1236 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 8, AUGUST 2012

Ricean K-Factor and SNR Estimation for M -PSK Modulated SignalsUsing the Fourth-Order Cross-Moments Matrix

Ines Bousnina, M. Bassem Ben Salah, Abdelaziz Samet, Member, IEEE, and Iyad Dayoub, Member, IEEE

Abstract—In wireless communication systems, the Ricean K-factor estimation is required in several applications such aslink budget calculations. While the SNR estimation can beused to measure the link quality. In this paper, the RiceanK-factor estimation for M-ary Phase Shift Keying (M -PSK)modulated signals with additive noise is studied. To this end, theFourth-Order Cross-Moments (FOCM) of the received signal areconsidered. First, the Signal-to-Noise Ratio (SNR) is computedby estimating the powers of the desired components of thesignal and the noise. Then, the K-factor is estimated using thekurtosis of the Ricean channel. Simulation results show that ourapproach outperforms the most recent developed estimator inthe literature.

Index Terms—Joint estimation, K-factor, SNR, Ricean chan-nel, Kurtosis, FOCM.

I. INTRODUCTION

IN wireless communications, the transmitted signal suf-fers from channel impairments due to the encountered

obstacles. Ricean distribution is often used to modelize theinduced multipath phenomenon [1]. One key parameter ofthe mentioned model is the K-factor, which is a measure ofthe severity of fading. Indeed, it is defined as the ratio ofsignal power in the Line-Of-Sight (LOS) component over thescattered power in the Non LOS components [2]. Therefore,the estimation of the Ricean K-factor is a good indicator ofthe channel quality. It is also essential in the link budgetcalculation. Moreover, to evaluate the effect of the additivenoise on the signal reception, the SNR is used as a measureof the link quality [3]. Many systems require the knowledgeof both the Ricean K parameter and the SNR (e.g. link budgetcalculation , adaptive receiver design and optimal powerloading of transmit diversity systems [4]). In our work, wepropose a low complexity approach to estimate simultaneouslythe Ricean K-factor and the SNR.

In the last decade, the K-factor estimation has been in-cluded in several studies [5]-[7]. In [5], a general class ofmoment-based and In-phase and Quadrature-phase (I/Q) basedestimators are introduced. In [6], the estimator is based on thestatistics of the Instantaneous Frequency (IF) of the receivedsignal (i.e. the ratio between the first moment and the ZeroCrossing Rate of the received signal IF is approximated by

Manuscript received March 29, 2012. The associate editor coordinating thereview of this letter and approving it for publication was A. Anpalagan.

I. Bousnina, M. B. Ben Salah, and A. Samet are with UR-CSE,Tunisia Polytechnic School, Carthage University, B.P. 743 - 2078, LaMarsa, Tunisia (e-mail: ines.bousnina@insat.rnu.tn, {bassem.bensalah, abde-laziz.samet}@ept.rnu.tn).

I. Dayoub is with the Univ. Lille Nord de France, IEMN-DOAE, CNRSUMR 8520, F-59313 Valenciennes, France (e-mail: iyad.dayoub@univ-valenciennes.fr).

Digital Object Identifier 10.1109/LCOMM.2012.061912.120708

a polynomial function). Finally, in [7], an analysis of theMaximum Likelihood (ML) estimator of the K-factor fromI/Q samples is presented.

On the other hand, the SNR estimation has been studiedsuch in [8]-[12]. The authors in [8]-[10] have consideredthe Rayleigh channel models. A few works treat the SNRestimation in the Ricean fading. In [11], a moment-basedestimator in Nakagami-m fading channels was derived. Thisestimator requires the knowledge of the Nakagami parameterm. In [12], a ML estimator for the SNR in a Ricean fadingchannel is proposed.

To estimate simultaneously the K-factor and SNR, onecould combine SNR and K-factor estimators among theabove mentioned ones. However, the cited K-factor estimatorsconsider only the channel coefficients. Consequently, theycannot be applied for modulated signals with additive noise. In[13], Chen and Beaulieu used the Auto-Correlation Function(ACF) of the received signal to estimate jointly the K-factorand SNR considering both Data-Aided (DA) and Non-Data-Aided (NDA) designs. To enhance the estimator accuracy, theACF values at two different time-lags are considered for eachparameter. To estimate the noise power, the authors assumethe perfect knowledge of the normalized maximum Dopplershift. Moreover, the NDA estimator can not be applied for M -PSK modulations (e.g. null ACF) which can be considered inseveral applications such as Code Division Multiple Access(CDMA) and Worldwide Interoperability for Microwave Ac-cess (WiMAX).

