ARTICLE IN PRESS

Contents lists available at ScienceDirect

Signal Processing

Signal Processing 90 (2010) 24–33

0165-16

doi:10.1

� Cor

E-m

fmercha

(F. Turc

(E. Griv

journal homepage: www.elsevier.com/locate/sigpro

Rayleigh fading channel simulator based oninner–outer factorization

Fernando Merchan �, Flavius Turcu, Eric Grivel, Mohamed Najim

IMS—Departement LAPS, Universite de Bordeaux 1, 351 cours de la Liberation, 33405 cedex, Talence, France

a r t i c l e i n f o

Article history:

Received 29 July 2008

Received in revised form

10 April 2009

Accepted 15 May 2009Available online 22 May 2009

Keywords:

Moving average processes

Autoregressive processes

Autoregressive moving average processes

Rayleigh fading channel

Inner–outer factorization

84/$ - see front matter & 2009 Elsevier B.V. A

016/j.sigpro.2009.05.010

responding author. Tel.: +33 5 40 00 66 74; fax

ail addresses: fernando.merchan@laps.ims-bo

n@yahoo.com (F. Merchan), flavius.turcu@lap

u), eric.grivel@laps.ims-bordeaux.fr

el), mohamed.najim@laps.ims-bordeaux.fr (M

a b s t r a c t

The paper deals with the design of Rayleigh fading channel simulators based on the

inner–outer factorization. The core of the approach is to approximate the outer

spectral factor of the channel power spectral density (PSD) by either finite-order

polynomials or rational functions. This, respectively, leads to MA or AR/ARMA

models. The parameter estimation operates in two steps: the outer factor,

which leads to a minimum-phase filter, is first evaluated inside the unit disk

of the z-plane. Then, we propose to compute the Taylor expansion coefficients of the

outer factor because they coincide with the model parameters. Unlike other

simulation techniques, this has the advantage that the first p parameters remain

unchanged when one increases the model order from p to p+1. A comparative study with

existing channel simulation approaches points out the relevance of our ARMA model-

based method. Moreover, the ARMA model weakens the oscillatory deviations from

the theoretical PSD in the case of AR models, or low peaks at the Doppler frequencies for

MA models.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

When designing communication systems basedfor instance on CDMA techniques [1] or when conceivingnew receivers and studying their performances,channel simulation is one of the steps to be carriedout.

In an environment with no direct line-of-sight betweentransmitter and receiver, the marginal distributions of thephase and of the amplitude of the channel process areuniform and Rayleigh, respectively [2]. In addition, thetheoretical power spectral density (PSD) of the real andimaginary parts of the fading channel samples is

ll rights reserved.

: +33 5 40 00 84 06.

rdeaux.fr,

s.ims-bordeaux.fr

. Najim).

U-shaped and has two infinite peaks at the normalizedmaximum Doppler frequency �f d:

Sthðf Þ ¼

1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2

d � f 2q if f

�� �� � f d

0 elsewhere

8>><>>: (1)

while the corresponding normalized discrete-time auto-correlation function is given by the zero-order Besselfunction of the first kind:

Rtheoh ðkÞ ¼ J0ð2pf djkjÞ 8 k 2 Z (2)

Given these assumptions, various families of channelsimulators have been proposed.

For instance, Jakes’ fading model [2] based on a sum-of-sinusoids makes it possible to generate time-correlatedwaveforms. However, as independent channels cannot beeasily simulated with this model, Dent et al. [3] propose toweight the sinusoids by orthogonal random codes such as

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F. Merchan et al. / Signal Processing 90 (2010) 24–33 25

Hadamard ones. In [4], Pop and Beaulieu report that Jakes’simulator is wide-sense nonstationary and hence, proposea modified simulator by adding random phases in the low-frequency oscillators. Nevertheless, in this simulator, theautocorrelation and the cross-correlations of the quad-rature components fail to match the desired propertieseven when the number of sinusoids tends to infinity [5].Therefore, Zheng and Xiao [6] propose a new simulatorthat introduces randomness to the path gain, the Dopplerfrequency and the initial phase of the sinusoids. Inaddition, the authors proved that the second-orderstatistics of this simulator could correspond to the desiredones.

As an alternative, filtering-based approaches can beconsidered:

On the one hand, the filter can be designed in thefrequency-domain. Thus, in [7], the author combines afiltering step and an inverse discrete Fourier transform(IDFT). Although Young and Beaulieu [8] manage toreduce the computational cost of this approach, allsamples have to be generated by using a single fastFourier transform. Due to the IDFT, this off-line simulationrequires large memory storage.

