Small Scale Characterization of Underwater Acoustic Channels

Transcription

1 Small Sale Charaterization of Underwater Aousti Channels Parastoo Qarabaqi Northeastern University Milia Stojanovi Northeastern University ABSTRACT Underwater aousti (UWA) hannel models provide a tool for prediting the performane of ommuniation systems prior to system deployment and are thus essential for system design. Small-sale modeling of the UWA hannel targets statistial haraterization of random effets suh as sattering and motion-indued Doppler shifting. In this study, eah transmission path is modeled as onsisting of miromultipath omponents that umulatively result in a omplex Gaussian multipliative distortion, whose time- and frequeny-orrelation are assessed analytially. Motion of the surfae and transmitter/reeiver displaements introdue additional variation whose temporal orrelation is modeled via Bessel-type funtions. Data obtained from three underwater experiments support the proposed model. Categories and Subjet Desriptors C.2.1 [Computer-ommuniation Networks]: Network Arhiteture and Design Wireless ommuniation General Terms Theory Keywords underwater aousti ommuniations, hannel modeling, small-sale fading 1. INTRODUCTION Underwater Aousti (UWA) ommuniation systems have to be designed to operate in a variety of onditions that differfrom thenominal onesduetothehanges insystemgeometry and environmental parameters. Beam traing tools, suh as Bellhop [3], provide an aurate deterministi piture oftheuwahannel for agiven geometry andsignal frequeny, but they do not take into aount random variation of the hannel. Numerous studies have been onduted to stohastially model the UWA hannel, e.g. [1, 4 1]. These Permission to make digital or hard opies of all or part of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for profit or ommerial advantage and that opies bear this notie and the full itation on the first page. To opy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speifi permission and/or a fee. WUWNet 12, Nov. 5-6, 212 Los Angeles, California, USA. Copyright 212 ACM /12/11... $15.. studies are usually based on the analyses of experimental aousti data olleted in a partiular loation. Some authors find Riean fading [4, 6, 7] or Rayleigh fading [1] to provide a good math for their measurements, while others find log-normal distribution [5, 8] or the K-distribution [9, 1], to be a better fit. The variety of the proposed statistial models is due to experiment-speifi properties, e.g. the deployment site and the type of signals used for probing, as well as the time intervals during whih the hannel is observed, i.e. small-sale versus large-sale phenomena. We distinguish two types of hannel variations: those that are aused by displaements spanning many wavelengths, and those that are aused by displaements on the order of one or a few wavelengths. The latter are referred to as the small-sale variations. Considering motion on the order of 1 m/s, and frequenies on the order of 1 khz, a wavelength is traversed during a sub-seond interval. Suh short intervals of time inidentally orrespond to typial ommuniation transations (a paket or a frame of pakets). Smallsale hannel variations an thus be thought of as those variations that our over a ommuniation transation. They are to be distinguished from the variations that are aused by larger system displaements that span many wavelengths and our on orrespondingly longer intervals of time. This paper follows a previous study [5] in whih a lognormal model was proposed for the large-sale variations of the aousti loss. In the present work, small-sale phenomena suh as miro-multipath formation due to sattering are the target of analysis. These phenomena may ause additional, possibly fast and omparatively large variations in the instantaneous signal power. The rest of the paper is organized as follows. In Se. 2, a theoretial model is proposed for small-sale hannel effets. Motion indued Doppler shifting is modeled in Se. 3. Experimental results that support the model are presented in Se. 4, and onlusions are summarized in Se SMALL-SCALE CHANNEL CHARACT- ERIZATION The UWA hannel is a multipath hannel whose impulse response an be represented as h(τ) = p h pq(τ τ p) (1) where q(τ) is the nominal impulse response of the referene path, h p are the s, and τ p are the path delays. These s and delays orrespond to a partiular largesale system realization, i.e. a partiular system geometry

