Parsimonious Correlated Nonstationary Models for ... - Semantic Scholar

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005

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Parsimonious Correlated Nonstationary Models for Real Baseband UWB Data Q. T. Zhang, Senior Member, IEEE, and S. H. Song, Student Member, IEEE

Abstract—Few ultrawide-band (UWB) models directly fit original baseband UWB data to simultaneously account for their correlation structure, non-Gaussianity, and nonstationarity. The difficulty arises from the fact that no relevant result is available, even in multivariate statistical analysis. It also arises from the attempt to pursue the details of the mechanism that is imagined to be responsible for the generation of UWB data. The consequence is the modeling complexity, which makes it difficult to handle the received signal correlation on the basis of a single realization. Accordingly, various partial characterization is used in the literature instead by virtue of second-order statistics (such as power delay profile), nonparametric characteristics (such as zero-crossing rate), or their combination. In this paper, we take a different philosophy, which believes that the information in the received UWB data itself, as long as fully exploited, plus some simple physical intuition should suffice for the model identification and its parameter estimation. A received UWB signal is decomposed into three factor processes and each is parsimoniously parameterized. The application of the new model to data regeneration and receiver design is illustrated by using the real UWB data acquired by Intel and the TimeDomain Corporation. Index Terms—Correlated baseband ultrawide-band (UWB)signal models, generation of correlated non-Gaussian UWB data, modeling UWB signals.

I. INTRODUCTION

U

LTRAWIDE-BAND (UWB) communications is emerging as a potential means for high data-rate wireless transmission in an indoor environment. Modeling UWB received signals is an indispensable step to the UWB receiver design and UWB data regeneration. Various modeling techniques have been proposed in the literature, which have their root in different philosophies. One technique relies on the geometrical theory of scattering and diffraction behind UWB propagation leading to a deterministic multipath channel model in the frequency domain [1]. A more popular framework is built on Turin’s multipath model, whose path gains and time delays are random variables. Theoretically, the joint probability density function (pdf) of these random parameters must be specified for a complete channel characterization. This task is too difficult to accomplish in practice. The difficulty arises from the current status in multivariate statistical analysis that few results are available on joint non-Gaussian distributions of relevance to UWB signals. Therefore, various partial characterization is adopted instead. Manuscript received December 2, 2003; revised April 2, 2004, September 6, 2004, and November 7, 2004. This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, under Project CityU 1206/01E. The review of this paper was coordinated by Dr. K. Dandekal. The authors are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TVT.2004.842460

For instance, the partial second-order information, in the form of the path loss and power delay profiles (PDPs), is used in [2], whereas an exponentially decaying function is introduced in [3] and [4] to account for the nonstationarity in the average PDP. The difference is that the former fits the amplitudes of each path by the Rician or gamma distribution, whereas the latter characterizes the fluctuation of the PDP about its mean curve by the Weibull distribution. Given that the characterization based on partial second-order statistics is incomplete and nonsystematic, different nonparametric techniques are suggested as complements. Reference [4] introduces the zero-crossing rate, hoping that it is helpful in retrieving the received UWB process from its second-order statistics. In [5], the arrival time of multipaths and the number of scatterers within a resolvable bin are employed besides the commonly used PDP and its mean value. The first two random variables are modeled as a combined Markov and Poisson process, whereas the third is governed by the gamma distribution. The treatment of the average PDP and small-scale fluctuation of path energy gains in [6] is much the same as that in [5], except that the total received energy is assumed to follow the lognormal distribution. The scattering and refraction process responsible for UWB propagation in an indoor environment is very complicated; one can certainly treat it as either a deterministic or stochastic mechanism, as taken by the geometrical theories of electromagnetics and random filter, respectively. This is the so-called principle of dualism in the theory of time series. We are playing a game with Nature. The latter produces a set of observed UWB data at the output of a black box and we need to infer on the mechanism dictating the black box behavior. The geometrical theory employs a simpler model that requires a much smaller number of unknown parameters, thereby simplifying the estimation procedure. The resulting model, however, is location dependent. The random filter theory, on the other hand, tries to find out the physical details of the black box and their statistical behavior and, thus, a large number of independent realizations from the same population are necessitated for the distribution estimation for all random parameters. This is difficult to do in practice, unless the spatial ergodic property is justified for UWB signals. Most of UWB random models focus on the power statistics of a UWB process, neglecting its more important correlation structure. From the application point of view, it is more important to model the original baseband UWB process rather than its partial second statistics (e.g., PDP) since the former is directly relevant to the receiver design and data regeneration. In this paper, no attempt is made to pursue the details of the black box’s mechanism. Rather, we will fully exploit the information in a single received UWB sequence itself to build a simple random model accounting for its correlation structure

