Design of SISO Wideband Mobile Radio Channel by INLSA Method

REV Journal on Electronics and Communications, Vol. 2, No. 1–2, January – June, 2012 19

Invited Article

An Improved Iterative Nonlinear Least Square ApproximationMethod for the Design of SISO Wideband Mobile Radio ChannelSimulatorsAkmal Fayziyev, Matthias Pätzold

Faculty of Engineering and Science, University of Agder

Correspondence: Matthias Pätzold, [email protected] communication: received 23 March 2012, accepted 3 April 2012

Abstract– In this paper, we present an improved version of the iterative nonlinear least square approximation (INLSA)method for designing measurement-based single-input single-output (SISO) wideband channel simulators. The proposedmethod aims to fit the time-frequency correlation function (TFCF) of the simulation model to that of a measured channel.The parameters of the simulation model are determined iteratively by minimizing the Frobenius norm, which serves as ameasure for the fitting error. In contrast to the original INLSA method, the proposed approach provides a unique optimizedset of model parameters, which guarantees a quasi-perfect fitting with respect to the TFCF. We analyze the performanceof the proposed method in terms of the goodness of fit to the measured data. The investigations will be carried out withrespect to the TFCF and the scattering function. We demonstrate that the proposed approach is a powerful tool for thedesign of measurement-based wideband channel simulators, which are important for the performance evaluation of mobilecommunication systems under real-world propagation conditions.

Keywords– Mobile fading channels, measurement-based channel models, channel simulators, single-input single-output,time-frequency correlation function, iterative parametrization.

This paper was presented in part at the 4th International Conference on Advanced Technologies for Communications (ATC2011), Danang, Vietnam.

1 Introduction

The reliability and the maximum achievable data rateof a communication system are heavily dependent onthe characteristics of the propagation channel. Hence,during the design and test phase, it is of utmost im-portance to have a realistic channel model to beginwith. A lot of research attention has been devoted tothe development of channel models the statistics ofwhich matches very closely that of a real propagationenvironment. The most recent comprehensive surveyon existing modeling approaches is reported in [1], [2].

In general, sophisticated channel models are basedon wideband scenarios, which incorporate the fre-quency selectivity of the physical channel. A widelyused approach to develop wideband channel modelsis to represent the time-variant transfer function by afinite sum of multipath propagation components [3],[4]. The efficient development of wideband channelmodels relies on a sophisticated method for deter-mining the primary model parameters, which are thepath gains, Doppler frequencies, and propagation de-lays. Hence, advanced and efficient parameter com-putation methods are required to accurately estimatethese model parameters from the measured channels.In last couple of decades, various high resolution pa-rameter estimation methods have been proposed to

derive the reference models [5]–[8]. These methodscan also be used as parameter computation techniquesfor designing measurement-based channel simulators.In [5], e.g., the multiple signal classification (MUSIC)algorithm has been applied for delay estimation. A jointangle and delay estimation (JADE) technique is pro-posed in [9]. A direction-of-arrival estimation methodusing the estimation of signal parameters via rota-tional invariance techniques (ESPRIT) algorithm hasbeen discussed in [6]. The unitary ESPRIT has beenproposed in [10] to improve the estimation accuracyof the ESPRIT algorithm. The ESPRIT method hasfurther been extended in [11] to enable a joint delayand azimuth estimation, while the unitary ESPRITtechnique has been used in [12] to perform a jointazimuth and elevation estimation. An application of theunitary ESPRIT algorithm to wide-band channel mea-surements is presented in [13].Owing to the high imple-mentation complexity of the aforementioned methods,maximum-likelihood-based algorithms are preferablefor solving the channel parameter estimation problem.These include the expectation maximization [7] and thespace alternating generalized expectation maximization(SAGE) [8] algorithms. Particularly the SAGE algorithmis widely used for channel parameter estimation due toits excellent performance in most scenarios. However,the SAGE algorithm often requires a large amount

1859-378X–2012-1202 c© 2012 REV

20 REV Journal on Electronics and Communications, Vol. 2, No. 1–2, January – June, 2012

of specular components, which makes the estimationprocess relatively complex. In [14], the authors re-ported a initialization sensitivity problem of the SAGEalgorithm. Several approaches have been proposedin [14]–[17] to mitigate the drawbacks of this algorithm.In [17], the authors proposed the iterative nonlinearleast square approximation (INLSA) method for thedesign of measurement-based wideband channel sim-ulators. The authors showed that the INLSA algorithmprovides a good match to empirical data by keepingthe complexity lower than that of the SAGE algorithm.The INLSA algorithm has further been refined in [18].

