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A new approach to wireless channel modeling using finite mixture models.

1 INTRODUCTION

At various stages of design, the performance of a communication system needs to be evaluated in the relevant communication channel. This evaluation can be completed most accurately through extensive field tests in environments similar to the ones in which the final product will be fielded. However, field data collections can be expensive and tedious. A cost-effective alternative is to use software simulations of the channel [1]. Accurate channel models are required to develop effective software simulations. Signals traveling through the channel experience different physical phenomena such as diffraction, scattering and reflection. These phenomena cause changes in the strength of the signal as it travels from the transmitter to the receiver. Channel propagation models predict the average signal strength and its variability at a given distance from the transmitter [1]. These models can be divided into large scale and small scale models. Large scale models characterize changes in signal power over large distances, while small scale models predict the rapid fluctuations in signal strength over short distances.

In small scale fading, the received signal is given by the vectorial addition of delayed copies of the actual signal (multipaths). Based on this effect, small scale fading can be characterized by the channel impulse response. The impulse response includes a complex gain term, an amplitude or magnitude component which affects the signal strength, and a phase component accounting for the phase shift in the signal. It is well established that the phase variations are uniformly distributed between 0 and 2[pi] [2]. The contribution of this research is limited to small scale models and specifically to the amplitude statistics of the impulse response.

The impulse response of a wireless channel can be represented as a discrete time series of time shifted and weighted delta functions (impulse train). The impulse response amplitudes are commonly modeled using non-mixture models such as Rayleigh, Lognormal, Weibull, Rice and Nakagami probability distribution functions for specific channel conditions. Rayleigh is widely used for amplitude statistics modeling due to elegant theoretical explanation and occasional empirical justification [2]. It has been observed that Rayleigh provides good fit when there is no single dominant multipath present among the arriving multipaths [3],[4],[5]. Rice distribution is used when there is a dominant multipath present in the various arriving multipaths when there is a line-of-sight between the transmitter and the receiver [6],[7]. Nakagami distribution has been observed to provide good data fit when the multipaths arrive with large time delay spreads [8],[9]. Even though Weibull and Lognormal distributions lack theoretic justification for their use in channel amplitude modeling, they provide excellent data fit in many cases [10],[11],[12],[13],[14],[15],[16]. In this research, we propose a new approach to model the impulse response amplitude statistics of a wireless communication channels using finite mixtures. Finite mixtures describe the probability distribution of a random variable as a weighted sum of component probability distribution functions. The development of a FMM based model is demonstrated using the ultrawideband impulse response amplitude data. The objective behind using FMMs is to harness the individual contributions from multiple component statistical channel characterizations to describe the existence of a multi-modal distribution of the data.

Section 2 of the paper discusses basics of small scale channel fading. Section 3 provides basics about UWB communication and existing channel models for UWB channels. Section 4 describes the UWB data collection methodologies and UWB data used in this paper for channel modeling. Section 5 presents background on the topic of FMMs and parameter estimation for FMMs and non-mixture models. Model selection techniques for picking the most appropriate channel model for UWB communication are described in section 6. Results of UWB channel modeling are presented in section 7, while section 8 discusses conclusions of the research conducted in this paper.

2 SMALL SCALE CHANNEL FADING

Small scale fading is caused by multipath propagation and Doppler spread. Multipath components are copies of the original signal that reach the receiver after reflection from the ground and various surrounding objects. They reach the receiver at different time delays and phase shifts and get added vectorially. The random phase shifts experienced by the individual multipath components determine the degree of attenuation or amplification in the received signal.

The impulse response of a channel can describe the small scale variations of the received signal based on the multipath characteristics of the channel. The multiple duplicates of the transmitted signal reach the receiver with different time delays after traversing different paths. Due to limited temporal resolution, practical receivers cannot isolate individual multipath components. Thus, several multipaths combine constructively and destructively within a single resolution bin of the receiver causing the received signal to fluctuate.