On the other hand, the filtering can be carried out inthe time-domain. This is the case of simulators based onAutoregressive Moving Average ARMA, AutoRegressive(AR) or Moving Average (MA) models. This approachmay be a priori questionable. Indeed, as the PSD of thereal and imaginary parts is bandlimited, the channelprocess should be deterministic, according to the Kolmo-gorov–Szego formula.1 However, unlike the sum-of-sinu-soids methods, since the linear stochastic models arequite simple and few parameters have to be estimated,these approaches are very popular both for channelsimulator design and Kalman-filter based receiver designin mobile communication system. Thus, in [9], the transferfunction associated to the ARMA model corresponds to a3rd order Butterworth low-pass filter. Nevertheless,choosing this kind of filter leads to a poor approximationof the channel properties. In [10] and [11], an ARMAprocess followed by a multistage interpolator is consid-ered. The authors in [10] report that this combinationmakes it possible to select low orders for the ARMAprocess. Nevertheless, only a very high down-samplingfactor can lead to a PSD which is never equal to zero andhence allows the simulated channel to match thetheoretical channel properties.

In various papers including [12] and [13], the authorssuggest using a pth order AR process because it cangenerate at most p resonances in the PSD [13]. However,when selecting a 2nd order AR model whose parametersare obtained by solving the Yule–Walker equations, thetwo PSD peaks are located at �f d=

ffiffiffi2p

instead of �f d [13].This factor 1=

ffiffiffi2p

is justified by comparing the Taylorseries expansions of both the theoretical autocorrelationfunction Rtheo

h ðkÞ and the 2nd order AR process. When theAR order is higher than 2, the channel autocorrelation

1 s2 ¼ exp½R 1=2�1=2 ln Sðf Þdf � where Sðf Þ denotes the PSD of the process

sðnÞ and s2 is the error variance of the optimal predictor of sðnÞ.

matrix used in the Yule–Walker equations becomes ill-conditioned [13]. To overcome this problem, Baddour andBeaulieu [12] suggest adding a very small bias s2 in themain diagonal of the autocorrelation matrix of thechannel (e.g., s2 ¼ 10�7 for f d ¼ 0:01). This leads tomodeling the channel by the sum of the theoretical fadingprocess and a zero-mean white process with variance s2

and hence makes the PSD of the process log-integrable.In [14], the authors model the channel by a finite-order

moving average process. The MA parameter estimationconsists in designing a FIR filter by means of the windowmethod. Indeed, the impulse response coefficients,namely the MA parameters, are estimated by first takingthe inverse Fourier transform of the square root of thetheoretical PSD of the channel, by windowing it and thenby shifting it in time. However, the first p MA parameterschange when increasing the model order from p to p+1.Durbin’s method [15], which is a standard estimationtechnique, can be also considered to estimate the MAparameters. Since this method turns the MA parameterestimation issue into a set of two of AR parameterestimation problems, the channel autocorrelation matrixused in the Yule–Walker equations may be ill conditioned.Hence, Baddour’s ad-hoc solution [12] must be consid-ered. In addition, like Verdin’s approach [14], the selectionof the model order has to be addressed.

To compensate for the drawbacks of the aboveapproaches, Grolleau et al. [17] recently suggest using asinusoidal stochastic model; it corresponds to a filteredversion of a sum of two sinusoids in quadrature at themaximum Doppler frequency and whose amplitudes areAR processes. Nevertheless, the estimation of the ARparameters is not necessarily straightforward and requiresmetaheuristics such as Genetic Algorithms when themaximum Doppler frequency is lower than 1=2p.

In this paper, to estimate the channel model para-meters and select the model order, we propose to considerthe inner–outer factorization, which has been used so farin information theory, system theory, control and filterdesign [16]. Thus, when dealing with MA modeling, thecorresponding transfer function can be expressed as theproduct of:

(1)

the inner function which can be seen as an all-passfilter;

(2)

the outer function which is causal and has its zeroes inthe closed unit disk in the z-plane; hence, thiscorresponds to a minimum-phase filter. In addition,the square absolute value of the outer function on theunit circle in the z-plane equals the theoretical PSD ofthe channel. Furthermore, among all functions whichhave the same absolute value on the unit circle in thez-plane, the outer function is the one with the largestabsolute value outside the unit disk.

In the following, we first suggest using an infinite-

order MA model whose transfer function coincides withits outer function. Since finite-order models are ratherconsidered in practical case, the approach we proposemakes it possible to tune the upper bound of the error, inabsolute value, between the theoretical PSD and the

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F. Merchan et al. / Signal Processing 90 (2010) 24–3326

simulated one. This tuning leads to the definition of acriterion for the model order selection. In addition, withour method, the first p parameters are unchanged whenincreasing the model order from p to p+1. It should benoted that our approach is also adapted to AR and ARMAmodels.