In this paper, we extend the INLSA method to allow ajoint estimation of the primary model parameters. TheINLSA algorithm is based on the expectation maximiza-tion approach [7]. In order to reduce the computationalcost, the INLSA algorithm computes the maximum-likelihood estimates of the channel model parametersiteratively. The iterative approach, however, may lead toa poor modeling of the TFCF of the measured channelat the origin. The value at the origin of the TFCFis of crucial importance for computing the character-istic quantities describing the statistics of the fadingchannel, such as the probability density function ofthe envelope, the level-crossing rate, and the averageduration of fades. The original INLSA method [17]offers two alternatives for optimizing the simulationmodel parameters. The first alternative computes thepath gains together with the Doppler frequencies bymodeling the measured temporal autocorrelation func-tion (TACF), and then computes the propagation delaysin a second step from the frequency correlation function(FCF) of the measured channel. This variant of theoriginal INLSA algorithm will be called the INLSA-T-Fmethod. The second alternative computes first the pathgains jointly with the propagation delays by modelingthe measured FCF, whereas the Doppler frequenciesare obtained in a second step from the TACF of themeasured channel. The second variant of the INLSAalgorithm will be referred to as the INLSA-F-T method.These kinds of separate optimizations of the simulationmodel parameters may lead to erroneous parameterestimations when evaluating the resulting channel sim-ulator with respect to the TFCF. Moreover, the INLSA-T-F and INLSA-F-T approaches result in different setsof estimated parameters for identical measured data.

In order to overcome the aforementioned drawbacksof the original INLSA algorithm, the proposed methodcomputes the simulation model parameters jointly byfitting the TFCF of the simulation model to that ofthe measured channel. The proposed method is calledINLSA-TF emphasizing the joint modeling of boththe temporal and frequency correlation functions ofreal-world channels. The proposed algorithm includesadditional error compensation steps in the iterativeestimation procedure to guarantee a perfect fitting atthe origin of the measured TFCF. As a result, theproposed method provides a unique set of optimizedmodel parameters while maintaining a perfect fittingat the origin. In order to demonstrate the performanceof the proposed INLSA-TF method, we implemented

the algorithm in MATLAB R© and compared the newapproach with the two known variants of the originalINLSA method. We focused on a number of issueswhich are of particular interest in channel modeling,such as the fitting accuracy with respect to the mostimportant correlation functions and the scattering func-tion of the measured channel. In addition to this, westudied the performance of the measurement-basedsimulation model as a function of the selected numberof propagation paths.

This paper is structured as follows. In Section 2, weintroduce the wideband channel simulation model. InSection 3, the parameter estimation problem is formu-lated. Section 4 describes the proposed parameter com-putation method. The application to measurement datais presented in Section 5. Finally, Section 6 concludesthis paper.

2 The Wideband Channel Simulation

Model

In this section, we describe the simulation model onwhich we apply the parameter computation methodpresented in this work. We also derive the TFCF ofthe simulation model under the assumption that theautocorrelation ergodicity conditions are fulfilled.

An important objective for the design of channelsimulation models is to guarantee that their statisticalproperties are as close as possible to real-world chan-nels. When using a measurement-based approach, wehave to consider that every measurement provides onlypropagation area specific information for a short periodof time. This means we have to select a proper chan-nel model the parameters of which can be estimatedfrom given snapshot measurements. For this reason,we adopt the widely accepted model from [19], whichrepresents the time-variant frequency response (TVFR)of multipath fading channel as a superposition of afinite number of N propagation paths in the form

H( f ′, t) =N

∑n=1

cnej(2π fnt−2π f ′τ′n+θn), (1)

where t and f ′ are the time and frequency variables,respectively. Each propagation path is characterizedby its path gain cn, propagation delay τ′n, Dopplerfrequency fn, and the phase shift θn. The number ofpropagation paths N determines the accuracy as wellas the complexity of the simulation model. We assumethat the elements of the set {θn} are independent andidentically distributed (i.i.d.) random variables, eachhaving a uniform distribution in the interval [0, 2π).Our aim is to determine the remaining sets of param-eters {cn}, { fn}, and {τ′n} of the channel simulationmodel described in (1) such that the simulation modelhas almost the same statistical characteristics as a givenmeasured channel. For our analysis, it will sufficeto consider the TVFR H( f ′, t) at discrete frequenciesf ′m = −B/2+ m∆ f ′ ∈ [−B/2, B/2], m = 0, 1, . . . , M− 1,and at discrete time instances tk = k∆t ∈ [0, T], k =