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and nonstationarity. Some prior physical intuition is also incorporated. We will directly work on received UWB signals rather than channel impulse responses, since the transmit pulse, regardless of its duration, is always spread in time by the transmit antenna. Furthermore, it is the characteristics of received signals that determines the receiver structure.

technique, but consider the gap over the rising portion of UWB data as a random fluctuation caused by a UWB realization. In this paper, we confine ourselves to cases without a line-of-sight component. III. STATISTICAL PRELIMINARIES AND DATA DESCRIPTION Before proceeding, let us introduce some necessary statistical knowledge and the data set to be modeled in the sequel.

II. MODELS is transmitted over a scatAssume that the monocycle tering channel, which is described by a discrete filter. Then, we can use the superposition principle to represent the signal received at the location with distance from the transmitter as , where is counted from the first scatterer at the given location and and are the random reflection gain and random time delay associated with the th scatterer, respectively. Since there are a large number of scatis usually samterers in an indoor environment and signal pled with a sampling interval much smaller than the transmitted pulse duration, we can make the following observations: 1) each sample is contributed from a random number of scatterers and 2) each scatterer can have contribution to adjacent samples. From these observation, it follows that correlation exists among adjacent samples. Since the number of scatterers that contribute to different samples is different and their cross-sections can vary dynamically over a large range, it also follows that there should be a factor component in the model to account for the large-scale fluctuation, in addition to a factor component that characterizes the local fluctuation caused by multipaths. Hence, given a se, , we quence of baseband UWB data can express it as (1) whose dependence on has been suppressed for simplicity. The assumptions made in this model are in order. is a second-order nonstationary process and its • nonstationarity is described by the deterministic exponen, consistent with empirical results tial function in wireless communications. is included to account for the fast • Component random fluctuation caused by local multipaths on one on the other. hand and for the correlation among • Component represents a slowly varying process over . a large dynamic range, which is not characterized by on the distance from the transmitter • Dependence of . is absorbed into the variance of The new model distinguishes from its counterparts previously proposed in the literature for UWB wireless channels in that it fully takes into account the nonstationarity of a received UWB process and its covariance structure. The inclusion of an exponentially decaying factor makes our model suitable for decaying delay profiles. Occasionally, there are some applications in which the received signal power may go up to a peak over a short period before decaying. These situations happen when the shortest path rays penetrating; for example, large concrete walls are more attenuated than other rays through doors. Under these circumstances, we can modify our models before applying it to UWB data. Alternatively, we may directly employ our modeling

A. Selection of Distribution The first problem we come across is to choose a right pdf for , . a given set of random samples, say Very often, the observer may have a preferred distribution in his mind on the basis of previous empirical results or certain physical understanding and, hence, uses it to fit the data samples. The appropriateness of the fitting is usually demonstrated by comparing a histogram obtained from the data with the pdf with the best selected parameters or more rigorously by a chi-square test. This approach, though widely adopted in the community of wireless communications, is somewhat subjective, since the start point of the procedure has a bias and tends to protect the preselected pdf for the data. In this paper, we, therefore, will instead use a more objective procedure that is very popular in statistical literature. In probability theory, most distribution functions fall into two major systems, namely, the Pearson’s system, developed by Karl Pearson in 1880s, and the Johnson’s system, developed by Johnson in 1949 [7]. The former includes the widely used Gaussian and gamma distributions in communications, whereas the latter includes the famous lognormal distribution. The selection of an appropriate pdf to fit a given set of random samples can be objectively indicated by the data set itself, rather than a subjective belief of the observer. Let denote the th central moment (about its mean) of the data. In the statistical theory, the information in the first four moments can be extracted to form the following three parameters:

(2) and (3) for the selection of a particular Pearson model for the given data set. The interested reader can be referred to [7] for more details. Classification of various distributions in the Pearson family corresponds to the partition of the parametric space spanned by , , and into regions with each corresponding to a particular distribution. In the Johnson’s system, such a partition is done and only. For example, in the Pearson’s on the basis of system, the regions for Gaussian and gamma distributions are , , and , respectively. The log-normal distribution in the Johnson’s system is specified by on a curve, which defines the nonlinear dependence of through the third variable such that and

ZHANG AND SONG: PARSIMONIOUS CORRELATED NONSTATIONARY MODELS FOR REAL BASEBAND UWB DATA

for some . For the TimeDomain UWB data to be pro. Over this range, the relation cessed, we find that and is well approximated by the linear equation between .

before We first handle the nonstationary envelope of component, which contains sign information. moving to its over a window of an appropriate To this end, we average length to remove the local fluctuation of

B. Determining the Model Order After the selection of an appropriate distribution for a given samples, say , we still need to data set of find a model to fit the data using as less number of parame. ters as possible, which is denoted here by This is the principle of parsimony, the name coined by Box and Jenkins [8]. Clearly, the maximum-likelihood (ML) principle alone cannot be used to determine the model dimension , since the ML function monotonically increases with the di. The decision on the model dimension has mension of its root in the Bayes framework ending up with the minimum description-length (MDL) criterion, which states that should be chosen such that MDL

(4)

is minimized [9]. The first term represents the minus maximum log-likelihood function (LLF) and the second term stands for the penalty for introducing free parameters in the model. Another criterion called AIC, which has a similar expression but with different penalty, is derived from the Boltzmann entropy principle. C. Data Set Description Parameter estimation in this paper will be conducted on the basis of the measurement data of the TimeDomain Corporation. The data was obtained in a modern office building by sending a pulse of approximately 1 ns duration as an excitation signal of the propagation channel. Multipath profiles were captured by digital sampling oscilloscope (DSO) of sampling rate 20.48 GHz, implying that the sampling time interval was 48.828 ps. In each room, 300-ns-long response measurements were made at 49 different locations in 1 yd [16], [19]. The data set for each location consists of 6143 samples. Shown on the top of Fig. 5 is a typical set of the TimeDomain’s data, which clearly indicates the second-order nonstationarity of UWB data.

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(5) with denoting the data length available after averaging. The window size should be chosen to be sufficiently large so that the fluctuation due to local multipaths can be removed, but sufis able to reflect the ficiently small so that the resulting large-scale variational behavior of the UWB signal. In TimeDomain’s experiments, the “clean wave” arriving via a 1-m-long line-of-sight path roughly lasts for 2 ns, implying that the resolution bin is of 2 ns or, equivalently, 0.6 m in distance. For a given time instant, all scatterers located within the corresponding resolution bin produce scattered waveforms that, when linearly superposed, cause the fast fluctuation of the received signal segment. Accordingly, a reasonable strategy to estimate the average scattering intensity of the scatterers within a bin is to average over the window size equal to the resolution bin. This suggests the use of equivalent to 2 ns in our case. In fact, a similar choice of window size is adopted in [6] for determining the PDPs. basically remains unSince the slowly faded factor changed over a window and the exponential factor varies slowly , we can take them out of the summation in compared to (5), yielding (6) where and is the mean value of the magnitude and is assumed to be constant over the index . Without of a loss of generality, we choose as an odd number so that is an integer. We take log of both sides of (6) to obtain (7) Define

(8) IV. MODELING AND ESTIMATION FOR The estimation of the unknown parameters in a model can be rigorously formulated in the framework of the ML principle is the product or Bayes estimation. In our case, however, of an exponential function and two complicated random processes, making it extremely difficult to tackle the problem in such frameworks. Therefore, we consider a suboptimal solution instead, which is based on the following observation. Namely, changes much faster than the largethe local fluctuation . In other words, basically remains scale variation of . This distincunchanged within an appropriate window for tion, when exploited, allows for the estimation of the unknown parameters for each components separately. A similar idea has been utilized to remove high-frequency ripples in the power estimation for handoff in cellular mobile communications [10].