2 and sound speed profile at a given time. A ommon way to model small-sale effets is to inlude a multipliative random fator into the overall h p. While suh a model is simple, it may be overly simplisti as it applies only to narrowband systems, while a more general model alls for modifiations to path filtering as well. Apart from experimental measurements, purely theoretial arguments may be invoked to model the sattering proess. Speifially, let us begin by onsidering a narrowband ase, in whih the transmitted signal s(t) = Re{u(t)e j2πft } has bandwidth f f. We will later extend this onept to the wideband ase by onsidering different frequenies f within a signal bandwidth B. Let us also fous on a single path, say path p. So far, we have modeled this path as having the gain h p and delay τ p. However, if sattering ours along this path, the arriving signal an be modeled instead as r p(t) = Q(f) i h p,is(t τ p δτ p,i) = Re{v p(t)e j2πft } where Q(f) = F {q(τ)}, v p(t) is the equivalent baseband reeived signal, and h p,i andδτ p,i desribe theintra-pathgains and relative delays of the miro-multipath omponents, respetively. Defining h p,i = h pγ p,i, the equivalent baseband signal is given by v p(t) = Q(f)h pe j2πfτp γ p,ie j2πfδτ p,i u(t τ p δτ p,i) i The delays δτ p,i orrespond to miro-multipath length differenes that an be on the order of the signal wavelength, λ = /f. Sine f f, the signal u(t) will remain pratially insensitive to suh delays, i.e. u(t δτ p,i) u(t). We an thus express the arriving signal as where v p(t) Q(f)h pe j2πfτp γ p(f)u(t τ p) (2) γ p(f) = i γ p,ie j2πfδτ p,i (3) This expression implies that the effet of sattering in a narrowband system is that of a multipliative distortion only. Colleting all the frequeny omponents of a wideband signal, and allowing for the possibility of time-varying path oeffiients and delays, the overall hannel transfer funtion is given by H(f) = Q(f) p h pγ p(f)e j2πfτp (4) Note that although we have dropped the time from notation, both the h p and the distortion oeffiient γ p vary with time, so stritly speaking we should write h p(t) and γ p(f,t) in the above expression. Allowing for the timevarying delays τ p as well, we have a general model H(f,t) = Q(f) p h p(t)γ p(f,t)e j2πfτp(t) (5) The fundamental differene between the two multipliative fators is that h p(t) varies slowly while γ p(f,t) may vary rapidly. Depending upon the system funtions on whih one is fousing, the appropriate time-dependene should expliitly be taken into aount. For example, if one assumes that large-sale power ontrol is in plae at the transmitter, then it suffies to fous on γ p(t) in order to evaluate the average probability of bit error. Conversely, if one is fousing on large-sale power ontrol, then the notion of average power is used tosignify thefat that small-sale fadingis not onsidered, i.e. averaging is performed over γ p. Speifially, one an define the instantaneous power gain orresponding to the p-th path as G p(f,t) = Q 2 (f)h 2 p(t) γ p(f,t) 2 (6) and the average power gain as } G p(f,t) = E γ { Gp(f,t) = Q 2 (f)h 2 p(t)e{γp(f,t)} 2 (7) 2.1 Probability density funtion The onstituent terms of the distortion oeffiient γ p(f) in expression(3) are treated as random variables, whose behavior is ditated by the amplitudes γ p,i and delays δτ p,i. While the amplitudes and delays may exhibit little variation, the phases φ p,i = 2πfδτ p,i mod2π will exhibit signifiant variation whenever the delays are on the order of a wavelength, ausing in turn a signifiant variation in γ p(f). Assuming that the onstituent terms are independent and identially distributed, the Central Limit Theorem implies a omplex Gaussian distribution for γ p(f). It is also possible to onsider a situation in whih there is a omponent whose delay is stable. In that ase, the distortion is modeled by γ p(f) = γ p, + i 1γ p,ie j2πfδτ p,i (8) where γ p, represents the relative oeffiient of the stable path (whose δτ p, = ). In general, the oeffiient γ p is omplex Gaussian with mean γ p(f) and variane σ 2 p(f) per omplex dimension, i.e. its magnitude γ p is Riean distributed. The orresponding Riean fator observed on the p th path is K p = γ p/2σ 2 p. The seond order statistis γ p and σ 2 p will be influened by the environmental onditions, e.g. the bottom and surfae roughness whih are in turn related to wave ativity. The path statistis γ p(f) and σ 2 p(f) an be determined experimentally, or analytially if the distribution of the onstituent terms is known. To illustrate an analytial approah, let us assume that the miro-path amplitudes are fixed at γ p, and γ p,i = µ p,i 1, and that the delays δτ p,i are zero-mean Gaussian, with variane σ 2 δ p. 1 The mean and variane are then obtained as γ p(f) = γ p, +µ ps pρ p(f) 2σ 2 p(f) = µ 2 ps p[1 ρ 2 p(f)] where S p is the number of intra-paths and ρ p(f) is defined as ρ p(f) = E{e j2πfδτ p,i } = e (2πf)2 σ 2 δp /2. In this model, we note that the seond-order statistis are determined by the funtion ρ p(f), whose value depends on the parameter fσ δp, i.e. the standard deviation of the miro-path delay normalized by the signal wavelength. Note that regardless of the distribution of δτ p,i, for high frequenies, suh that fσ δp 1, the model redues to the standard one in whih the phases φ p,i are uniformly distributed. 1 Gaussian-distributed intra-path delays orrespond to Gaussian-distributed displaements of sattering points.