A. Fayziyev, M. Pätzold: An Improved INLSA Method for the Design of Mobile Radio Channel Simulators 21

0, 1, . . . , K − 1, where B and T denote the measuredfrequency bandwidth and the observation time inter-val, respectively. The time sampling interval ∆t andthe frequency sampling interval ∆ f ′ are characteristicquantities of the channel sounder used to collect themeasurement data. Consequently, the TVFR H( f ′, t)can be represented as a discrete TVFR H[ f ′m, tk]. Forsimplicity’s sake, it is assumed that the parametersof the simulation model described by (1) fulfill thefollowing conditions

fn 6= fm, τ′n 6= τ′m, ∀ n 6= m, (2)

E{ej(θn−θm)} = 0, ∀ n 6= m, (3)

where E{·} represents the statistical averaging operator.The condition (3) is always fulfilled as {θn} are i.i.d.random variables, which are uniformly distributed overthe interval [0, 2π]. As a result of the imposed con-straints (2) and (3), the channel model in (1) is autocor-relation ergodic with respect to time and frequency [20].In this case, the discrete TFCF R[ν′p, τq] of the simulationmodel can be deduced from a single realization of thediscrete TVFR H[ f ′m, tk] as follows

R[ν′p, τq] = 〈H[ f ′m, tk]H∗[ f ′m + ν′p, tk + τq]〉

=N

∑n=1

c2nej2π(τ′nν′p− fnτq), (4)

where τq = 0, ∆t, . . . , (K − 1)∆t and ν′p =0, ∆ f ′, . . . , (M − 1)∆ f ′. The notation 〈·〉 denotesaveraging over time and frequency.

The preceding equation shows that R[ν′p, τq] dependson the number of paths N, path gains cn, propagationdelays τ′n, and Doppler frequencies fn, but not on thephase shifts θn. The TACF rtk [τq] is obtained from theTFCF R[ν′p, τq] by setting ν′p to zero, i.e.,

rtk [τq] = R[0, τq] =N

∑n=1

c2ne−j2π fnτq . (5)

The FCF r f ′m [ν′p] of the simulation model can be deter-

mined from the TFCF R[ν′p, τq] by setting τq to zero,i.e.,

r f ′m [ν′p] = R[ν′p, 0] =

N

∑n=1

c2nej2πτ′nν′p . (6)

For mathematical convenience, the discrete TFCFR[ν′p, τq] will be represented by an M × K time-frequency correlation matrix (TFCM) R. It is straight-forward to show that the TFCM R can be formulatedas

R =N

∑n=1

c2nPn, (7)

where Pn is an M×K matrix and is defined as follows:

Pn =

N∑

n=1ej2π(τ′nν′0− fnτ0) · · ·

N∑

n=1ej2π(τ′nν′0− fnτK−1)

.... . .

...N∑

n=1ej2π(τ′nν′M−1− fnτ0) · · ·

N∑

n=1ej2π(τ′nν′M−1− fnτK−1)

(8)

3 Problem Formulation

In this section, we describe the measured channel anddiscuss its correlation properties. We also formulate theproblem of determining the simulation model parame-ters.

Let H( f ′, t) denote the TVFR of a real-world channelmeasured at discrete frequencies fm, and at discretetime instances tk. In this case, the TVFR H( f ′, t) can berepresented in the form of a discrete function, whichwill be denoted by H[ f ′m, tk]. Further, we assume thatthe measured TVFR H[ f ′m, tk] is wide-sense station-ary [21] and autocorrelation ergodic with respect totime and frequency. This allows us to obtain the discreteTFCF R[ν′p, τq] of the measured channel from H[ f ′m, tk]by averaging over time and frequency. Thus,

R[ν′p, τq] =1

KM

K−1

∑k=0

M−1

∑m=0

H[ f ′m, tk]H∗[ f ′m + ν′p, tk + τq].

(9)The TACF rtk [τq] and the FCF r f ′m [ν

′p] of the measured

channel are obtained from R[ν′p, τq] in (9) by settingν′p = 0 and τq = 0, respectively, i.e., rtk [τq] = R[0, τq]

and r f ′m [ν′p] = R[ν′p, 0]. For mathematical convenience,

we represent the discrete TFCF H[ f ′m, tk] by an M× KTFCM R.