In this paper, we propose a new approach based on theFOCM of the received signals. This method does not requirethe knowledge of the Doppler shift or the transmitted data.We derive closed-form expressions for the desired parametersafter an analytical development. In fact, the SNR is obtainedusing the subdiagonal terms of the FOCM matrix, then, theK-factor is estimated using the kurtosis of the Ricean channel.The performances of the estimator are examined in terms ofthe Root-Mean-Squared Error (RMSE). Numerical results arepresented and compared to those obtained by [13].

The paper is organized as follows. In section 2, we describethe considered signal model. Then, we use the FOCM matrixof the received signals to develop our algorithm allowing toestimate the K-factor and the SNR. In section 3, simulationresults are discussed. Finally, section 4 concludes this paper.

II. DERIVATION OF THE FOCM-BASED ESTIMATOR

Let us consider an uplink transmission in the case of aSingle Input Multiple Output (SIMO) system where 2-Darbitrary antennas are set at the receiver as illustrated in Fig.1. The expression of the baseband received signal at the ith

1089-7798/12$31.00 c© 2012 IEEE

BOUSNINA et al.: RICEAN K-FACTOR AND SNR ESTIMATION FOR M-PSK MODULATED SIGNALS USING THE FOCM MATRIX 1237

Fig. 1. Multipath propagation in Ricean fading channel.

antenna element (i = 1 · · ·Na) is done by:

yi(t) = hi(t)a(t) + wi(t), (1)

where Na is the number of antenna elements. hi(t) are theRicean channel coefficients, a(t) is the M -PSK transmit-ted signal, and wi(t) is an Additive White Gaussian Noise(AWGN) of zero mean and variance N0. The noise compo-nents at all antenna elements are temporally and spatially un-correlated with equal average power N0, (E[|wi(t)|2] = N0),where E[.] represents the statistical expectation. The secondorder moment of the transmitted signal a(t) is normalized,i.e., E[|a(t)|2] = 1. The channel coefficients hi(t) can bemodeled as the sum of a LOS component and a Rayleighchannel coefficients (hi(t)), called also the diffuse component.So, the expression of the considered channel coefficients is [8]:

hi(t) =

√PiK

K + 1exp {j (ωDcos(θ0)t+ φ0)}+

√Pi

K + 1hi(t),

(2)where Pi is the received signal power at the ith antennaelement, ωD is the maximum Doppler spread, θ0 and φ0 are,respectively, the angle of arrival and the phase of the LOScomponent. The channel coefficients hi(t) have zero meancomplex Gauss value. Let us consider the same noise powerN0 and the same K-factor at each antenna element of thearray.Our purpose is to estimate the Ricean K-factor by estimatingfirst the SNR, ρi, at each antenna element given by:

ρi =E[|hi(t)|2]E[|a(t)|2]

E[|wi(t)|2] =Pi

N0. (3)

Then we use the FOCM of the received signal envelope at theith and jth antenna elements to estimate the powers of bothdesired components and additive noise. The FOCM is definedas following (we will omit the temporal index (t) to simplifythe computation):

M4(i, j) = E[yiy

Hi yjy

Hj

], (4)

where (.)H is the transconjugate operator. We suppose that thechannel coefficients hi, the transmitted signal a and the noisewi are independents. The expression of the FOCM matrix in(4) becomes:

M4(i, j) =

{KaPiPj + PiN0 + PjN0 +N2

0 , for i �= j, (5)

KRKaP2i + 4PiN0 +KwN

20 , for i = j. (6)

where KR =E[|hi|4]E[|hi|2]2 , Ka =

E[|a|4]E[|a|2]2 and Kw =

E[|wi|4]E[|wi|2]2

are respectively the kurtosises of the Ricean channel, thetransmitted signal and the noise. For M -PSK modulations witha complex additive noise, the associated Kurtosies becomeKa = 1 and Kw = 2 [9]. In our approach, we use thekurtosis of the Ricean channel gain to estimate the K-factorby establishing the relationship between the two parameters[3]:

KR = 2− K2

(K + 1)2. (7)

We can note that the estimation of Pi and N0 is requiredto deduce the kurtosis of the Ricean channel. Considering theterms on the upper matrix of the eq. (5), we obtain:

Pi(k, l) +N0(i, k, l) =

√M4(i, k)M4(i, l)

M4(k, l), (8)

where k, l, i = 1...Na and k �= l. The difference between thepowers of the desired components of the received signal at theith and jth antenna elements is:

Pi(k, l)−Pj(m,n) =

√M4(i, k)M4(i, l)

M4(k, l)−√

M4(j,m)M4(j, n)

M4(m,n),

(9)

where m,n, j = 1...Na and m �= n. The powers of usefulsignals can be also extracted from (6). The terms on thediagonal of the matrix M4 can be rewritten as:

M4(i, i) = (KR − 2)P 2i + 2(Pi +N0)

2. (10)

By substituting (8) in (10), we obtain:

(KR − 2)P 2i (k, l) = M4(i, i)− 2

M4(i, k)M4(i, l)

M4(k, l). (11)

Let us consider the ratio Pi(k, l)/Pj(m,n):

Pi(k, l)

Pj(m,n)=

√√√√ M4(i, i)− 2M4(i,k)M4(i,l)M4(k,l)

M4(j, j)− 2M4(j,m)M4(j,n)M4(m,n)

. (12)

From (9) and (12), we can estimate the powers of the usefulsignals at the jth antenna element:

Pj(i, k, l,m, n) =

√M4(i,k)M4(i,l)

M4(k,l)−√

M4(j,m)M4(j,n)

M4(m,n)√M4(i,i)−2

M4(i,k)M4(i,l)

M4(k,l)

M4(j,j)−2M4(j,m)M4(j,n)

M4(m,n)

− 1

,

(13)where M4(i, j) = 1

Ns

∑Ns

i,j |yi(n)|2|yj(n)|2 and Ns is thenumber of the received signal samples.

Let Nc the number of all possible combination of(i, k, l,m, n) in (13). The final Pj estimation is then equalto the average:

Pj =1

Nc

∑k,l,m,n

Pj(i, k, l,m, n). (14)

1238 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 8, AUGUST 2012

Once the powers of the useful signals are estimated, the powerof the noise can be deduced using (8):

N0 =2

Na(Na − 1)

∑k �=l �=j

l>k

(√M4(j, k)M4(j, l)

M4(k, l)− Pj

).

(15)where Na(Na−1)

2 is the number of all possible combination of(j, k, l) in (15). The SNR at the jth antenna element is then:

ρj =Pj

N0

. (16)

The Ricean channel kurtosis can now be estimated by using(6):

KR(j) =M4(j, j) − 4PjN0 − 2N2

0

P 2j

, j = 1...Na. (17)

The K-factor is estimated from the estimated value of theKurtosis using (7), the final estimated value is the averageover the antenna elements:

K =1

Na

Na∑j=1

⎛⎝

√2− KR(j)

1−√2− KR(j)

⎞⎠ . (18)

III. SIMULATION RESULTS

As mentioned before, most K-factor estimators consideronly channel coefficients. There are very limited publicationswhere modulated signals and additive noise are considered.In fact, only Chen and Beaulieu have proposed a joint K-factor and SNR estimator which deals with this problem [13].This is why we do not consider other algorithms as bench-marks. However, it requires the a priori knowledge of theDoppler shift and the transmitted symbols in the case of M -PSK modulation. We illustrate below the performance of ourestimator with comparison to the one developed in [13]. 1000Monte-Carlo simulations with Ns=1024 samples are carriedout. We consider a Quadrature Phase Shift Keying (QPSK)signal transmitted through a Ricean channel. A uniform lineararray with five elements spaced by half-wavelength is used(Na = 5). The channel coefficients are modelized as in (2). Wehave chosen for example the powers of the useful componentsas follows P = [15 12 8 5 2]. The Doppler frequency is set toFd = 40 Hz. Two scenarios corresponding to small and highSNR values are considered. In the first one, the average SNRat the Na antenna elements is 10 dB while in the second is20 dB (e.g. N0 = 0.65 and N0 = 0.065). Several values ofthe powers of the useful signals, the K-factor and the SNRare considered to cover the maximum of possible scenarios.For the DA-ACF estimator, we use two time-lags (L = 1 andL = 30) to estimate respectively the K-Factor and the SNR,as mentioned in [13].