The remainder of the paper is organized as follows:Section 2 presents the proposed approach to estimate themodel parameters. In Section 3, the method is derived forthe MA, AR and ARMA models. It also presents how toselect the model order for MA and AR models. Section 4describes the numerical implementation of the estimationmethod. In Section 5, a comparative study illustrates thebenefits of the proposed ARMA methods with respect tothe methods presented in [12,14,15,17]. Conclusions arepresented in Section 6.

2. Mathematical overview of the method

Let xðnÞ be a real regular process and Pðf Þ denote itsPSD. It is known that, whenever Pðf Þ is log-integrable, theprocess PSD can be factored as follows:

Pðf Þ ¼ jHðzÞj2jz¼expðj2pf Þ ¼ HðzÞH�ðzÞjz¼expðj2pf Þ (3)

for causal functions HðzÞ ¼Pþ1

k¼0bkz�k. The functions HðzÞ,called spectral factors of Pðf Þ, can also be seen as thetransfer functions of a system driven by a zero-meanwhite noise, with output xðnÞ.

In the following we will rather deal with causalfunctions of z instead of z�1, in order to fit into the settingof Hardy spaces [18]. Therefore, instead of HðzÞ, we willconsider the spectral factors:

FðzÞ ¼Xþ1k¼0

bkzk (4)

It also satisfies (3) as long as fbkgk�0 are real, since

jHðzÞj2 ¼ jFðz�1Þj2 ¼ jF�ðzÞj2 for z ¼ expðj2pf Þ (5)

A spectral factor FðzÞ is not unique in general; they are allof the type [18]:

FðzÞ ¼ UðzÞF0ðzÞ (6)

where U is any inner function, i.e. jUðzÞjz¼expðj2pf Þ ¼ 1, andF0 is the unique outer function satisfying (3). For the sakeof simplicity, we consider the particular case whereUðzÞ ¼ 1, meaning that the spectral factor F is outer.According to (6), selecting any other function will onlyaffect the argument of HðzÞ.

The estimation methods we propose operate in two steps:

(1)

Estimation of outer the factor F0: the factor F0 iscomputed on a circle of radius ro1 in the z-plane bymeans of the following Poisson kernel representation:

F0ðrej2puÞ ¼ exp

Z 1=2

�1=2

1

2ð2pÞ1þ rej2pðu�f Þ

1� rej2pðu�f Þ

� �log jPðf Þjdf

(7)

(2)

Estimation of the coefficients fbkgk�0: according to (6),the outer function can be expressed either in terms

of the coefficients fbkgk�0 or by using the Taylorseries.

F0ðzÞ ¼ F0ð0Þ þFð1Þ0 ð0Þ

1!zþ � � � þ

FðnÞ0 ð0Þ

n!zn þ � � � ¼

Xþ1k¼0

bkzk

(8)

Hence, by using the Cauchy formula on the circle ofradius r in the z-plane, the coefficients fbkgk�0 can beobtained as follows:

bk ¼FðkÞ0 ð0Þ

k!¼

1

rk

Z 2p

0F0ðrej2puÞe�jk2pu du 8 k 2 N (9)

Since Pðf Þ is integrable, the coefficients bk

� �k�0

arenecessarily square summable, meaning that thefinite-order polynomials

FpðzÞ ¼Xp

k¼1

bkzk (10)

converge to F0 in the L2 norm when p!þ1.

The methods we propose in the remainder of the paperdeal with three types of finite-order approximations of thespectral factor F0 and lead to MA, AR and ARMA models:

(1)

The pth order MA model, denoted MA(p), is the moststraightforward and corresponds to the estimation ofFp in (10)

(2)

A pth order AR model, denoted AR(p), is obtained byapplying the above finite-order estimation to theinverse of Pðf Þ.

(3)

An ARMA model of order p and q, denoted ARMA(p,q),model is derived by approximating F0 with rationalfunctions of order ðp;qÞ:

Fp;qðzÞ ¼

Ppk¼0bkz�k

1þPq

k¼0akz�k(11)

The following sections describe each approach andhow to implement them in practical cases.

3. Inner–outer factorization for MA, AR and ARMAchannel modeling

3.1. MA model for Rayleigh channel simulator

Let xðnÞ be an infinite-order MA process defined asfollows:

xðnÞ ¼Xþ1k¼0

bkuðn� kÞ (12)

where fbkgk�0 are the real MA parameters and uðnÞ is a zero-mean white noise sequence with unit variance. If xðnÞ has itsPSD equal to the channel’s theoretical power spectral densitySth, the corresponding transfer function HðzÞ ¼

Pþ1k¼0bkz�k

satisfies:

Sthðf Þ ¼ jHðzÞj2jz¼expðj2pf Þ (13)

Since parameters fbkgk�0 are real, we can rewrite (5) asfollows:

jHðzÞj2 ¼ jFðz�1Þj2 for z ¼ expðj2pf Þ (14)

ARTICLE IN PRESS

2 This choice of modified PSD for the AR model is due to the

difficulty of dealing with very large values of the inverse of the PSD. In

Section 4, this modified PSD is compensated by filtering the resulting AR

process by a low-pass filter.