With reference to the simulation model describedby (1), the problem at hand is to determine the setof parameters P = {N, cn, fn, τ′n} in such a way thatthe statistical properties of the simulation model matchthose of the measured channel. Here, the desired sta-tistical properties are described by the TFCM R of themeasured channel. The problem of determining the setof parameters P can now be formulated as follows:

Given is the measured discrete TVFR H[ f ′m, tk] of areal-world channel. Determine the set of parameters P ={N, cn, fn, τ′n} of the channel simulation model described by(1), such that the TFCM R of the simulation model is asclose as possible (in the Frobenius norm sense) to the TFCMR of the measured channel, i.e.,

P = arg minP

∥∥R− R∥∥

F (10)

where ‖ · ‖F denotes the Frobenius norm.As it was pointed out in [17], the problem for-

mulation in (10) is valid if the measured channel isautocorrelation ergodic.

4 The Proposed Parameter Computation

Method

This section focuses on the description of the INLSA-TFalgorithm proposed for computing the parameters ofthe measurement-based channel simulator. By referringto the problem formulation discussed in Section 3 andassuming that the discrete TFCM R of the measuredchannel is known, we measure the model error betweenthe measured channel and the simulation model by

E(P) =∥∥∥R−

N

∑n=1

c2nPn

∥∥∥F. (11)


The objective is to minimize the Frobenius normin (11), such that the TFCM R of the simulation isfitted as close as possible to the given TFCM R ofthe measured channel. Owing to the complexity ofthis optimization problem, the implementation of themaximum-likelihood estimation of model parametersthat minimizes the error in (11) is intractable. Therefore,the proposed method utilizes the iterative approach tooptimize the simulation model parameters. The itera-tive parameter computation method starts with N = 1and arbitrarily chosen initial values for f (0)1 and τ

′(0)1 .

At every iteration i (i = 0, 1, 2, . . . ), the algorithmcomputes the complete set of model parameters P .The parameter computation procedure is carried outseparately for each path l (l = 1, 2, . . . , N) by using theauxiliary error matrix as follows:

Y(i)l = R−

N

∑n=1,n 6=l

(c(i)n

)2P(i)

n (12)

c(i+1)l = arg min

cl

∥∥∥Y(i)l − c2

l P(i)l

∥∥∥F

(13)

f (i+1)l = arg min

fl

∥∥∥∥Y(i)l −

(c(i+1)

l

)2P(i)

l

∥∥∥∥F

(14)

τ′(i+1)l = arg min

τ′l

∥∥∥∥Y(i)l −

(c(i+1)

l

)2P(i+1)

l

∥∥∥∥F

(15)

where the M × K matrix P(i+1)l is recomputed by

substituting the optimized values of f (i+1)l in (8). In

this algorithm, the iterative computation of the set ofparameters P continues as long as the relative changein the model error E(P) is larger than a predefinedthreshold level ε. When the threshold level ε is reached,the iteration stops, and the number of propagationpaths N is increased by one, i.e., N + 1 → N. Theinitial values of c(0)N , f (0)N , and τ

′(0)N are set to zero. After

this, the iterative parameter computation procedure iscarried out again starting from (12). This process isrepeated until no perceptible progress can be made byincreasing N or the maximum number of paths Nmaxis reached.

It is straightforward to show that (13) results in aminimum if c(i+1)

l is computed as

c(i+1)l =

√√√√<{

y(i)l

}T<{

p(i)l

}+=

{y(i)

l

}T={

p(i)l

}(

p(i)l

)Hp(i)

l

, (16)

where the operators {·}T and {·}H denote the trans-pose and complex-conjugate transpose, respectively.The symbol y(i)

l refers to a column vector formed bymapping the elements in Y(i)

l such that all columnsin Y(i)

l are stacked on top of each other. Analogously,the symbol p(i)

l denotes a column vector, which isformed by applying the same mapping scheme to P(i)

l .However, in order to guarantee a perfect fitting at theorigin ν′p = 0 and τq = 0 of the TFCF R[ν′p, τq], we

replace the optimization step in (13) by

c(i+1)l =

√√√√<{

y(i)l

}T<{

p(i)l

}+=

{y(i)

l

}T={

p(i)l

}(

p(i)l

)Hp(i)

l

, l 6= N

R[0, 0]−N−1∑

l=1

(c(i+1)

l

)2, l = N.