The received signal or channel output y(t) can be expressed as the convolution of the transmitted signal or channel input x(t) with the impulse response of the channel h(t,[tau]) as represented in equations 1 and 2.

y (t) = x(t)* h(t, [tau]) (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The multipath signals arriving at the receiver are a series of attenuated, time delayed, and phase shifted replicas of the transmitted signal. Thus a functional form of the impulse response explaining such a phenomenon is a weighted sum of time shifted delta functions. This function is expressed in equation 3.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

In equation 3, the independent variable t indicates that the impulse response can vary with time and i represents the multipath number. [[alpha].sub.i](t,[tau]) and [[tau].sub.i](t) are the amplitude and excess delay of the ith multipath component. The excess delay of the ith multipath is the arrival time of that multipath with respect to the first arriving multipath. The phase changes due to free space propagation and other channel characteristics are encapsulated in the complex exponential argument 2[pi][f.sub.c][[tau].sub.i](t) + [[phi].sub.i](t,[tau]). It is common to represent the entire phase term as a single variable [[theta].sub.i](t,[tau]). The equation then can be written as in equation 4.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

If it is assumed that the channel impulse response is static in time, the equation further reduces to the form shown in equation (5):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

A probing pulse p(t) is transmitted to measure the impulse response of the channel. The response of the channel to the probing pulse is measured as the power delay profile. Power delay profiles describe the power contained in the impulse response at various time delays and can be represented as in equation 6.

P(t,[tau]) [approximately equal to] k[[absolute value of [h.sub.b] (t;[tau])].sup.2] (6)

Power delay profiles are generally calculated by averaging the channel impulse response at different spatial locations. The resultant profiles can be rewritten as a time independent variable P([tau]) as shown in equation 7.

P([tau]) [approximately equal to] k[[absolute value of [h.sub.b] ([tau])].sup.2] (7)

where k is the gain relating the transmitted power in the probing signal to the received signal.

3 ULTRAWIDEBAND CHANNEL MODELING

In this research, to demonstrate the procedure and effectiveness of the FMM modeling techniques, specifically for the amplitude of the impulse response, [absolute value of [[alpha].sub.n]], the UWB channel is used as a case study. An UWB system has characteristics very different from a conventional narrowband system. Therefore, more scrutiny of existing channel model accuracy is required. This section gives a brief introduction to ultrawideband systems and discusses the reasons why investigations into UWB channel models is warranted.

3.1 Ultrawideband systems

Ultrawideband communication systems use extremely narrow (short duration) pulses as building blocks for communicating between the transmitter and receiver [17]. The pulses are typically in the hundreds of picoseconds with duty cycles less that 0.5%. This results in UWB systems requiring very low average transmission power.

A communication system can be classified as UWB in one of two ways.

* If the fractional bandwidth [B.sub.f] of the sytem is greater than 20%

* The bandwidth of the system, BW, is greater than 500MHz irrespective of its [B.sub.f] [18].

The fractional bandwidth, Bf, of a system is the ratio of the bandwidth BW to the center frequency [f.sub.c] given by equation 8 [18].

[B.sub.f] = BW/[f.sub.c] x 100% = ([f.sub.h] - [f.sub.l])/(([f.sub.h] - [f.sub.l])/2) x 100% (8)

where [f.sub.h] and [f.sub.l] are the lowest and highest -10 db cutoff frequencies.

3.2 Modeling of small scale statistics of ultrawideband signals

The time resolution at the receiver is inversely proportional to the bandwidth of the signal. Thus, narrowband signals have wide time resolutions. This poor time resolution results in a large number multipath signals getting combined within a particular time resolution bin. Given the random nature of the channel characteristics, the receiver can see either a constructive or destructive multipath superposition. Since the received signal is a combination of a large number of multipaths, irrespective of the distribution of the individual multipaths, the resultant signal can be assumed to have a complex Gaussian distribution due to central limit theorem [19]. Thus, the amplitude of the signal of narrowband signals can be assumed to be Rayleigh distributed [20].

