-parametric ones usually required less computational complexity for long data records. Hereinafter, the spectral properties are estimated by the averaged periodogram non parametric approach using an FFT. Contrary to [5], a threshold is setted in order to lower the computational cost.

The averaged periodogramY gives an estimate of the PSD of the received sequence by:

Y= 1 K

K−1X

k=0

|F F T(y^k)|² (12) from which we get the set of frequencies F. Therefore, using (10) in (12) estimated band-edge frequencies Wi are those which average power are above threshold while scanning the PSD (12). Then, in each band-edge, we compute the PSD cumulative sum, set to as: Ye of size NF F T. Therefore, suc- cessive steps described below allow other spectral components estimation.

The central frequency f₀ⁱ in each detected frequency band, defined asf₀ⁱ=F(nopt)is obtained by solving the following optimization problem:

nopt= argmin

1≤n≤NF F T

n eY−Y(Ne F F T)/2o

(13) Moreover, the band-edge Wi estimation can be im- proved by computing the signal energy as follows: E^Ye =

e

Y₊(Nw, nopt)−Ye₋(−Nw, nopt), whereNw=min(NF F T− nopt+1, nopt)is the set of frequencies containing95%of signal energy, vectorYe₊(Nw, nopt)andYe₋(−Nw, nopt)contain values of Ye from indices nopt to nopt+Nw+ 1 and from indices nopt−Nw+ 1tonopt, respectively. Then, the refined bandwidth defined asWfi=F(nopt+n^′_opt)−F(nopt−n^′_opt) is estimated by solving the following optimization problem:

n^′_opt= argmin

1≤n^′≤NF F T

{E^Ye>0.95·Ye(NF F T)} (14) At last, the number of subcarriers used can be approximated as: M = ⌊Wfi·Tu⌋, where ⌊ ⌋ stands for the immediately lower integer operator. The following subsection describesTu

estimation method.

C. Temporal parameters estimation

The temporal expression of (5) within each analysis window kis considered. Then, an estimation of the channel-corrupted received signal autocorrelation functionRb^knr is computed as:

Rb^knr(θ) = 1 TF

Z TF

0

ynr

k(t)(ynr

k)^∗(t−θ)dt (15) Then, we compute the second-order moment of (15) using K windows, set to asΦ(θ):

Φ(θ) =En

|Rdnr(θ)|²o

= 1 K

K−1X

k=0

|Rb^knr(θ)|² (16) where E(·) is the estimated expectation of (·). Hence, Φ(θ) is a measurement of the fluctuations ofRbcp(θ). Indeed, from assumptions made in section II, we first get

Rbnr(θ)≃Rbs_nrs_nr(θ) +Rbbb(θ) (17) where the subscript snr stands for the received noise-free signal at then^thr antenna. Equation (17) proves that the global fluctuations are composed of fluctuations due to the noise- free signal and that of the AWGN. Thus, (17) justifies to successively investigate respective both noise-free signal and AWGN contributions to the global fluctuations. The analysis of the fluctuationsΦ(θ)of estimators of the correlation instead of the analysis the correlation allows to estimate temporal parameters as decribed in the sequel

1) Analysis of the noise contribution: Assuming a receiver filter with a flat frequency response in [−fWi/2, +fWi/2]

and zero outside, and with assumptions made in section II, fluctuations due to the AWGN can be characterized by their mean MΦb and standard deviation σΦb as





MΦb=En

|Rbbb(θ)|²o

= ^{M σ}

4 b

e

W TF

(a) σΦb =q

2

KMΦb =q

2 K

M σ_b⁴

e

W TF

(b)

(18) Expression (18) proves that noise contribution is flat over all values of θ, and can be lowered by increasing K or TF, according to the computing available power.

2) Analysis of the channel-corrupted signal contribution:

Since symbols are assumed independent and centered, i.e., En

Rbs_nrs_nr(T_u^′)o

= 0, and then we get channel-corrupted signal characteristics as

Mⁿ_Φ^r=Ebn

|Rbs_nrs_nr(Tu^′)|²o

=M(1 + ∆Tu)Tu

LTF

σs⁴_nr (19)

σ_Φⁿ^r = r2

KM^cp_Φ = r2

K

(1 + ∆Tu)Tu

TF

σ_s⁴_nr (20) where σs²_nr represents the channel-corrupted signal power at the n^th_r antenna. Equation (19) shows that, on average, high equispaced amplitudes peaks of fluctuations occur forθ multiple ofTu and lower ones occur at multiple ofTcp; while (20) evidences that any increase inK or TF allows to lower fluctuations standard deviation.

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Number of received antennas

1 2 3 4

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Figure 3. Extracted correlation matrixR, SNR =−5dB,Nt= 2,Nr= 4.

0.5 1 1.5 2 2.5 3 3.5 4 4.5

0 1 2 3 4 5 6 7 8 9

Number of Received antennas

MatrixReigenvalues

Eigenvalues

Predicted eigenvalues threshold

Figure 4. Estimation of the number of transmit antennas, SNR =−5dB, Nt= 2,Nr= 4.

At last, computing (19) and (20) leads to an insensitive parameters estimation to receiver phase and frequency offsets.