(17)The preceding equation implies that the cN is computedsuch that exact fitting of R[ν′p, τq] is always preservedif ν′p = 0 and τq = 0.

The procedure above is coined INLSA-TF. It shouldbe pointed out here that when using the INLSA-TFmethod, the path gains are optimized in combinationwith the corresponding Doppler frequencies and prop-agation delays, which result in an unambiguous andunique triple (cn, fn, τ′n) for each propagation pathn = 1, 2, . . . , N. This is in contrast to the INLSA-T-Fand INLSA-F-T methods for which we obtain only theunique tuples (cn, fn) and (cn, τ′n), respectively. Since theoptimization process in the proposed method is carriedout by modeling the TFCF of the measured channel, theINLSA-TF method provides an excellent performancewhen evaluating the channel simulator with respect tothe TFCF. As we will see, the INLSA-T-F and INLSA-F-T methods cannot provide an accurate modeling ofthe TFCF of the measured channel, especially close tothe origin. The results of a comparative study of thesethree approaches are demonstrated in the next section.

5 Application to Measurement Data

In this section, we demonstrate the usefulness and theperformance of the proposed INLSA-TF algorithm. Toaccomplish these two objectives, we used measurementdata collected in an urban area in the 2.1 GHz frequencyband. In the measurement campaign, the transmitterantenna was mounted on a trolley and acted as amobile part. The receiver antenna was stationary andmounted on the roof of a building at a height of 20 m.The performed measurements were double-directionalmultipath measurements conducted using a widebandchannel sounder. The key parameters of the measure-ment equipment are:• time sampling interval: ∆t = 0.02 s,• frequency sampling interval: ∆ f ′ = 195 kHz,• bandwidth: B = 100 MHz,• observation time interval: T = 0.5 s.For comparative purposes, we first demonstrate the

channel estimation capabilities of the proposed INLSA-TF algorithm and the two original variants of theINLSA method using the same measurement data. Thethreshold level ε of the proposed algorithm was empir-ically set to ε = 0.01 in all cases. Figure 1 illustrates thenormalized residual error norm ‖R− R‖F/‖R‖F versusthe number of propagation paths N.

As can be seen from the results shown in this figure,the proposed INLSA-TF method performs significantlybetter than the two original INLSA approaches. Thedifferent performances of the INLSA-T-F and INLSA-F-T originate from the fact that the estimated sets of


0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

Number of propagation paths, N

Norm

alizedresidualerrornorm

Simulation model, INLSA−T−F

Simulation model, INLSA−F−T

Simulation model, INLSA−TF (proposed method)

Figure 1. Normalized residual error norm ‖R− R‖F/‖R‖F versusthe number of propagation paths N.

0

0.05

0.1

0.15

0

2

4

6

8

0

0.2

0.4

0.6

0.8

1

Frequency lag, ν

′p(MHz)

Time lag,τq (s)

Normalized

TFCF,|R

[ν′ p,τq]|

Figure 2. Absolute value of the normalized TFCF |R[ν′p, τq]| of themeasured channel.

model parameters are not identical when these methodsare used. As an aside, we want to mention that theINLSA-TF algorithm as well as the two original ver-sions (INLSA-T-F and INLSA-F-T) are initialization in-sensitive. This means that, for any chosen initial valuesof the Doppler frequencies and path delays, we alwaysget the same graph for the normalized residual errornorm. We also realize from the inspection of Figure 1that the normalized residual error norm associated withthe INLSA-TF method is approximately 10 % if N = 30,whereas the error norm drops below 3 % if N = 80. Byreferring to these results, the number of propagationpaths Nmax = 80 is considered to be sufficient andadopted for further simulations as a stopping criterionin the iterative optimization procedure.

The important difference between the proposed andthe original INLSA method is that the former oneaims to model the TFCF of real-world channels, whilethe latter one focuses only on the behavior of theTFCF along the principal axes, meaning the TACF andthe FCF. In order to demonstrate this difference, wehave plotted the TFCFs of the measured channel andthe simulation model by using the INLSA-TF and theINLSA-T-F in Figures 2, 3, and 4, respectively. FromFigures 2–4, we can realize that the INLSA-TF methodperforms significantly better than the original INLSA-T-F. It is readily apparent from comparing the results in

0

0.05

0.1

0.15

0

2

4

6

8

0

0.2

0.4

0.6

0.8

1

Frequency lag,

ν′p(MHz)

Time lag,τq (s)

Normalized

TFCF,|R

[ν′ p,τq]|

Figure 3. Absolute value of the normalized TFCF |R[ν′p, τq]| ofthe simulation model with the parameters estimated by using theproposed INLSA-TF method.