On the other hand, the extremely wide bandwidth of UWB makes the time resolution at the receiver much narrower than conventional narrowband systems. This means that only a few multipaths combine together in a given time bin [21]. Therefore, the use of central limit theorem cannot be justified and the assumption that the received signal will have a complex Gaussian distribution does not hold true. Thus, the amplitude distribution of the ultrawideband signal cannot be assumed to be Rayleigh distributed. Correspondingly, an impulse sent through the ultrawideband channel may not be received as a Rayleigh distributed series of discrete time samples. In other words, the impulse response of the channel may not be Rayleigh distributed. This has resulted in increased interest and research effort to determine, impulse response amplitude, [absolute value of [a.sub.n]], distribution for UWB communication channels. Chong et al. used Kolmogorov-Smirnov (K-S) test and Chi-square test to suggest that Weibull distribution can be used to model ultrawideband tap amplitudes in the 3-10 GHz range [22]. Cassioli et al. indicate that Nakagami distribution provides the best fit for ultrawideband amplitude statistics [23], while Forester et al. suggest the use Lognormal distribution [24]. For wireless channel modeling, Taneda et al. introduced the use of information theoretic model selection techniques by applying Minimum Description Length (MDL) to find channel tap amplitudes statistics [25]. Schuster et al. introduced the use of another model selection technique, Akaike's Information Criterion (AIC), to identify appropriate models for ultrawideband channel amplitude statistics [26]. Choudhary et al. extended the use of a time series model selection technique, the Accumulative Prediction Error, to model UWB channel amplitude statistics [27]. Finite mixure models (FMMs) take advantage of the vast number of applicable distributions for the UWB channel impulse response amplitude by forming a weighted combination of the individual distributions. The effectiveness of this approach will be detailed in this treatment. Specifically, in this research we estimate FMMs for the UWB channels using the data collected by Schuster et al [26] and NIST [28]. The FMMs under consideration use combinations of Rayleigh, Nakagami, Lognormal, Weibull and Rice distributions. Stochastic Estimation Maximization is used to estimate the parameters of the FMMs. Model selection techniques such as the Akaike's Information Criterion (AIC) are used to identify the best FMM combination for the UWB channel model and to quantify the model complexity versus accuracy tradeoff.

4. CHANNEL MEASUREMENT

Measuring the communication channel characteristics can be achieved in the frequency domain via a transfer function or in the time domain through direct impulse response measurement. In time domain channel measurement, an impulse signal is transmitted and the impulse response of the channel is measured using a digital sampling oscilloscope (DSO) [29]. Conversely, frequency domain (FD) measurement is accomplished by "sweeping" the channel with a series of narrowband signals and measuring the frequency response of the channel in the band of interest using a vector network analyzer. The data used in this paper, was derived from frequency domain channel measurements [26]. In frequency domain channel measurement, the channel sounding is done using a series of sinusoids with frequencies in the band of interest. The radio frequency RF sounding signals are transmitted from the transmitter of a vector network analyzer (VNA) and captured by the receiver of the VNA placed at a distance d from its transmitter. The requirement for FD channel measurement is that the channel be static during the period of measurement. In essence, this means that the coherence time of the channel should be greater than the time required to make the measurement.

4.1 Data measurement in typical university environment

One set of data used in this research was measured and reported by Schuster et al. in [26]. Figure 1 illustrates the set up for the data collection. An HP 8722D VNA is used to measure the channel transfer function. To measure the transfer function, the VNA is operated in a stepped frequency mode with an IF bandwidth of 300 Hz. The sweep time was 9.8s. Each transfer function is recorded at 3201 equally spaced frequencies in the band from 2 GHz to 8 GHz. As a result, the transfer functions are measured with a frequency resolution of 1.875 MHz. Skycross SMT-3TO10M UWB antennas are used as transmitting and receiving antennas. The transmitted sounding signal has a 25dbm power level. The transmitter (Tx) and receiver (Rx) of the VNA are placed such that there exists a LOS between Tx and RX. The Tx-Rx distances of d = 15.4m, 18.4m, 21.2m, 24.3m, and 27.2m are used to collect the channel data. At each distance d, a 9x5 grid is created each separated with a spacing of 7 cms in both dimensions. At each location of the grid, the receiver is placed and the channel transfer function is measured. This process results in 45 transfer functions being measured. Next the entire grid is shifted by 50cms and another set of 45 transfer functions are measured. Thus, for each d, a total of 90 transfer functions are measured. The individual transfer functions are inverse Discrete Fourier Transformed to obtain the impulse response of the channels. Thus for each d, 90 impulse responses are obtained and each impulse response has 3201 points. Sample impulse responses from this data set are shown in figure 2. In summary, there are 90 impulse response measurements at each of the five values of d.

4.2 Data measurement in typical industrial environment

The National Institute of Standards and Technology (NIST) data set used in this research was measured in an indoor industrial environment [28]. The receiver of a vector network analyzer was placed on a circular turn table with radius of 24 cms as shown in figure 3. The distance from the center of the turn table and the transmitter is d. The transmitter sends out sine waves in the 2-8 GHz frequency range in frequency increments of 1.2MHz.