F. Merchan et al. / Signal Processing 90 (2010) 24–33 27

Following the approach presented in the previous section, thefunction FðzÞ can be expressed as the product of its inner andouter factors. Taking the same assumption as above, i.e. theinner factor equal to 1, the MA parameters fbkgk�0 coincidewith the coefficients of the Taylor series of the outer functionof FðzÞ.

We propose to estimate the outer function FðzÞ and theMA parameters by using, respectively, (7) and (9).

However, to use (7) the PSD must be log-integrable, i.e.,the PSD cannot be bandlimited. This is not the case of thetheoretical channel PSD Sth

ðf Þ. In [13], the authors takeinto account that issue by adding a white noise of variances2 to the original process. Here, we also slightly modifythe channel PSD and express it as the product of twofactors defined as follows:

SUðf Þ ¼

1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2

d � f 2q if jf j � f d

1 elsewhere

8>><>>: (15)

and

Sflatðf Þ ¼

1 if jf j � ð1þ xÞf d

� elsewhere

�(16)

where the parameter � guarantees the log-integrabilitycondition whereas x allows the compensation of the offsetof the frequency peaks at f d. In counterpart, it affects thedecay of the PSD at f d. In the following, Fmod

0 denotes theouter function computed with (7) where Pðf Þ is replacedby the modified PSD, namely SUSflat.

In practical case, a truncated version of the MA modelFpðzÞ ¼

Ppk¼0bkzk is considered. In order to estimate the

modeling error we will use the L2 norm on the unit circle,i.e.

J ¼

Z p

�pjSthðejyÞ � jFpðe

jyÞj2jdy where ejy ¼ ej2pf (17)

Note that Fp also depends on �. It is clear that Fp does notconverge to Sth, since Fp converges to � outside½e�j2pf d ; ej2pf d � while Sth is zero on this interval. However,given a prescribed error d0, one can choose both � and p

such that Jod0. We indicate here how one can calculate anupper bound of the modeling error for a given order p.However, we do not address the question of selecting anoptimal order for a given modeling error.

The error J can be upper-bounded as follows:

J �

Z p

�pjSthðejyÞ � jFmod

0 ðejyÞj2jdy

þ

Z p

�pjjFmod

0 ðejyÞj2 � jFpðe

jyÞj2jdy (18)

According to Eqs. (1), (15) and (16), the first term in theinequality above is equal to 2p�ð1� 2f dÞ, while, thesecond satisfies:

Z p

�pjjFmod

0 ðejyÞj2 � jFpðe

jyÞj2jdy

�

Z p

�pjFmod

0 ðejyÞ2 � Fpðe

jyÞ2jdy

¼

Z p

�pjFmod

0 ðejyÞ � FpðejyÞj � jFmod

0 ðejyÞ þ FpðejyÞjdy

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ p

�pjFmod

0 ðejyÞ � FpðejyÞj2 dy

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ p

�pjFmod

0 ðejyÞ þ FpðejyÞj2 dy

s

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2�Xp

k¼0

jbkj2

vuutffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXp

k¼0

4jbkj2 þ

Xþ1k¼pþ1

jbkj2

vuut

� 2E

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2�Xp

k¼0

jbkj2

vuut (19)

where E ¼R p�p jF

mod0 ðe

jyÞj2 dy ¼ 2pð1þ �ð1� 2f dÞÞ.So, one has:

J � 2p�ð1� 2f dÞ þ 2E

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2�Xp

k¼0

jbkj2

vuut (20)

At that stage, given a prescribed error boundd042p�ð1� 2f dÞ, there is always a minimum value forthe model order p that satisfies:

0oE2�Xp

k¼0

jbkj2oðd0 � 2p�ð1� 2f dÞÞ

2ð2EÞ�2 (21)

In the next section, the approach for the case of ARprocesses is presented.

3.2. AR model for Rayleigh channel simulator

Let us now consider the infinite-order AR process xðnÞ

defined by:

xðnÞ ¼ �Xþ1k¼0

akxðn� kÞ þ uðnÞ (22)

In that case, FARðzÞ ¼

Pþ1k¼0akzk verifies:

Sthðf Þ ¼ jHðzÞj2jz¼expðj2pf Þ ¼

1

jFARðz�1Þj2

�����z¼expðj2pf Þ

(23)

To evaluate the outer function of FARðzÞ and the AR

parameters, Pðf Þ is replaced by 1=Sthðf Þ in Eq. (7). Then, the

AR parameters can be estimated by using the methoddescribed in Section 2 based on (9).