0

0.05

0.1

0.15

0

2

4

6

8

0

0.2

0.4

0.6

0.8

1

Frequency lag, ν

′p(MHz)

Time lag,τq (s)

Normalized

TFCF,|R

[ν′ p,τq]|

Figure 4. Absolute value of the normalized TFCF |R[ν′p, τq]| of thesimulation model with the parameters estimated by using the INLSA-T-F method.

Figure 2 and Figure 4 that the main focus of the originalINLSA algorithm falls only on the TACF and the FCF.

As mentioned in Section 1, the separate parameteroptimization approach implemented in the originalINLSA algorithm may lead to incorrect estimations ofthe simulation model’s parameters. To highlight thisproblem, we have illustrated the performance of thetwo variants of the original INLSA method in Fig-ures 5 and 6. As can be seen in those figures, if theINLSA-F-T approach is used, the difference betweenthe TACF curves of the simulation model and themeasured channel is significant, whereas the FCF ofthe simulation model closely matches the FCF of themeasured channel. However, there is a significant gapbetween the FCFs close to the origin. The discrepancybetween the FCFs of the simulation model and themeasured channel is relatively small if the INLSA-T-F method is applied. This characteristic feature of theINLSA-F-T method close to the origin is a result of theused iterative approach for simplifying the maximum-likelihood optimization problem. We can also concludefrom the inspection of Figures 5 and 6 that the TACFand the FCF of the simulation model are well fittedto the corresponding correlation functions of the mea-sured channel if the simulation model’s parameters arecomputed by using the proposed INLSA-TF method.


0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.2

0.4

0.6

0.8

1

Time lag, τq (s)

Norm

alizedTACF,|r

t k[τ

q]|

Measured TACF




Figure 5. Absolute value of the normalized TACFs∣∣rtk [τq]

∣∣ (mea-sured channel) and

∣∣rtk [τq]∣∣ (simulation model) for different param-

eter computation methods.

0 1 2 3 4 5 6 7

0.2

0.4

0.6

0.8

1

1.2

Frequency lag, νp (MHz)

Normalized

FCF,|r

f′ [ν′ p]|

Measured channel




Figure 6. Absolute value of the normalized FCFs |r f ′m [ν′p]| (measured

channel) and |r f ′m [ν′p]| (simulation model) for different parameter

computation methods.

Impressively, the proposed method provides an exactcorrespondence between the correlation functions ofthe simulation model and the measured channel at theorigin, as it was expected from (17).

In our performance analysis, we have also incor-porated the study of the scattering function S(τ′, f )of the simulation model, which is obtained by ap-plying the two-dimensional discrete Fourier transformto the TFCF R[ν′p, τq]. In Figure 7, we have depicteda comparison between the scattering function of themeasured channel and that of the simulation modeldesigned by using the proposed INLSA-TF algorithm.The corresponding results obtained by applying theoriginal INLSA method are shown in Figure 8. Anexcellent agreement between the scattering functionsof the measured channel and the simulation model isobserved if the proposed INLSA-TF method is used (seeFigure 7). The results presented in Figure 8 demonstratethe poor performance of the INLSA-T-F and the INLSA-F-T methods in terms of modeling the scattering func-tion of the measured channel.

00.05

0.10.15

0.20.25 −30

−20−10

010

2030

0

0.005

0.01

0.015

Doppler frequenc

y, f (Hz)

Propagation delay, τ ′

(µs)

Scatteringfunction,S(τ

′,f)

Measured channel


Figure 7. Comparison between the scattering functions S(τ′, f )(measured channel) and S(τ′, f ) (simulation model) designed byusing the INLSA-TF method.

00.05

0.10.15

0.20.25 −30

−20−10

010

2030

0

0.005

0.01

0.015

Doppler frequenc

y, f (Hz)

Propagation delay, τ ′

(µs)

Scatteringfunction,S(τ

′,f)

Measured channel



Figure 8. Comparison between the scattering functions S(τ′, f )(measured channel) and S(τ′, f ) (simulation model) designed byusing the INLSA method.