and rewrite (7) (9) We also need to parameterize the random process . Given a random sequence of samples, one can certainly characterize it by using its -by- covariance matrix. From the application point of view, however, it is inconvenient or even impossible samples to estimate more than unknown parameto use ters. Therefore, a parsimonious model must be used instead. It is shown by the Wold theorem that a stationary process with rational power spectrum can be uniquely modeled by an autoregressive moving average (ARMA) process [11]. Fuller [12] further shows that an ARMA process can be approximated by an AR process to any accuracy, as long as a sufficiently large model

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order is used. Suppose that an AR such that

model is used for

(10) where is an independent and identically distributed (i.i.d.) error process with zero mean and variance . To proceed to the issue of parameter estimation, let superscript denote transposition and let us define vectors

Fig. 1. Nearly linear relationship between and for (k ) suggests the adoption of the log-normal distribution for (k ).

(11) The use of these notations in (10) allows us to rewrite (12)

Here, the summation are taken from to . The Hessian matrix is of symmetry and the remaining submatrices can be easily determined. In each iteration, remember to represent

We will denote the all-one column vector of order by . The unknown parameters should be estimated to minimize the sum of squared errors defined by

(17)

(13)

and update them, as well as other variables , and , with the new estimate . Once is estimated as , we can determine the variance of for AR(p) model model error

Note that and are unobservable variables and should be and unknown parameters and represented in terms of in the estimation. Clearly, is a nonlinear function in and we can employ Newton–Raphson method for its optimization, as given by (14) The Hessian matrix is defined by the second derivative of with respect to and the column vector is formed by its derivative. Specifically

(15)

where

,

,

, and

(16)

(18) Here, we use hat and subscript to indicate that the parameters are estimated on the basis of the AR(p) model. It remains to determine the model order , which can be done by using the MDL criterion

(19) where is the number of unknown parameters in the model. The results for the data set obtained from a typical room suggest that an AR(1) is used with the following parameters: (20) The knowledge of allows us to determine the exponentially decaying factor. Furthermore, the small value of reveals remains basically unchanged within a window, but that changes from one window to another with a small correlation. . Having obtained the Let us return to the pdf issue of model parameters, it is easy to determine for which the and can be evaluated. We put the values of values of , so obtained in a single plot resulting in Fig. 1 where the data points correspond to 49 locations in a typical room. The line reflects the unique feature of log-normal distribution and all 49 data points, signified by asterisks, scatter closely along the line,


ESTIMATION

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TABLE I FADING PARAMETER m ON DIFFERENT LOCATIONS: MEAN AND STANDARD DEVIATION (STD)

OF

Fig. 2. Use a (k , ) plot to indicate the distribution of x(n).

demonstrating that the log-normal distribution is indeed a good , . To examine this asmodel for the data set sertion in a more rigorous framework of hypothesis testing, we on the basis of 147 points of test the linearity obtained in three rooms. It turns out that the linearity is accepted with a 75% confidence interval. V. MODELING AND ESTIMATION FOR To separate

from

, define vector

(21)

It follows from (1) and (6) that

can be estimated as

We examined various data acquired in six different rooms with results tabulated in Table I. All the results obtained by using real data to (23) show that falls between 0.5 and 0.6. For most rooms, is very close to 0.5. In about ten rooms tested so far, most are very close to and only two rooms are exceptional: one room with close to 0.6 and the other slightly exceeding it. A. Gaussian Model

(22) is the th entry of . where To model , we calculate the values of and from and the typical results for the TimeDothe data samples of main’s data from an arbitrarily chosen room is shown in Fig. 2. (denoted by asterisks and circles, The tested data points respectively), line up closely in the neighborhood of the straight and , which represents the Gaussian distribulines tion. The test indicates that the Gaussian distribution is a good . To further verify the assertion of Gaussian dismodel for tribution for , we examine its magnitude in a wider is really Gaussian, framework of Nakagami distribution. If would be . Various then the fading parameter of estimators for have been examined in [13]. An estimator for with accuracy close to the MLE is given by (23) where is the th moment and quite larger than unity and

Fig. 3. Covariance structure of x(n): TimeDomain versus Intel UWB data.