Thus, at the receiver, transfer function of the channel is measured at 4801 frequencies. An inverse Fourier transform is applied to the transfer function to obtain the impulse response containing 4801 impulses. For a particular value of d, the turn table is rotated 96 times in increments of 360/96 degrees and at each location the impulse response is measured. Thus, a total of 96 impulse responses are measured. Impulse response data used in this research corresponds to values of d equal to 13.3m, 18.2m, 21.37m, 24.72m, and 30.06m. Sample impulse responses from NIST data set are shown in figure 4. In summary, there are 96 impulse response measurements at each of the five values of d.

5. PARAMETER ESTIMATION FOR NONMIXTURES DISTRIBUTIONS AND FINITE MIXTURES

This section first describes the functional form of the non mixture models and equations to estimate the parameters of associated distributions. Next, finite mixtures are introduced and the process estimating parameters of the mixtures using Stochastic Expectation Maximization (SEM) is described.

5.1 Non-mixture models and parameter estimation for non-mixture models

The choice of the five probability density functions (PDFs) is guided by their popularity for non mixtures modeling of communication channels. It should be noted that the process of statistically modeling the channel is essentially the identification of the parameters associated with the statistical function that best describe the data. The parameters of the non mixtures models can be estimated using the maximum likelihood estimation (MLE) technique.

5.1.1 Rayleigh distribution

The Rayleigh probability density function (pdf) is given by equation (9).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where x is the data and o is parameter of the distribution.

The maximum likelihood estimate of the Rayleigh distribution is given by equation 10:

[??] = [square root of 1/2N [N.summation over (1=1)] [x.sup.2.sub.1]] (10)

5.1.2 Nakagami distribution

The Nakagami pdf is expressed as shown in equation 11.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[GAMMA] represents the gamma function, m represents the shape parameter and [OMEGA] represents the scale parameter. The estimate of m is given by equation 12.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where E([x.sup.4]) = 1/N [[SIGMA].sup.N.sub.i=1][x.sup.4.sub.l] is the fourth order moment. The parameter Q is the second moment as is estimated as in equation 13 [31].

[??] = E ([x.sup.2]) = 1/N [N.summation over (i-1)] [x.sup.2.sub.i] (13)

5.1.3 Rice distribution

The Rice pdf is represented as shown in equation 14.

p(x) = x/[[sigma].sup.2] exp(-[[x.sup.2] + [v.sup.2]/2[[sigma].sup.2]])[I.sub.0](xv/[[sigma].sup.2]) (14)

When describing channels, the rice distribution is commonly expressed in terms of the rice K factor given by equation 15:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where the relationship between [sigma], v and K, [OMEGA] is given by equations 16 and 17 [32].

[v.sup.2] = K[OMEGA]/(K +1) (16)

[[sigma].sup.2] = [OMEGA]/2(K + 1) (17)

The parameter [OMEGA] is the second moment as is estimated as shown in equation 18.

[??] = E([x.sup.2]) = 1/N [N.summation over (i=1)][x.sup.2.sub.i] (18)

The value of K can be estimated by solving the following non-linear equation 19 [33]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

5.1.4 Weibull distribution

The Weibull pdf is given by equation 20:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

The parameter a is called the shape parameter while the parameter b is called the scale parameter. Linear estimators are one of a number of possible methods used to estimate the parameters of the Weibull distribution. In this technique, the estimators are linear combinations of the data [34]. Thus, the task of estimating the Weibull parameters is converted to the problem of estimating the parameters [lambda] and [delta] of the extreme value parameter. The parameters of the extreme value distributions are related to the shape and scale parameters of the Weibull distributions as [lambda] = ln(b) and [delta]=1/a

5.1.5 Lognormal distribution

The lognormal distribution is given by equation 21:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

[sigma] and [micro] are the parameters of the lognormal distribution. It should be noted that if the distribution of x is lognormal, then the distribution of ln(x) of is normally distributed with mean [micro] and standard deviation [sigma]. The parameters of the lognormal distribution are given by equation 22 and 23.

