A MAXIMUM ENTROPY APPROACH TO ULTRA-WIDEBAND CHANNEL MODELING

R. L. de Lacerda Neto, A. Menouni, M. Debbah

Institut Eurecom2229 route des Cretes - BP 19306904 Sophia Antipolis - France

B. H. Fleury

Aalborg UniversityFredrik Bajers Vej 7, A3

DK-9220 Aalborg - Denmark

ABSTRACT

In this paper, a unified framework for Ultra-Wideband chan-nel (UWB channel) modeling based on the maximum entropyapproach is provided. For a given set of constraints, a consis-tent model which takes into account the information at handis obtained. Two cases are considered: channel power knowl-edge and knowledge of the covariance matrix. The channelpower delay spectrum is also derived and the impact of thechannel bandwidth is assessed through information theoreticconsiderations. The information variation slope with respectto the bandwidth is also studied, based on wideband measure-ments performed at Institut Eurecom1.

1. INTRODUCTION

UWB systems radiate waveforms that are formed by a se-quence of “very short” pulses, i.e., pulses with durations aboutfew hundred picoseconds. Such signals are free of sine-wavecarriers and do not require IF processing because they canoperate at baseband. Moreover, the system uses very lowpower spectral density, below the thermal noise of the re-ceivers, which is inherently difficult to be detected and doesnot cause significant interference with other systems. Thesebasic properties of the radio UWB system make it an idealcandidate for commercial, short-range, low power, low costindoor communication systems such as Wireless Local AreaNetwork (WLAN) and Personal Area Network (WPAN).

Although appealing, the efficiency of UWB systems isstill questionable. Indeed, for large bandwidths, channel un-certainty limits the achievable rates of power constrained sys-tems and therefore the capacity depends crucially on the modelat hand. In fact, recent results [1] show that the capacity is afunction of how the number of channel paths scale with thebandwidth (linear, sub-linear,..). Furthermore, although thelarge bandwidth is supposed to increase the rate, channel es-timation of all the channel paths becomes a real bottleneck asone may dedicate a large fraction of the rate to estimate moreand more channel coefficients.

1This work is part of the European Network of Excellence NEWCOM,Contract IST NoE 507325.

This contribution aims at analyzing how channel uncer-tainty scales with bandwidth in UWB systems. Equivalently,the number of parameters necessary to predict (or also repre-sent) the wideband channel is determined. In this respect, asound framework for UWB channel modelling [2] based oninformation theoretic tools is provided. Based on a given setof constraints (measurements, power), a channel model is de-rived based on the maximum entropy approach and validatedthrough measurements performed at Institut Eurecom. Withinthis setting, channel uncertainty has a straightforward mean-ing through the notion of entropy that is analyzed with respectto the bandwidth.

Previous studies [3] have already analyzed channel uncer-tainty scaling through the number of significant paths. How-ever, in many cases, additional criteria (such as AIC, MDL..)have to be considered as, for noisy measurements, the notionof significant paths is subjective. Note finally that previouscontributions have also focused on characterizing the wide-band channel through a limited number of parameters (ARmodels with few coefficients [4]). The benefit of such char-acterizations rely on the fact that it is possible to reproducethe channel frequency behavior using only a small number ofparameters.

2. MAXIMUM ENTROPY MODELING (MEM)

The problem of modelling wideband channels is crucial forthe efficient design of wireless systems. Indeed, the wirelesschannel suffers from constructive/destructive interference sig-naling and therefore yields a randomized channel for whichone has to attribute a joint probability distribution for the chan-nel frequency response. In this contribution, we would liketo provide some theoretical grounds to model the widebandchannel based on a given state of knowledge. In other words,knowing only certain things related to channel (power, mea-surements), how to translate that information into a model forthe channel? This question can be answered in light of theBayesian probability theory [5] and the principle of maximumentropy. The principle of maximum entropy is at present theclearest theoretical justification in conducting scientific infer-ence: we do not need a model, entropy maximization creates

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a model for us out of the information available. Choosingthe distribution with greatest entropy avoids the arbitrary in-troduction or assumption of information that is not available.Note that this approach has been successfully used in spec-trum analysis [6] and signal interpolation problems [7] [8].