IV. NUMERICAL RESULTS

Simulations have been carried out considering a (Nr = 4, Nt = 2)-MIMO system, with a QPSK-DS-CDMA modulation. M = 64 subcarriers with 16 allocated for the CP, and the signal duration is equal to 436 µs. The channel model used is the COST207RAx6 [15]. Following parameters were set:L=127(Complex GOLDsequence),Fc=150M Hz, Fe=300 M Hz,TF=2 µs, K = 300; then each data stream contains Ns=TF/Te samples. The performance in terms of probability of detection was studied for different SNR at the receiver input. From (18) and (19), the SNR and fluctuations peaks detection threshold are given by :SN R=^M

nr Φ

σ²_Φ

b

, and ζ=MΦb+ 3·σΦb. Hence, the probability of detectionPd, i.e., the probability for the fluctuations average amplitude to be above the threshold ζis defined as Pd=P(Mⁿ_Φ^r > ζ).

Fig. 3 shows an extracted correlation matrixRof size(Nr× Nr) while computing the received correlation matrix R, as discussed in subsection III-A.

As expected, applying the PET method to matrix R led

1 1.5 2 2.5 3 3.5 4

x 10⁹ 0

0.005 0.01 0.015 0.02 0.025

Frequency (GHz)

Power Spectrum density (PSD)

Estimated PSD Estimated band−edge Power threshold η

Figure 5. Filtered periodogram with the estimated power thresholdη.

0 1 2 3 4 5

0 5 10 15 20 25 30 35 40 45 50

Time (in µs)

Fluctuations amplitude

Fluctuations Φ(θ) Noise theoretical mean Noise theoretical maximum

Figure 6. Fluctuations of correlation estimators with CP, SNR = -5 dB.

to the number of transmit antennas estimation. We deduce Nt= 2 as illustrated on Fig. 4. The value ofNtcorresponds to the eigenvalue index from which its behaviour does not reach the noise PET model, as detailed in [12].

Fig. 5 shows an example with two MC-DS-CDMA signals, one around 1.62 GHz and the other around 3.3 GHz. One can see that the proposed method provides good estimates of the spectral properties of the received signal. Note that if two or more signals share the same bandwidth, their parameters estimation, i.e., their differenciation is performed through the analysis of the fluctuations measurement. The second band- edge of interest estimation is performed through the analysis of the centered spectrum bandwith. We deduceWfi=100M Hz, for the signal around1.62 GHz.

Fig. 6 illustrates the fluctuations curve, which is in ac- cordance with theoretical developments. Using the average

Table I

COMPUTATIONAL COST COMPARISON.

Proposed approach O 4·K·NF F T·log2(NF F T) +N_s³+Ns

Algorithm in [5] O K·NF F T·log2(NF F T) +Ns²(Ns−1)!

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spacing between high amplitude peaks and low amplitude ones respectivly, we get Tes = 0.51µs andTecp= 0.1275 µs, thus fT^′u= 0.6375µs. Therefore, with the assessmentWfi, we get M=63, which is very close to the real number of subcarriers.

A last, Table I gives the computational cost of the proposed scheme, which is suitable for the context addressed in this paper.

V. COMMENTS ON THE PROPOSED METHOD

First, let us point out that the temporal parameter estimation is not based on the correlation itself, but rather on the analysis of fluctuations of correlation estimators. Therefore, the curve of fluctuations obtained highlights regularly spaced peaks as explained in the relevant section. Which is not the case when analyzing the correlation function, especially the correlation of spread spectrum signals [11].

The method herein presented can be performed iteratively to its low complexity. The interest of the iterative process is to eliminate successively the contribution of signals detected in previous iterations. On the other hand it also avoids the propagation of estimation error of frequency bands due to interfering signals.

Indeed, initially low-power signals are not taken into account in estimating the frequency bands because of the high level threshold. However, in the next iteration, after identification of both temporal and frequency components, a new threshold is computed after subtracting the contribution of identifified signals.

Furthermore, when multiple interfering signals appear in the same frequency band, they are separated during the process of temporal components estimation. The interested reader can find more details in [11].

VI. CONCLUSION

The reconfigurability of embedded radio devices for wireless train to ground communications is a major issue for railway industry in order to avoid the development of proprietary communication systems and to take advantage of the latest development in wireless telecommunication standards. Spec- trum monitoring is one of the first steps in the reconfigurability process of the radio devices. In this paper, we presented a blind estimation technique aimed at MIMO-MC-DS-CDMA transmission parameters estimation for vehicular applications.

The described scheme leads to an efficient estimation of symbol periods, cyclic prefix duration. Furthermore, theoretical results proved its insensitivity to phase and frequency offsets, and also to channels gains. Simulations results given for different scenarios confirm theoretical developments. Moreover, this method can be used in several applications in the field of cognitive radios such that spectrum monitoring, dynamic spectrum management, power control strategies (detect and avoid) through the assessment of transmission parameters.

ACKNOWLEDGEMENT

This work was performed in the framework of the CISIT project (Campus Interdisciplinaire de recherche, d’Innovation Technologique et de formation à vocation Internationale centrè sur la Sécurité et l’Intermodalité des Transports de surface) and MOCAMIMODYN project (MOdélisation des CAnaux MIMO DYNamiques) supported by the North Region, the FEDER and the National Research Agency (ANR).

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