. Here, is an integer will be used in this paper.

As indicated in the tests based on the TimeDomain’s data, Gaussian distribution is an appropriate pdf for most scenarios. , we need to take into acJust as for the factor component for a complete statistical count the covariance structure of obtained from Timedescription. The autocorrelation of Domain’s measurement is depicted in Fig. 3 with Intel’s result also included for comparison. Clearly, significant correla. Suppose that an AR(q) tion does exist among the data of model is used as follows: (24) where the hat over the ’s has been dropped for notational simdenotes the i.i.d. zero-mean Gaussian innovaplicity and . We have totally untion process with variance known parameters in the AR( ) model. The estimation is much simpler compared to what we have shown in the previous section. Specifically, the model coefficients can be estimated from -bysample correlation matrix of and the can be estimated by using the MDL criterion. The reader is referred to Haykin [9] for details. Typically, the AR(4) model is selected for the TimeDomain’s data.

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Fig. 5. Original UWB signal versus its regenerated counterpart. Fig. 4. Excellent fit between x(n) and its AR-based linear predictor indicating the effectiveness of AR modeling.

As an illustration, in Fig. 4 we show a typical plot for , which clearly indicates that is a correlated stationary process. The dashed–dotted line signifies the one-step forward linear prediction based on the AR model, which has a good agreement with the original data. B. Nakagami Model , If is indeed considerably deviate from 0.5, say, using a Nakagami model. It should be one should fit emphasized that Nakamgami distribution is only for the magnitude data, but not the original one. The difficulty lies in finding a simple model, such as AR, to account for the correlation among Nakagami data, since the closure property is no longer applicable. Unlike Gaussian distribution, a linear combination of Nakagami variates is no longer a Nakagami variable. The difficulty can be circumvented if we consider a correlated Nakagami sequence to have been synthesized from a number of independent correlated Gaussian processes for which AR modeling is suitable. It is shown in [14] that the correlation matrix for the Gaussian processes can be determined from their Nakagami counterpart whereby an efficient algorithm is derived. The reader can employ such an algorithm for paramprocess; detail is not eterizing a Nakagami distributed pursued here due to space limitations. VI. APPLICATIONS The UWB models derived in this paper can be used for various purposes, such as the synthesis of UWB data in a laboratory environment and the receiver design. A. UWB Data Synthesis Suppose we are given a set of UWB measured data and want to generate a sequence with the same statistical behavior. We first decompose the measured process into three factor components and then follow these steps. Step 1) Estimate parameter whereby the exponentially decaying component is determined.

Fig. 6. Distribution comparison of original and regenerated UWB data.

Step 2) Select an appropriate AR model order for process and then estimate its model coefficients and the variance of the error process. The parameter set so obtained allows for the reconstruction of the component reflecting the large-scale behavior of the UWB signal. Step 3) Estimate the fading parameter for the third compobefore we are able to choose a right model nent for . If is close to 0.5, we use AR model to . If well exceeds 0.5, we concharacterize with a correlated sider fitting the magnitude of Nakagami process. Step 4) Multiply the three components leading to the process we need. Finally, we scale the process so that it has the total energy, as specified. The regenerated data, as shown in Fig. 5 resemblances its real counterpart in appearance. A more qualitative comparison can be made if we consider the two sequences as realizations of two corresponding stochastic processes and estimate their marginal pdfs. The results are depicted in Fig. 6 for comparison. A good match is observed. A more accurate measure for pdf matching


Fig. 8. Fig. 7.

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Energy-capture capability of three receivers using different templates.

Autocovariance comparison of original and regenerated UWB data.