In addition, the modified2 PSD is equal to SUðf Þ.

Furthermore, one can calculate an upper bound of themodeling error G for a AR model of order q 1=FqðzÞ ¼

1=Pq

k¼0akzk as follows:

G ¼

Z 2pf d

�2pf d

j1=SthðejyÞ � jFqðe

jyÞj2jdy where ejy ¼ ej2pf

�

Z p

�pjjFmod

0 ðejyÞj2 � jFqðe

jyÞj2jdy

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW2�Xq

k¼0

jakj2

vuut 2W (24)

where W ¼R p�p jF

mod0 ðe

jyÞj2 dy ¼ p3f 2d þ 2pð1� 2f dÞ.

ARTICLE IN PRESS

Table 1Evaluation of error Jd and quality measures Gmean and Gmax for various simulators.a

Number of parameters Jd Gmean ðdBÞ Gmax ðdBÞ

f d ¼ 0:01 f d ¼ 0:05 f d ¼ 0:01 f d ¼ 0:05 f d ¼ 0:01 f d ¼ 0:05

50

Proposed ARMA 0.8826 0.2495 1.0483 0.4397 1.1128 0.4556

ARMA (window meth.)b 0.5698 0.2164 1.0712 0.3994 1.1334 0.4208Proposed AR 0.6979 0.3318 10.3672 6.3017 10.4001 6.3171

Baddour AR [12] 0.5570 0.6317 1.5935 2.4487 1.7659 2.6832

Proposed MA 0.9764 0.4263 4.0557 6.1530 4.2903 6.4728

Verdin MA [14] 0.6040 0.3591 3.2063 3.0014 3.4206 3.1852

Durbin MA [15] 0.9859 0.4961 4.6373 10.4704 4.8668 10.7523

SS [17] 0.7652 0.4132 3.7491 0.7306 3.7874 0.8953

120

Proposed ARMA 0.4116 0.1251 0.5989 0.1680 0.6163 0.1735

ARMA (window meth.)b 0.2805 0.1173 0.3679 0.1592 0.4056 0.1643Proposed AR 2.6900 0.3014 16.6011 6.0031 16.7239 6.0468

Baddour AR [12] 0.8008 0.3382 2.1303 0.7204 2.3946 0.7940

Proposed MA 0.6435 0.2960 7.5843 3.9450 7.8907 4.1729

Verdin MA [14] 0.5307 0.3428 3.0998 2.0063 3.3135 2.1541

Durbin MA [15] 0.8234 0.3762 8.4128 9.4176 8.6786 9.7032

300

Proposed ARMA 0.2303 0.0664 0.2785 0.0515 0.3450 0.0538ARMA (window meth.)b 0.1533 0.0685 0.0617 0.0581 0.2624 0.0609

Proposed AR 1.7068 0.2889 14.6827 5.6942 14.7964 5.7240

Baddour AR [12] 0.5695 0.2475 0.5434 0.1830 0.7038 0.2514

Proposed MA 0.4308 0.1718 5.1412 1.4344 5.4138 1.5508

Verdin MA [14] 0.3831 0.1963 1.9042 0.9865 2.0559 1.0713

Durbin MA [15] 0.5850 0.2489 11.8706 7.4949 12.1374 7.7233

a For MA(p) and AR(p) models the number of parameters is the order p. For ARMA (p, q) models the number of parameters is p+q. For the proposed

simulator, we set: N ¼ 4096. In addition, L ¼ 131 072 and D ¼ 1024.b See Remark in II.C.

0.046 0.048 0.050

10

20

30

40

50

60

normalized frequencies0.046 0.048 0.05

0

10

20

30

40

50

60

normalized frequencies

Theoretical PSD

Proposed MA (1000)Durbin-MA (1000) [15]

Theoretical PSD

Proposed MA (1000)Verdin-MA (1000) [14]

Fig. 1. PSD of MA-based simulators with f d ¼ 0:05.

F. Merchan et al. / Signal Processing 90 (2010) 24–3328

ARTICLE IN PRESS

0.03 0.035 0.04 0.045 0.050

10

20

30

40

50

60

70

80

90

100

110

normalized frequencies

Theoretical PSDProposed ARMA (2,48)Komninakis ARMA [11]

Fig. 3. PSD of simulators with f d ¼ 0:05 and 50 parameters.

F. Merchan et al. / Signal Processing 90 (2010) 24–33 29

Therefore, by considering a prescribed bound r040,the model order q can be chosen such as:

0oW2�Xq

k¼0

jakj2or0ð2WÞ�2 (25)

In the next section, we consider the case of ARMAprocesses.