2.1. MEM for channel power knowledge

Suppose in the following that the modeler has no knowledgeabout the frequency response of the wideband channel andwould like, with this limited knowledge, to determine thenumber of clusters in the environment. The modeler knowshowever that the channel carries some power P and is sta-tionary during the channel modelling phase.

Let {hi}i∈Z be the sequence of samples at frequenciesiδf (δf is the frequency resolution) of the channel frequencyresponse. In this case, the spectral autocorrelation function isdefined as:

R(k) = E[hih∗i+k] (1)

and the power delay spectrum is defined as

P (τ) =∞∑

k=−∞R(k)e−j2πτk, (2)

where τ = τTs

is the normalized delay and τ is the delay inseconds.

In the following, we suppose that∫ τmax

2− τmax

2P (τ)dτ is the

power of the channel equal to P and τmax is the maximumdelay. The modeler would like to take into account the powerconstraint without additional information that is not available.Maximizing the entropy of the process guarantees such a set-ting as one finds the sequence of autocorrelations that makethe impulse response as white as possible. In other words,such an extrapolation places the least amount of structure inthe channel.

For a Gaussian random process, with power delay spec-trum P (τ), the entropy H is given by

H = log(πe) +∫ 1

2

− 12

log(P (τ) + ε)dτ, (3)

where ε is an arbitrary small constant (ε ≥ 0) used to regular-ize the non-regular Gaussian process (hi).

The modeler would like to maximize H under the con-straint that the received power in a given delay interval isknown. This statement can simply be expressed if one triesto maximize the following expression using Lagrange multi-pliers with respect to R(k):

C = H − µ0(∫ τmax

2

− τmax2

P (τ)dτ − P ) (4)

∂C

∂R(k)=

∫ 12

− 12

1P (τ) + ε

∂P (τ)∂R(k)

dτ − µ0

∫ τmax2

− τmax2

e−j2πτkdτ,

=∫ 1

2

− 12

1P (τ) + ε

e−j2πτkdτ +µ0

j2πk

[e−j2πkτ

] τmax2

− τmax2

,

=∫ 1

2

− 12

1P (τ) + ε

e−j2πτkdτ − µ0τmaxsinc(kπτmax),

= 0.

Let Q(τ) = 1P (τ)+ε and qk =

∫ 1/2

−1/2 Q(τ)e+j2πkτ dτ . Wehave

∫ 12

− 12

Q(τ)ej2πτkdτ = µ0τmaxsinc(kπτmax) (5)

⇔ qk = µ0τmaxsinc(kπτmax). (6)

Thus, Q(τ) =∑∞

k=−∞ µ0τmaxsinc(kπτmax)e−j2πkτ , whichis constant for all −τmax

2 ≤ τ ≤ τmax

2 . As a consequence,

P (τ) ={

Pτmax

; −τmax

2 ≤ τ ≤ τmax

2

0 ; elsewhere.

and R(k) = P sinc(kπτmax) for all k. P (τ) is constant andvanishes on [− 1

2 , + 12 ]/[− τmax

2 , + τmax

2 ]. In other words ifthere is no knowledge except the maximum delay, the modelgives an infinite number of clusters and the power is equallysplit across the different clusters. The methodology can beeasily extended if the modeler has knowledge of the band-width (which determines the number of correlation coeffi-cients R(k)) used.