is the Kullback–Leibler distance [15] that, for a random variable with the true pdf and hypothesized pdf , is given by (25) The value of ranges between zero to infinity. The values close to zero indicate a good match between the two pdfs. We partiinto 50 intervals and approximated the tioned the range of integral by a sum and calculated the probabilities that the real and regenerated data fall into these intervals. It turns out that , demonstrating, in one way, the validity of our model. We can further compare their covariance structures by focusing on the small portion of first 1000 samples, so that local stationarity is justified. Such treatment allows us to compute the sample correlation coefficients, ending up with results graphed in Fig. 7. Again, two correlation curves are very close, further demonstrating the effectiveness of our new model. B. Receiver Design Most UWB receivers consider a received UWB process to be a deterministic signal consisting of various delayed versions of the transmitted monocycle. The philosophy for receiver design, then, is to seek a good template that matches the received signal as much as possible. In fact, the randomness of path lengths and the possible distortion caused by scatterers [1] make it almost impossible to use a single template to match different segments of the received process. A compromise scheme is, therefore, to use the waveform received at 1 m apart from the transmitter [16], called the “clean” pulse, or the typical idealized Gaussian waveform [17] as the template for correlation. The clean pulse, along with the ML estimates of paths’ time delays and channel gains, is employed to synthesize the received waveform for use in a RAKE receiver. We take a different philosophy here by treating a received UWB signal as a stochastic process. The factor-based model developed before enables us to estimate the covariance matrix whose eigenvector, corresponding to the largest eigenvalue, can

be employed to build an eigen-filter. Fig. 8 compares the energycapture performance of three receivers that employ the eigenfilter “clean” pulse and Gaussian waveform for local correlation, respectively. The eigen-filter-based receiver outperforms its competitors. The three receivers have the same type of correlation structure. Their distinction only lies in different waveforms used for correlation. A “clean” pulse is obtained from onsite measurement, Gaussian waveform from theoretical calculation, and eigenvector from the sample covariance matrix of received data. For a more detailed description of the eigen-filter receiver, the reader is referred to [18]. VII. FITTING INTEL DATA AND MODEL REVISIT The new modeling technique was illustrated before through the TimeDomain data. In fact, it is equally well applicable to the UWB data sets by Intel. The Intel Laboratories conducted measurements on nearly 900 residential channels in the frequency domain over the band from 2 to 8 GHz. The frequency resolution was 3.75 MHz. Three kinds of operational environments were investigated: Townhouse, chamber and office. The distance between the transmitter and receiver ranges from 1.05 to 14.799 m. The transmitter and receiver were placed on the same floor or adjacent floors allowing for 0–4 walls between them. Unlike Timedomain data which records 49 realizations for each single room, the Intel measurement was not so regular. The Intel data are given in the frequency domain, and need inverse Fourier transform if processing is to be performed on a time sequence. For detailed description, see [19]. The data acquired for short distances basically contains few multipath components. Therefore, we choose data sets with transmitter and receiver separation greater than 4 m. Selected for our model fitting are about 20–30 arbitrary data sets, which correspond to 0–4 walls between the transmitter and receiver, representing different operaand are tional environments. The distribution test for shown in Figs. 9 and 10, whereby one can draw an assertion similar to that for the TimeDomain data. For a given UWB data set by Intel, we can regenerate a sequence in the same way as described before. The resulting sequence and its pdf and covariance structure are depicted in Figs. 11–13, respectively. Again,

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Fig. 9.


Fading parameter m for jx(n)j for Intel data. Fig. 11. Original UWB signal versus its regenerated counterpart: Intel data.

Fig. 10.

Lognormal distribution test for (k ): Intel data. Fig. 12. Distribution comparison of original and regenerated UWB sequence: Intel data.

a good agreement between the statistical features of the real and simulated data is observed. Having fit the TimeDomain and Intel UWB data satisfactorily by using the proposed procedure, we would like to get some further understanding of our proposed model. When we partition the UWB data into segments, each having a duration equal to corresponds to twice the transmit pulse, random variable the average power of the th segment. There are various objects in an indoor environment, which constitute a three-dimensional has its corresponding portion in the reflection field. Each field and the former’s intensity depends on various physical and geometrical characteristics of the latter, such as cross section, relative angle, and material. These features basically remain unchanged within each segment, but vary from one segment to another. The overall variation, however, can be very large, thereby . The factor calling for a lognormal distribution for fitting accounts for both sign and correlation incomponent formation in the original UWB data. Its correlation structure reflects the combined effect of the transmit signal and random channel. The random delays cause the random superposition of

received signals which, along with the characteristic variation of scatterers in space, determines the approximate Gaussianity of . Therefore, as we observe before, the estimated fading parameter for most rooms is very close to 0.5. Random superposition can destroy the correlation structure of the transmit monocycle. This explains why the eigen-filter vector of the sample can provide a better match, on covariance matrix of average, to the received UWB signal than the clean pulse or Gaussian waveform, as evident in Fig. 8. VIII. CONCLUSION In this paper, we considered real UWB data as a nonstationary correlated stochastic process. We used a deterministic exponential component to account for the decaying trend in the UWB power profile. The removal of the exponential decaying function results in a stationary process that, in turn, can be factored into two components with one accounting for the large behavior and another for local fluctuation due to multipaths. We utilized the