3.3. ARMA model for Rayleigh channel simulator

Let us now consider the ARMA process xðnÞ defined by:

xðnÞ ¼ �Xq

k¼1

akxðn� kÞ þXþ1k¼0

bkuðn� kÞ (26)

The corresponding transfer function:

HðzÞ ¼

Pþ1k¼0bkz�k

1þPq

k¼1akz�k¼

BðzÞ

AðzÞ(27)

satisfies: jHðzÞjjz¼expðj2pf Þ ¼ Sthðf Þ.

Given (1), we propose to define denominator AðzÞ as asecond-order polynomial with roots equal to re�j2pf d , withr close, but inferior to 1. Therefore, the numerator BðzÞ canbe defined as follows:

jBðzÞj2jz¼expðj2pf Þ ¼ Sthðf ÞjAðzÞj2jz¼exp j2pfð Þ (28)

To evaluate the outer function of Bðz�1Þ, Pðf Þ in Eq. (7) isreplaced by the right side of Eq. (28). Then, the parametersfbkgk�0 can be estimated by using the method presented inSection 2 based on (9).

Moreover, since a finite polynomial approximation ofthe outer factor of BðzÞ has no roots equals to re�j2pf d , theresulting transfer function still admits two poles equal tore�j2pf d . This leads to a better fitting of the model PSD in asmall neighborhood of Doppler frequencies.

Remark. An alternative way to estimate the MAparameters of this ARMA model consists in using thewindow method for the design of FIR filters, as used in

0.034

0.036

0.038 0.0

40.0

420.0

440.0

460.0

48 0.05

0.052

0

50

100

150

normalized frequencies

Theoretical PSDProposed ARMA (2,298)Baddour AR (300) [12]

Fig. 2. PSD of simulators with f d ¼ 0:05 and 300 parameters.

[14] for a MA model. Thus, we also propose to obtain theMA parameters by first taking the inverse Fourier trans-form of the square root of the right side of Eq. (28), bywindowing it and then by shifting it in time.

The purpose of the next section is to estimate the outerfunction and its Taylor coefficients, in practical cases.

4. Implementation

To illustrate the way the simulation method can beimplemented, let us consider the MA model for the sake ofsimplicity.

Since the integrand in (7) cannot be computedanalytically, we suggest estimating the values of the outerfunction for a finite number of points on a circle of radiusro1 in the z-plane.

For this purpose, let us introduce N values of thediscrete PSD SU

ðm=NÞ:

SUðm=NÞ ¼

1 m ¼ �N=2; . . . ;�md � 1

1

pf d

mdp2� arcsin

md � 1

md

� �� �m ¼ �md

1

pf d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðm=mdÞ

2q �md þ 1 � m � md � 1

1

pf d

mdp2� arcsin

md � 1

md

� �� �m ¼ md

1 m ¼ md þ 1; :::;N=2� 1

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

(29)

where md ¼ f dN

is the largest frequency index that doesnot exceed the maximum Doppler frequency. The re-sponse for m ¼ md is chosen such that the area under aninterpolation of the spectrum coefficients is equal to thearea under the continuous-time spectrum curve given in(1), as shown in [8].

Given f d, N must be chosen high enough so that thediscrete version of the PSD can be relevant. Then, for agiven radius r, the outer function FU of SU can becomputed by using the discrete version of the Poisson

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F. Merchan et al. / Signal Processing 90 (2010) 24–3330

integral formula (7) for �N=2 � noN=2:

FUðn=NÞ ¼ exp

XN=2�1

m¼�N=2

1

2N

1þ rej2pðn�mÞ=N

1� rej2pðn�mÞ=N

� �logðSU

ðm=NÞÞ

24

35

¼ exp1

2N

1þ rej2pn=M

1� rej2pn=M

� �� logðSU

ðn=NÞÞ

� �(30)

where * denotes the convolution.Thus, (30) can be implemented through a fast

convolution algorithm [19]. Moreover, the outer function

0.03 0.035 0.04 0.045 0.050

10

20

30

40

50

60

70

normalized frequencies

10

10

10

1

Theoretical PSDProposed ARMAARMA (window meth.)

Fig. 5. PSD of proposed ARMA methods with f d ¼ 0:05 an

0.03 0.035 0.04 0.045 0.050

10

20

30

40

50

60

70

80

90

100

110Theoretical PSDProposed ARMA (2,48)SS (50) [17]

Fig. 4. PSD of simulators with f d ¼ 0:05 and 50 parameters.