2.2. MEM for covariance channel knowledge

In the following, we suppose that the modeler has knowledge(through measurements) of a finite number of frequency au-tocorrelation coefficients R(k). The number of coefficients isdetermined by the measured bandwidth as well as the mea-surement resolution. Based on this knowledge, the modelerwould like to derive a wideband model taking into account ac-count the previous constraints and not more, i.e, the modelerwould like to extrapolate the missing autocorrelation coeffi-cients for deriving the power delay spectrum function. Usingthe same methodology as Section 2.1, the following theoremdue to Burg [9] can be obtained:

Theorem 1 The maximum entropy rate stochastic process{hi}i∈Z that satisfies the constraints

E[hihi+k] = R(k), k = 0, 1, ....., N, for all i, (7)

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is the N -th order Gauss-Markov process of the form

hi = −N∑

k=1

akhi−k + Zi, (8)

where the Zi are i.i.d. ∼ N(0, σ2) and a1, a2, ..., aN , σ2 arechosen to satisfy Equation (7).

Remark: The theorem does not assume the hi to bewide-sense stationnary.

A process satisfying (8) is also called autoregressive oforder N (AR(N)). The coefficients a1, a2, ..., aN , σ2 are ob-tained by solving the Yule-Walker equations

R(0) = −N∑

k=1

akR(−k) + σ2, (9)

R(l) = −N∑

k=1

akR(l − k), l = 1, 2, ..., N. (10)

Fast algorithms such as the Levinson and Durbin algo-rithm have been devised which exploit the special structureof these equations to efficiently calculate the coefficients a1,a2, ...., aN from the covariance samples R(0), ..., R(N). Thepower delay spectrum of the N th order Gauss-Markov process(8) is given by

P (τ) =σ2∣∣∣1 +

∑Nk=1 ake−i2πkτ

∣∣∣2. (11)

In general, from a finite set of N frequency measurements[hl

1, ...hlN ] over a bandwidth of Nδf (l is the lth channel re-

alization), there are many ways to estimate the spectral auto-correlation coefficients. In the following, the sample autocor-relation function is defined as:

RN (k) =1L

1N − k

L∑l=1

N−k∑i=1

hli(h

li+k)∗, k ≥ 0.

For a given N, we estimate and determine the autocorre-lation function RN(k), the autocorrelation coefficients a

(N)k

and the power delay spectrum PN (τ). As a consequence, theestimated entropy is given by:

HN = log(πe) +∫ 1

2

− 12

logσ2∣∣∣1 +

∑Nk=1 a

(N)k e−i2πkτ

∣∣∣2dτ.

(12)The roots of the power delay spectrum (11) determine the

number of significant clusters. Practically, although the rootsmay exist, some may be not significant and therefore unnec-essary to model. In order to assess the number of significantmodelling coefficients, we will analyze in Section 3 the infor-mation HN contained in the process with respect to N .

3. MEASUREMENT DESCRIPTION

3.1. Measurement setup

The measurements were carried out in the Mobile Commu-nications Laboratory of Institut Eurecom, which is a typi-cal lab environment, rich in reflective and diffractive objects.The measurement device used is a wideband Vector NetworkAnalyzer (VNA) which allows complex transfer function pa-rameter measurements in the frequency domain, extendingfrom 10MHz to 20GHz. This instrument has low inherentnoise, less than -110dBm for a measurement bandwidth of10Hz, and high measurement speed, less than 0.5ms/point.The maximum number of equally-spaced frequency samples(amplitudes and phase) per measurement is 2001. The mea-surement data were acquired and controlled remotely usingRSIB protocol over Ethernet permitting off-line signal process-ing and instrument control in MATLAB.

In order to perform truly wideband measurements withsufficient resolution, several bands was concatenated by us-ing consecutive measurements. In this contribution, measure-ments were performed from 3GHz to 9GHz by concatenatingthree groups of 2001 frequency samples per 2GHz sub-bands(3-5GHz,5-7GHz,7-9GHz). This yields a 1MHz spacing be-tween the frequency samples. Systematic and frequent cal-ibration (remotely controlled) was employed to compensatethe undesirable frequency-dependent attenuation factors thatmight affect the collected data. The wideband antennas em-ployed in the measurements are omnidirectional in the ver-tical plane and have an approximate bandwidth of 7.5GHz(varying from 3.1GHz to 10 GHz). They were not perfectlymatched across the entire band, with a Voltage Standing WaveRadio (VSWR) varying from 2 to 5.