Fig. 13. Autocovariance comparison of original and regenerated UWB sequence: Intel data.

statistical method, which is based on the first four moments, to determine appropriate distributions for each component and utilized AR modeling to parameterize their covariance structures. A systematic procedure has been derived for distribution selection, model-order determination, and parameter estimation. The new technique was then applied to UWB data by the TimeDomain and Intel ending up with the following findings. • The large-scale component is well characterized by a correlated log-normal process for which AR modeling offers parsimonious description. The model order is typically equal to 1. is a correlated • A good model for local fluctuation Gaussian process in most of scenarios for which AR modeling is also applicable. Occasionally, local fluctuation is better to be fit by a correlated Nakagami model with the fading parameter ranging over (0.6, 0.7). How to handle the case with well deviating from 0.5 in order to keep the sign information of is the issue worth further investigation. The new model is used to generate UWB data and the results confirm its capability in fitting the distribution and correlation structure specified by the TimeDomain data. The same assertion is true of UWB data by Intel. The new model also provides a foundation for formulating the design of UWB receivers in the framework of detecting stochastic signal in white noise. REFERENCES [1] R. C. Qiu, “A study of the ultra-wideband wireless propagation channel and optimum UWB receiver design,” IEEE J. Sel. Areas Commun., vol. 20, no. 9, pp. 1628–1637, Dec. 2002. [2] S. S. Ghassemzadeh, R. Jana, C. W. Rice, W. Turin, and V. Tarokh, “A statistic path loss model for in-home UWB channels,” presented at the IEEE Conf. Ultra Wideband System and Technologies, Baltimore, MD, May 21–23, 2002. [3] J. Kunisch and J. Pamp, “Measurement results and modeling aspects for the UWB radio channel,” presented at the IEEE Conf. Ultra Wideband System and Technologies, Baltimore, MD, May 21–23, 2002. [4] Á. Álvarez, G. Valera, M. Lobeira, and R. Torres, “New channel impulse response model for UWB indoor system simulations,” presented at the IEEE Vehicular Technology Conf. (VTC’03), Apr. 2003.

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Q. T. Zhang (S’84–M’85–SM’95) received the B.Eng. degree from Tsinghua University, Beijing, China, the M.Eng. degree from South China University of Technology, Guangzhou, China, both in wireless communications, and the Ph.D. degree in electrical engineering from McMaster University, Hamilton, ON, Canada, in 1986. He held a research position and was an Adjunct Assistant Professor at McMaster University. In January 1992, he joined the Satellite and Communication Systems Division, Spar Aerospace, Ltd., Montreal, QC, Canada, as a Senior Member of Technical Staff. At Spar Aerospace, he participated in the development and manufacturing of the Radar Satellite (Radarsat). He was subsequently involved in the development of the advanced satellite communication systems for the next generation. He joined Ryerson Polytechnic University, Toronto, Toronto, ON, Canada, in 1993 and became a Full Professor in 1999. In 1999, he spent a one-year sabbatical at the National University of Singapore, Singapore, and is now with the City University of Hong Kong, Kowloon, Hong Kong. His research interest is on transmission and reception over fading channels with current a focus on wireless multiple-input–multiple output (MIMO) and ultrawide-band (UWB) systems. Dr. Zhang is an Associate Editor for the IEEE COMMUNICATIONS LETTERS.

S. H. Song (S’02) received the B.S. and M.S. degree from Tianjin University, Tianjin, China, in 2000 and 2002, respectively, and is currently working toward the Ph.D. degree at the City University of Hong Kong, Kowloon, Hong Kong. His research interests include ultrawide-band (UWB) systems and statistical signal processing for wireless communication.