[??] = 1/N [N.summation over (i=1)]ln([x.sub.i]) (22)

[??] = 1/N [N.summation over (i=1)][(ln([x.sub.i]- [??]).sup.2] (23)

5.2 FMMs and parameter estimation of FMMs

Finite mixtures provide an effective way of modeling data that is multimodal in nature and cannot be accurately described by a non-mixture model [35]. Examples of such variables would be the second impulse from the Schuster et al. impulse response at d = 15.4m or the tenth impulse of the NIST data at d = 21.53m shown in figure 5.

The FMMs represent the PDF of a variable X as a weighted sum of constituent PDFs as represented in equation 24 [36]:

p (x|[THETA]) = [M.summation over (k=1)] [w.sub.k][p.sub.k] (X | [[theta].sub.k]) (24)

p(X|[THETA]) represents the finite mixture model of variable X.

[THETA] = [([w.sub.1],[[theta].sub.1]), ([w.sub.2],[[theta].sub.2]), ... ([w.sub.M],[[theta].sub.M])] are the parameters of the mixture model. Further, [p.sub.k]{X|[[theta].sub.k]) represents the PDF of the kth constituent distribution and [w.sub.k] represents the mixing coefficient of that PDF

In this paper, the Stochastic Estimation Maximization (SEM) is used to estimate the FMM parameters. SEM is an iterative technique similar to the conventional Expectation Maximization Technique (EM). SEM estimates both the mixing coefficients as well as the parameters associated with the constituent PDFs of FMM. However, unlike the EM algorithm, the SEM technique does not require that the exact number of the FMM components be specified. Only the maximum number of components in the mixture needs to be known. The SEM algorithm is a three step process as described below [37].

Step 1: As in the basic EM algorithm, in the nth iteration, the first step is the Expectation step (E-step). This involves finding the posteriori probability or the probability that X is generated by the kth constituent PDF with parameter [[theta].sub.k]. The probability can be calculated as shown in equation 25.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

In the first iteration, an initial guess is made for the mixing coefficients wk and the component parameters [[theta].sub.k].

Step 2: The second step is the Stochastic step (S-step). In this step, based on the posteriori probabilities, regroup Xinto Kgroups [X.sub.1], [X.sub.2], ..., [X.sub.k]. For example, if max{P([[theta].sub.1]|[x.sub.1]), P([[theta].sub.2]\[x.sub.1]), P([[theta].sub.k]\[x.sub.1])} is P([[theta].sub.2]\xi), then place the sample xi in group X2 (where max{} is a function that finds the highest value among its input arguments).

Step 3: The third step is the maximization step (M-step) which involves updating the mixing coefficients and parameters of the constituent PDFs for the (n+1)th iteration. The mixing coefficients, [W.sup.n+1.sub.k], are given by the ratio of the samples in [X.sup.k] in the nth iteration to the total samples N. The parameters of the kth constituent PDF are estimated using the maximum likelihood estimation technique based on the data in [X.sub.k] are given by equation 26.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

The new value of [[theta].sub.k] for the (n+1)th iteration is the value of the parameter that maximizes the loglikehood of the kth component given the data [X.sub.k]. In the above equation [x.sup.n.sub.i,k] is the ith element in [X.sub.k] and M represents the number of elements in [X.sub.k] for the nth iteration.

In the E-step of each subsequent iteration, the updated values of w and [theta] computed in the M-step of the previous iteration are computed. The process is repeated until the estimates of w and [theta] in M-steps of successive iterations do not change significantly.

6. Model Selection Techniques

Since this research introduces the use of FMMs, a seemingly more complex model, the question of whether FMMs are worth the complexity they bring to the modeling process needs to be addressed. To answer this question, appropriate statistical metrics or model selection techniques are used. Given the data and a group of potential PDFs, model selection techniques can be used to select a PDF that provides the best representation for the data. A good model selection technique should be capable of evaluating the trade off between the "goodness" of the fit provided by a candidate model and the simplicity of the model. Model selection techniques are generally designed to penalize complex models [38]. A complex model will only emerge as a winning model if its data fitting ability outweighs it structural or parametric complexity. The penalty for complexity arises from the idea that complex models tend to overfit the currently available data since they may be strongly influenced by noise in the data.

Model selection techniques (including AIC) that have roots in information theory, typically, are variations of or are derived from the Kullback-Leibler (KL distance). The KL distance I(f,p) is an information theoretic measure of how closely a distribution p(x) approximates the true but unknown distribution f(x). The KL distance is given by the formula in equation 27 [39].