Fflat of Sflat is expressed by:

Fflatðn=NÞ ¼ exp

Z 1=2

f¼�1=2

1

2ð2pÞ1þ rej2p n=N�fð Þ

1� rej2p n=N�fð Þ

!log jSflat

ðf Þjdf

(31)

To calculate the Taylor coefficients, the following discreteversion of Eq. (9) is considered:

bðkÞ ¼1

Nrk

XN�1

n¼0

FUðn=NÞ � Fflat

ðn=NÞe�j2nkðn=NÞ (32)

0 0.1 0.2 0.3 0.4

-15

-10

-5

00

normalized frequencies

Theoretical PSDProposed ARMAARMA (window meth.)

d 120 parameters in linear and logarithmical scale.

0 100 200 300 400 500 600 700 800 900 1000-0.5

0

0.5

1

Lag

TheoryBaddour AR(120) [12]Proposed ARMA (2,118)

Fig. 6. Autocorrelation functions for methods with f d ¼ 0:01 and 120

parameters.

ARTICLE IN PRESS

Table 2Values of pole modulus r for the proposed ARMA simulator.

Number of parameters f d Poles modulus r minimizing criteria

Jd D ¼ 200 D ¼ 512 D ¼ 1024

Gmean Gmax Gmean Gmax Gmean Gmax

50 0.05 0.99410 0.99280 0.99270 0.99330 0.99330 0.99370 0.99380

0.10 0.99240 0.99170 0.99160 0.99230 0.99240 0.99290 0.99300

0.15 0.99140 0.99100 0.99100 0.99190 0.99200 0.99250 0.99260

120 0.01 0.99780 0.99695 0.99690 0.99705 0.99700 0.99715 0.99715

0.05 0.99700 0.99645 0.99640 0.99665 0.99665 0.99685 0.99685

0.10 0.99630 0.99600 0.99600 0.99630 0.99625 0.99655 0.99660

0.15 0.99580 0.99100 0.99560 0.99605 0.99605 0.99640 0.99645

300 0.01 0.99905 0.99885 0.998800 0.99880 0.99875 0.99880 0.99880

0.05 0.99880 0.99865 0.998650 0.99860 0.99860 0.99870 0.99865

0.10 0.99920 0.99920 0.99920 0.99915 0.99910 0.99910 0.99910

0.15 0.99910 0.99905 0.999050 0.99910 0.99910 0.99915 0.99910

Table 3Effect of parameter N in the proposed methods.a

Method N J Gmean ðdBÞ Gmax ðdBÞ

Proposed ARMA 4096 0.1251 0.1680 0.1735

2048 0.1231 0.1652 0.1705

1024 0.1211 0.1627 0.1676

512 0.1321 0.1766 0.1811

256 0.1815 0.2495 0.2545

128 0.4990 0.6878 0.6958

Proposed MA 4096 0.2960 3.9450 4.1729

2048 0.2962 3.8539 4.0793

1024 0.2966 3.6969 3.9179

512 0.3093 3.7416 3.9663

256 0.3473 4.0266 4.2646

128 0.5931 4.0372 4.2639

Proposed AR 4096 0.3014 6.0031 6.0468

2048 0.3111 5.7253 5.7676

1024 0.3340 5.2206 5.2599

512 0.4077 6.9288 6.9754

256 0.7342 10.7168 10.7601

128 0.7063 8.0198 8.0408

a With f d ¼ 0:05 and number of parameters 120.

3 Simulations showed that when ro0:965 and r40:998, the square

of the absolute value of the estimated transfer function diverges from the

theoretical PSD.

F. Merchan et al. / Signal Processing 90 (2010) 24–33 31

It should be noted that (32) can be implemented as aweighted FFT of FU

ðn=NÞ � Fflatðn=NÞ.

Therefore, the algorithm requires OðN log NÞ operations.

Remark 1. When dealing with AR modeling, SUðm=NÞ is

replaced by its inverse in eq. (30) and only FUðn=NÞ is

considered in Eq. (32). Furthermore, a low-pass filteringstage is added. The L-length impulse response pðnÞ isdefined by:

pðnÞ ¼ 2f d

sinð2pðn� L=2Þf dÞ

2pðn� L=2Þf d

(33)

where L is high enough to weaken Gibb’s oscillations atdiscontinuities in the frequency domain.

Remark 2. In the case of ARMA models, SUðm=NÞ is

replaced by the product SUðm=NÞjAðzÞj2jz¼expðj2pm=NÞ in (30).

5. Comparative study

In this section, we compare the proposed simulatorswith:

(1)

the AR-based simulator [12], (2) the MA-based simulator given in [14], (3) Durbin’s method for MA parameter estimation [15]

and,

(4) the stochastic sinusoidal (SS) model simulator [17].