6 Summary and Conclusions

In this paper, we introduced the INLSA-TF methodfor designing measurement-based wideband channelsimulators. The proposed method is an extension ofthe INLSA method. The INLSA-TF method aims toiteratively compute an optimized set of parametersdescribing the channel simulator by modeling the em-pirical TFCF of the given measurement data.

A performance analysis of the proposed method hasbeen conducted by applying the INLSA-TF algorithm tomeasurement data collected in an urban environment.We have compared the correlation properties of thesimulation model with those of the measured channel.A comparison has also been made with respect to thescattering function.

The obtained results demonstrated that the INLSA-TF algorithm precisely estimates the simulation modelparameters and provides an excellent fitting to thestatistics of real-world channels. Finally, it has beenshown that the INLSA-TF method significantly out-performs the two original versions. Owing to the lowcomplexity and an excellent performance, the INLSA-TF method can be considered as a powerful tool for syn-thesizing real-world channels and can therefore be usedfor the performance analysis of mobile communicationsystems under real-world propagation conditions.


Acknowledgment

The authors would like to thank Elektrobit Corporationfor the kindly provided measurement data.

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Akmal Fayziyev was born in Tashkent,Uzbekistan, in 1984. He received his Bach-elor’s degree in Telecommunications fromTashkent University of Information Technolo-gies, Uzbekistan, in 2005. In August 2007, hereceived the Master of Science (M.Sc.) degreein Engineering from Yeungnam University,South Korea. From 2007 to 2009, he worked asa core network department engineer at UCell,a cellular GMS/3G operator in Uzbekistan.Since July 2009, he is working on his PhD

thesis under the supervision of Professor Matthias P atzold at theUniversity of Agder, Grimstad, Norway. His current research interestsinclude multipath fading channel modeling and channel parameterestimation.

Matthias Pätzold was born in Engelsbach,Germany, in 1958. He received the Dipl.-Ing.and Dr.-Ing. degrees from Ruhr-UniversityBochum, Bochum, Germany, in 1985 and 1989,respectively, all in Electrical Engineering. In1998, he received the habil. degree in Com-munications Engineering from the TechnicalUniversity of Hamburg-Harburg, Hamburg,Germany. From 1990 to 1992, he was withANT Nachrichtentechnik GmbH, Backnang,Germany, where he was engaged in digital

satellite communications. From 1992 to 2001, he was with the De-partment of Digital Networks at the Technical University Hamburg-Harburg. In 2001, he joined the University of Agder, Grimstad,Norway, where he is a full professor for Mobile Communications andthe Head of the Mobile Communications Group. He authored andco-authored more than 230 technical journal and conference papers.His publications received nine best paper awards. He is author ofthe books "Mobile Radio Channels - Modelling, Analysis, and Sim-ulation" (in German) (Wiesbaden, Germany: Vieweg, 1999), "MobileFading Channels" (Chichester, U.K.: Wiley & Sons, 2002), which hasbeen translated into Chinese in 2008, and "Mobile Radio Channels"(Chichester, U.K.: Wiley & Sons, 2011). Prof. P atzold has been theTPC Chair of ISWCS’07, the TPC Co-Chair of PIMRC’13, ATC’12,ATC’11, ATC’10, ICCE’10, and the Local Organizer of KiVS’01. Hehas been actively participating in numerous conferences serving asTPC member for more than 20 conferences within the last 3 years. Heserved as Keynote Speaker, Tutorial Lecturer, and Session Chair formany reputed international conferences. He is an Associate Editorof the IEEE Vehicular Technology Magazine and a member of theInternational Advisory Board of the Radio Electronics Associationof Vietnam (REV). He edited several special issues, including thespecial issue on "Wireless Future" (Springer, 2009), the special issueon "Trends in Mobile Radio Channels: Modeling, Analysis, andSimulation" (IEEE Vehicular Technology Magazine, 2011), as wellas the special issue on ’Modeling and Simulation of Mobile RadioChannels’ (Hindawi Publishing Corporation, to appear in 2012). Heis a Senior Member of the IEEE. His current research interests includemobile radio communications, especially multipath fading channelmodelling, multiple-input multiple-output (MIMO) systems, coopera-tive communication systems, vehicular-to-vehicular communications,mobile-to-mobile communications, and coded-modulation techniquesfor fading channels.

Documents

Design of SISO Wideband Mobile Radio Channel by INLSA Method