I (f,p) = [integral] f(x)log(f(x)/p(x))

= [integral] f(x)[log(f(x)) - log(p(x))] (27)

I(f,p) represents the loss information when p(x) is used instead of fx), with -log(p(x)) representing information. Model selection algorithms based on this information theoretic measure attempt to choose a distribution that has a very small value of I(f,q). This means that model selection algorithms choose models that minimize the KL distance. Since the parameters, p(x), of the distribution under test are estimated using MLE from the data, the I(f,p) is also an estimated distance represented by [??](f,p)

This section provides an overview of three model selection techniques: Akaike's Information Criteria (AIC) [41], Minimum Description Length (MDL) [42] and Accumulative Prediction Error (APE) [43]. These three model selection techniques are used in this research to find the best model (PDF) for the channel impulses.

6.1 Akaike's Information Criterion (AIC)

Akaike's Information Criterion further simplifies the KL distance to make it easier for practical implementation and is given by equation 28 [41].

AIC = -2 log(p(x| [theta]) + 2K (28)

where K is the number of parameters in the distribution p(x). The smaller the value of AIC the closer the p(x) represents the unknown true distribution f(x). To compare the relative fit of the distributions under consideration, a difference, [[phi].sub.j], is computed. [[phi].sub.j], is difference between the AIC value of each distribution and the minimum AIC value in the set. Thus difference value for the distribution with minimum AIC value is zero. The AIC weights for a particular distribution are computed as shown in equation 29.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

where [[SIGMA].sup.M.sub.m=1] [w.sub.m] = 1 and M is the number of distributions under test. Higher values of [w.sub.j] indicate that the distribution provides a better the fit to the data. Thus, among the group of distributions under consideration, the distribution with the highest value of [w.sub.j] can be chosen as the model best representing the data.

6.2 Minimum description length

The MDL algorithm loosely states that the most appropriate model to describe a data set is the one which uses the least number of "code words" or the shortest description of the data [42]. MDL can be seen as finding the model that compresses the data the most and effectively eliminates models that overfit. MDL involves choosing a model which has the smallest description length (DL) for the data set. DL is calculated as shown in equation Error! Reference source not found.30.

DL = - log(L([??] | x)) + 1/2 K log(N)

= -log(p(x | [??]))+ 1/2 K log(N) (30)

K is the number of parameters in the distribution under test and N is the number of data samples. Thus, it can be seen that DL is a sum of two expressions. The first term measures the how well the model under evaluation fits the data and the second term is measure of the complexity of the model or the number of parameters. A better data fit and reduced model complexity will result in a smaller description length. This results in the increase of the description length.

6.3 Accumulative prediction error (APE)

Wagenmakers et al. state that the sole requirement of APE is that the models under consideration are capable of generating a prediction for the next unseen data point [43]. In practical implementations, the observed data is split into two parts: the seen or the training data set and the unseen or testing data set. Consider a data set of N observations given by, x = {[x.sub.i]} i = 1,2, ... N. Initially S observations are placed in the training data and the remaining N - S in the testing data. Based on the training data, the parameters [??] of the distribution under test are estimated using MLE. One data point is picked from the testing set and the value of the log conditional probability ln p(x|0) for that point is calculated. The larger the probability value, the better the distribution predicts the data point and hence the smaller the prediction error. This data point is then transferred to the training data set, increasing its size by one. Based on this training data set, the parameters [??] of the distribution under testing are recalculated. Another point is picked from the testing set and the value of the log conditional probability ln p(x|[??]) is calculated for that data point. The data point is then transferred into the training data set. This process is repeated till all N - S points in the testing data set are covered. The APE value is then given by the sum of log probabilities calculated for all N - S data points of the testing data set.