For the channel simulators we present in this paper, theparameter r is assigned3 to 0.98 whereas � and x are set to0.0125 and 0.025, respectively. These values are chosenexperimentally for a compromise between the maximumDoppler frequency offset and the PSD decay. For theARMA-based model, different values of the pole modulus,namely r, is chosen in each scenario.

To compare the various simulators, we introduce threecriteria. The first criterion is the mean error Jd defined asfollows:

Jd ¼1

L

XL=2�1

n¼�L=2

jSthðn=LÞ � jHðn=LÞj2j (34)

The second and third criteria are two quality measures,used for instance in [8,12]. The mean power andmaximum power margins Gmean and Gmax are defined by:

Gmean ¼1

s2XD

tracefCXC�1

X CXg (35)

Gmax ¼1

s2X

maxfdiagfCXC�1

X CXgg (36)

where CX and CX are the D D covariance matrices of thetheoretical channel and the simulated channel processes,

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0.044 0.046 0.048 0.050

10

20

30

40

50

60

70

0.044 0.046 0.048 0.050

10

20

30

40

50

60

70

Theory

N = 4096N = 2048

N = 1024

Theory

N = 512N = 256

N = 128

Fig. 7. PSD of ARMA method for different values of N with f d ¼ 0:05 and 120 parameters.

F. Merchan et al. / Signal Processing 90 (2010) 24–3332

respectively, and s2X is set to unity in this case. These

measures are given in dB (i.e. GdB¼ 10 log 10ðGÞ). Thus,

perfect performance corresponds to 0 dB for both mea-sures. Note that Gmean and Gmax measure the similarity ofthe theoretical and the simulated autocorrelation function(ACF) whereas the error Jd evaluates the differencebetween the theoretical and the simulated PSD. Thereader is referred to [20] for more information aboutthese criteria.

According to various tests we carried out and givenTable 1 and Fig. 1, the proposed MA-based method andVerdin simulator [14] provide close results and outper-forms Durbin-based method [15].

According to Table 1 and Figs. 2, 3 and 4, the ARMA-based proposed approaches, i.e. using inner–outer factor-ization and using the window method, outperform theother solutions. More particularly, in Fig. 2, the PSDobtained with the ARMA (2,298) is very close to thetheoretical PSD both in low frequencies and in theneighborhood of the Doppler frequency whereas Bad-dour’s AR-based approach4 shows important oscillationsin the range ð�f d; f dÞ and has a maximum peak offset atDoppler frequency.

According to Fig. 5, the PSD of both ARMA-basedproposed approaches are very close to each other forf � f d. For f4f d the PSD of the approach using thewindow method has values closer to the theoreticalPSD than the PSD of the approach using inner–outerfactorization.

4 The AR-based approach requires Oðp2Þ operations when using

Levinson recursion [21].

According to Fig. 6, the ACF of Baddour’s AR processpresents important deviations from theoretical ACF,especially for lags superior to the order of the model.The ACF the proposed ARMA-based approach presentssmaller differences from the theoretical one.

The SS approach was tested for the case of 50parameters. It leads to the second best performance5 forf d ¼ 0:05.

To choose the root modulus, namely r, for theproposed ARMA-based approach, we have tested valuesvarying from 0.9910 to 0.99990 with step of 5 10�5.Table 2 presents the values minimizing the error criterionJd and quality measures Gmean and Gmax for different valuesof f d and different number of parameters.

As mentioned in Section 4, the number of samples N ofthe discrete PSD used in the proposed methods must bechosen high enough. For instance, we consider theproposed simulators with f d ¼ 0:05 and 120 parameters.According to Table 3 and Figs. 7 and 8, we obtain veryclose results when setting N to f4096;2048;1024 or 512g.For No512, the difference with respect to the theoreticalPSD clearly increases.

6. Conclusions

In this paper, we have investigated the relevance ofinner–outer factorization for Rayleigh fading channelsimulator. We have proposed a new method to estimatethe model parameters of AR, MA and ARMA models. The

5 Given the frequencies tested, i.e.: f d ¼ f0:01;0:05g, genetic algo-

rithms were used to estimate the AR parameters.

ARTICLE IN PRESS

0.01 0.02 0.03 0.04 0.050

2

4

6

8

10

12

14

16

18

20

0.01 0.02 0.03 0.04 0.050

2

4

6

8

10

12

14

16

18

20

Theory

N = 4096N = 2048

N = 1024

Theory

N = 512N = 256

N = 128

Fig. 8. PSD of MA method for different values of N with f d ¼ 0:05 and 120 parameters.

F. Merchan et al. / Signal Processing 90 (2010) 24–33 33

comparative study we have carried out confirms that ourARMA-based simulator outperforms the other approaches.

Acknowledgements

This work was partially supported by the FrenchMinistry of Foreign Affairs and the National Bureau ofScience, Technology and Innovation of Panama (SENACYT).

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