3.2. Measurement environment

The data analyzed in Section 4 were collected in a Line ofSight (LoS) setting with measurements performed at spatiallydifferent locations. The experiment area is set by fixing thetransmitting antenna on a mast at 1m above the ground andclose to the VNA. The receiving antenna moves then to dif-ferent locations on a horizontal linear grid of side 50 cm insteps of 5cm. The transmitter antenna’s height was varied by5cm up to 20cm after completion of the measurements at var-ious receiver positions. The transmitter and the receiver wereseparated by a distance of six meters.

4. RESULTS

In the following, the scaling of channel uncertainty with re-spect to the bandwidth is analyzed. In Fig. 1, the variation ofHN is plotted for the LoS case as well as the Gaussian case(zero mean i.i.d frequency samples are generated in this case)versus Nδf (the frequency band is [3000, 3000+Nδf]MHz).As one can observe, the channel uncertainty decreases with

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bandwidth. However, in the Gaussian case, additional infor-mation provided by the frequency samples does not lower theuncertainty as the samples are completely independent. Re-markably, the results show that in UWB settings and for agiven channel representation complexity (here, the slope ofthe entropy), there is an optimum number of parameters tobe chosen. In other words, AR modelling based on a limitednumber of parameters is adequate. In this respect, we haveplotted in Fig. 2 and Fig. 3 the power delay spectrum basedon 500MHz and 6GHz measurements. The results show thatwith increasing bandwidth, one is certainly able to capture thesmall variations but a great amount of the information is al-ready included in the 500MHz band. Note that the delay of20 nanoseconds corresponds to the 6 meter distance betweenthe transmitter and the receiver. Moreover, in both cases, theAR model fits the measured power delay spectrum response.

5. CONCLUSION: INFORMATION VERSUSCOMPLEXITY

In section 4, we have shown that although in modelling, oneshould take into account all the information at hand, thereis a compromise to be made in terms of model complexity.Each information added will not have the same effect on thechannel model and might as well more complicate the modelfor nothing rather than bring useful insight on the behaviorof the propagation environment. In this respect, entropy isa useful measure and the slope decrease characterizes howinformation scales with bandwidth. In particular, in wide-band schemes, we have shown that it is possible to reproducethe channel frequency behavior with a limited number of co-efficients since the channel uncertainty decreases with band-width. We have also provided a sound methodology to modelchannels when additional channel constraints are given.

6. REFERENCES

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[5] E. T. Jaynes, Probability Theory: The Logic of Science, Cam-bridge, 2003.

[6] G. L. Bretthorst, Bayesian Spectrum Analysis and ParameterEstimation, Ph.D. thesis, Washington University, St. Louis,1987.

0 500 1000 1500 2000 25002.9

2.95

3

3.05

3.1

Bandwidth (MHz)

En

tro

py

(Bas

e 2)

AWGNMeasured

Fig. 1. Entropy variation with respect to the bandwidth.

0 0.025 0.05 0.075 0.1 0.125 0.150

20

40

60

80

100

120

140

160

180

τ (µs)

Po

wer

Del

ay S

pec

tru

m

Fig. 2. Power delay spectrum with 500Mhz bandwidth.

0 0.025 0.05 0.075 0.1 0.125 0.150

20

40

60

80

100

120

140

160

180

τ (µs)

Po

wer

Del

ay S

pec

tru

m

Fig. 3. Power delay spectrum with 6Ghz bandwidth.

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[8] P. A. Regalia, M. Mboup, and M. Ashari, “A Class of first-and second-order interpolation problems in Model Reduction,”Archiv fr Elektronik und Ubertragungstechnik, vol. 49, pp. 332–343, September 1995.

[9] T. Cover and J. Thomas, Elements of Information Theory, Wiley,1991.

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