7. UWB channel model selection results

For each Tx-Rx distance d, non mixtures and FMM based models were estimated for each significant impulse of the impulse response based on their 90 samples (Schuster et al.) or 96 sample (NIST data). FMMs consisted of at most 3 component PDFs drawn from the five PDFs (Rayleigh, Rice, Nakagami, Weibull and Lognormal). SEM was used to estimate the FMMs for each significant impulse. Thus for each impulse, a total of 25 models were estimated: 20 FMMs (3-component and 2 component FMMs drawn from 5 PDFs) and 5 homogeneous models. The models were ranked using AIC model selection technique for each impulse. Analysis of the model selections indicate that none of the models (FMM or non-mixtures) was consistently ranked first. Hence the identification of the most appropriate model for the impulse response is done based on the number of impulses for which a model ranked among the top 3 out of 25 competitors. Table 1 through table 3 list the percentage of impulses for which a model belonging to non-mixtures, 2-component FMM, and 3-component FMM is ranked among the top 3 for university environment data. It can be seen that the Lognormal distribution is the best among non-mixture since it is ranked among the top 3 for over 36% of the impulses. Among the 2-component FMMs, Rayleigh-Lognormal FMM stands out since it is ranked among the top 3 for almost 70 % of the impulses. Among the 3-component FMMs, the FMM composed of Nakagami, Weibull, and Rice distributions performs the best with it being ranked among the top 3 for almost 20% of the impulses. The results of the tables are summarized in figure 7 where the performance of best three models is illustrated. It can be seen that for the university data the Rayleigh-Lognormal FMM performs best in terms of the number of impulses for which it is ranked among the top 3. This indicates that the Rayleigh-Lognormal FMM can be an effective alternative to the conventional non-mixtures for modeling channel impulse response. Similarly, tables 4 through table 6 list the percentage of impulses for which a model belonging to non-mixtures, 2-component FMM, and 3-component FMM is ranked among the top 3 for industrial environment data. As in the case of university data, for non-mixtures and 2-component FMMs, Lognormal and Rayleigh-Lognormal FMM perform the best. For 3-component FMMs Nakagami-Lognormal-Weibull FMM performs the best. The bar chart in figure 8 summarizes the tables and indicates that the Rayleigh-Lognormal FMM is the best model since it is ranked among the top 3 models for over 62% of impulses. As in the case of university data, the Rayleigh-Lognormal FMM performs is a more effective alternative to conventional non-mixtures for in modeling wireless channel impulse responses for industrial environments as well.

Figures 6 and 7 are shown for further analysis in to the performance of the Rayleigh-Lognormal FMM in terms of normalized AIC weights. In figure 6, if a vertical line is drawn at any time instant, three symbols will be encountered: the circle representing the AIC weight of the best model for the impulse arriving at that instant of time, the triangle representing the AIC weight of the third best model for impulse arriving at that instant of time and the asterisk representing the AIC weight of the Rayleigh-Lognormal FMM for the impulse arriving at that instant of time. Note that AIC weights are normalized AIC values which assign higher values to better models. When the asterisks in the plot intersect the circles, it indicates that the Rayleigh-Lognormal FMM is the best model for the impulse arriving at that instant of time. Many such intersections of circles and asterisks can be seen. Also it can be seen that the asterisks like on or above the triangle symbols for most time instants. This means that most of the AIC weights of the Rayleigh-Lognormal FMM are either first, second or third highest in value for most of the impulses. This basically indicates that the Rayleigh-Lognormal FMM is among the top 3 models for a majority of impulses. A similar conclusion can be drawn for industrial data based on figure 7.

8. Conclusions

In this research, a new approach to modeling wireless channels using finite mixtures is presented. The process and effectiveness of FMMs is demonstrated by modeling the amplitude statistics of the ultrawideband channel impulse response. The ultrawideband data represented two types of channel: industrial and typical college environment. FMMs used here were composed of distributions drawn from five commonly used non mixture channel models, namely Rayleigh, Nakagami, Rice, Weibull and Lognormal distributions. The component distributions of FMMs were chosen based on the wide usage in channel modeling literature. The various 2-component and 3-component FMMs were estimated and compared among themselves and with non-mixtures models using statistical metrics such as AIC. The statistical metrics indicate that the FMM composed of Rayleigh and Lognormal distributions consistently ranked high in their ability to provide accurate statistical description for most of the impulses of channel impulse response. This research has thus established Rayleigh-Lognormal FMM as an alternative to conventional non-mixture FMMs. Scrutiny of the weighting coefficients of the Rayleigh-Lognormal FMM indicate that the FMM has higher contributions from the Rayleigh component for higher amplitude impulses of the impulse response, while the FMM takes higher contribution from the Lognormal component for the low amplitude impulses. Following the Rayleigh-Lognormal FMM, the Nakagami-Lognormal FMM performed second best and the Lognormal non-mixture performed third best in terms of the percentage of impulses for which they ranked among the top three models. From these observations, it can be inferred that Lognormal distribution is an important contributor to "true" data model of the UWB channel data. The Rayleigh and Lognormal non mixtures showed trends similar to the Rayleigh-Lognormal FMM coefficients in that Rayleigh performed well for high amplitude impulses and Lognormal performed well for low amplitude impulses. It is also observed that the non-mixture Rayleigh and Lognormal parameters are smoother in terms of variations than that of the Rayleigh-Lognormal FMM. This could be because of the complexity of the FMMs have tendency to over fit the data. Visual inspection of the histograms of the impulses showed that many of the impulses had two modes. This can be linked to the success of the 2-component FMM in modeling the data. The poor performance of the 3-component FMMs can be attributed to their increased complexity that gets penalized by the model selection techniques. Also, the number of data samples was limited; hence the estimation of the parameters of the 3-component FMMs may not have been accurate. The statistical samples available for this research were very limited. The parameter estimation process and the model selection process will be more accurate with the inclusion of other UWB data sets with large number of statistical samples.