First, we check respectively the estimation of both receivedsignals and noise powers at each antenna. Fig. 2 shows acomparison between the RMSE of the useful signal power P3

(i.e. at the third antenna element) obtained by our approachand the DA-ACF based estimator proposed in [13]. It isclear that the OFCM-based approach outperforms the DA-ACF one. For instance, in the range of K ∈ [4...10], we

0 1 2 3 4 5 6 7 8 9 1010

−2

10−1

100

101

102

True value of K

RM

SE

(P

3)

FOCM, avg

=10 dB

FOCM, avg

=20 dB

DA−ACF, avg

=10 dB

DA−ACF, avg

=20 dB

Fig. 2. RMSE of the power of the useful signal at the 3rd antenna.

0 1 2 3 4 5 6 7 8 9 1010

−2

10−1

100

101

102

True value of K

RM

SE

(N

0avg

)

FOCM, avg

=10 dB

FOCM, avg

=20 dB

DA−ACF, avg

=10 dBB

DA−ACF, avg

=20 dB

Fig. 3. RMSE of the noise power N0.

obtain a difference of 1 dB between the RMSEs given bythe two estimators. The simulation results associated to theother antenna elements show the same behavior. In Fig. 3,we compare the RMSE of the noise power N0 from the twomethods. It is clear that the proposed FOCM-based estimatorshows lower error estimation in particular for high K values.Consequently, the SNR estimation using the FOCM presentsmore accurate results as illustrated in Fig. 4. For the K-factorestimation, as shown in Fig. 5, our estimator achieves a lowerRMSE than the DA-ACF-based estimator for both consideredSNR values especially for large K-factor values. We note thatfor small values of the K-factor (K < 1.5), the DA-ACFpresents better results. In fact, when K → 0 the left term of(11) tends to zero as well. In this case, the ratio Pi(k,l)

Pj(m,n) in(12) leads to erroneous estimation of the useful signal power,Pj , which affects the K-factor estimation when this latter issmall. This is why, our FOCM method shows high RMSE forsmall K values. This result corroborates with the Cramer-RaoBound for moment-based estimators developed in [5]. Indeed,for K = 0 the Cramer-Rao Bound tends to infinity as shownby simulation results presented in [5].

BOUSNINA et al.: RICEAN K-FACTOR AND SNR ESTIMATION FOR M-PSK MODULATED SIGNALS USING THE FOCM MATRIX 1239

0 1 2 3 4 5 6 7 8 9 1010

−1

100

101

102

True value of K

RM

SE

( 3)

FOCM, 3=10dB

FOCM, 3=20dB

DA−ACF, 3=10dB

DA−ACF, 3=20dB

Fig. 4. RMSE of SNR at 3rd antenna.

0 1 2 3 4 5 6 7 8 9 1010

−2

10−1

100

101

102

True value of K

RM

SE

(K

)

FOCM, avg

=10dB

FOCM, avg

=20dB

DA−ACF, avg

=10dB

DA−ACF, avg

=20dB

Fig. 5. RMSE of K-factor.

The new algorithm presents accurate estimates with a com-plexity order of (NsNa +Nc)(Na − 1) floating points oper-ations. As one can notice, the computational complexity canbe significantly reduced if we consider limited combinationsfor the useful signals powers, Pj . In other terms, Nc must bechosen in a way that the number of iterations of the algorithmis decreased.

IV. CONCLUSION

In this paper, a new approach was proposed to estimatesimultaneously the Ricean K-factor and the SNR for SIMOSystems. We presented an analytical development of ouralgorithm using the FOCM matrix. Simulation results werealso proposed and compared with the ACF-based methodpresented in the literature. The comparison results prove ouralgorithm efficiency. Our method allows a better RMSE forhigh K values. In addition, our algorithm does not require anya priori knowledge of the Doppler shift or any pilot symbolsunlike the ACF-based estimator.

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