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Divya Choudhary (1) and Aaron L. Robinson (2)

(1) Department of Electrical and Computer Engineering, Christian Brothers University, Memphis, TN, USA

(2) Department of Electrical and Computer Engineering, University of Memphis, Memphis, TN, USA

(1) dchodhry@cbu.edu

(2) alrobins@memphis.edu

Table 1. Performance of non-mixture models (University
Environment)

Non-Mixture model              % of impulses for which
                               model ranked in top 3

Lognormal                      36.8
Rice                           6.7
Weibull                        3.9
Rayleigh                       3
Nakagami                       1.1

Table 2. Performance of 2-component FMMs
(University Environments)

2-component FMMs               % of impulses for which
                               model is ranked in top 3

Rayleigh-Lognormal             69
Nakagami-Lognormal             54
Weibull-Lognormal              8.3
Rice-Lognormal                 3.9
Nakagami-Weibull               3.5
Nakagami-Rice                  3.4
Rayleigh-Weibull               2.9
Nakagami-Rayleigh              1.9
Rice-Rayleigh                  1.9
Rice-Weibull                   0.5

Table 3. Performance of 3-component FMMs
(University Environment)

3-component FMMs               % of impulses for which
                               model ranked in top 3

Nakagami-Rice-Weibull          19.3
Nakagami-Lognormal-Weibull     10.6
Rice-Lognormal-Weibull         8.3
Rayleigh-Lognormal-Weibull     3.9
Rice-Rayleigh-Lognormal        3.5
Nakagami-Rayleigh-Rice         3.4
Nakagami-Rayleigh-Weibull      2.9
Nakagami-Rayleigh-Lognormal    1.9
Nakagami-Rice-Lognormal        1.9
Rice-Rayleigh-Weibull          0.5

Table 4. Performance of non-mixture models (Industrial
Environments)

Non-Mixture model              % of impulses for which
                               model ranked in top 3

Lognormal                      46.1
Rice                           4.2
Weibull                        3.7
Rayleigh                       3.7
Nakagami                       0.6

Table 5. Performance of 2-component FMMs (industrial
environment)

2-component FMMs               % of impulses for which
                               model is ranked in top 3

Rayleigh-Lognormal             62.1
Nakagami-Lognormal             51.2
Weibull-Lognormal              33.1
Rice-Lognormal                 21.0
Nakagami-Weibull               2.1
Rayleigh-Weibull               1.4
Nakagami-Rice                  0.9
Nakagami-Rayleigh              0
Rice-Rayleigh                  0
Rice-Weibull                   0

Table 6. Performance of 3-component FMMs (industrial
environments)

3-component FMMs               % of impulses for which
                               model is ranked in top 3

Nakagami-Lognormal-Weibull     20
Rice-Rayleigh-Lognormal        11
Rayleigh-Lognormal-Weibull     8.8
Nakagami-Rice-Weibull          7.9
Rice-Lognormal-Weibull         5
Nakagami-Rice-Lognormal        4
Nakagami-Rayleigh-Rice         3.3
Nakagami-Rayleigh-Weibull      3
Nakagami-Rayleigh-Weibull      1
Rice-Rayleigh-Weibull          0
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Author:Choudhary, Divya; Robinson, Aaron L.
Publication:International Journal of Digital Information and Wireless Communications
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Date:Apr 1, 2014
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