3 ρ p ( f) σ δ p =.2ms σ δ p =.5ms f [khz] Figure 1: Frequeny orrelation term ρ p( f) for different standard deviations of the miro-path delays. 2.2 Correlation in the frequeny domain Frequeny-orrelation of the small-sale path oeffiients is desribed by the funtion E { γ p(f + f,t)γ q(f,t) }. Assuming that the sattering proess is independent between different paths, i.e. that refletion points of different paths are suffiiently far apart, we have that E{γ p(f + f)γ q(f)} = γ p(f + f) γ q(f) +δ p,qe{[γ p(f + f) γ p(f + f)][γ q(f) γ q(f)] } Note that although the paths exhibit unorrelated sattering, the above funtion is not zero in general, due to the non-zero mean values. In order to evaluate the frequeny-orrelation, the distribution of the intra-path delays δτ p,i has to be known. For the Gaussian distributed delays with zero mean and variane σ 2 δ p, frequeny-orrelation is obtained as E{[γ p(f + f) γ p(f + f)][γ p(f) γ p(f)] } = µ 2 ps pρ p( f)[1 ρ p( 2f(f + f))] µ 2 ps pρ p( f) (9) where the approximation holds well for σ δp > 1/ 2max {f,f + f}. Fig. 1 illustrates the funtion ρ p( f). We note that depending upon the standard deviation of the path delays, there may be more or less orrelation between the small-sale oeffiients γ p within the signal bandwidth. In our analysis of experimental data in Se. 4, we will assume that full orrelation exists aross the signal bandwidth, i.e. that a single value γ uffies to desribe small-sale effets. However, we note that this assumption may not hold always, and, hene, our results should be regarded as preliminary. Note also that while the partiular funtion of Fig. 1 pertains to Gaussian distributed delays, one an expet a similar onlusion to hold for a different distribution as well. 2.3 Correlation in the time domain Time-orrelation of the sattering oeffiients is desribed by E{γ p(f,t+ t)γ p(t)}. This funtion aptures the effet of motion within the sattering field whih influenes the oeffiients γ p(f,t) through the time-varying miro-path delays. To assess the time orrelation funtion, power spetral density of δτ p,i(t) has to be known. Without loss of generality, let us assume that the Gaussian-distributed delays δτ p,i(t) obey a first-order auto-regressive proess (AR-1): δτ p,i(t+ t) = α δp δτ p,i(t)+w δp,i (t) where w δp,i (t) N(,σ 2 δp(1 α 2 δp)), α δp = e πb δp t and B δp is the 3 db width of the psd of δτ p,i(t). The timeorrelation funtion is obtained as E{[γ p(f,t+ t) γ p(f)][γ p(f,t) γ p(f)] } where we have defined = µ 2 ps pe (1 α δp )(2πf)2 σ 2 δp [1 e α δp (2πf)2 σ 2 δp] 2σ 2 p(f)e πbp(f) t (1) B p(f) = (2πfσ δp ) 2 B δp and the approximation holds under the assumption that t 1/B δp. Note that the expression (1) implies that γ p(f,t) an equivalently be modeled as a Gauss Markov AR-1 proess. 3.MOTION-INDUCED DOPPLER SHIFTING Motion of the transmitter/reeiver or any refletion points in the hannel leads to time-varying path delays τ p. Fousing on the small-sale phenomena, we are interested in the variations that our over short intervals of time (e.g. subseond intervals). During suh intervals, it is reasonable to assume that any motion ours at a onstant veloity, i.e. that it is only the veloity and not aeleration that matters. When that is the ase, the path delays are modeled as τ p(t) = τ p a pt where a p = v p/ is the Doppler fator orresponding to veloity v p. 2 Note that the Doppler fators a p may vary over longer periods of time and a more general quasi-stationary model an be introdued to address this fat. We will keep this fat in mind, but drop time indexing for simpliity. At least three types of motion influene the Doppler fator, (i) unintentional transmitter/reeiver motion (drifting, whih gives rise to a Doppler saling fator a dp ), (ii) intentional transmitter/reeiver motion (vehiular motion, whih gives rise toa vp)and(iii) waves (surfae motion, whihgives rise to a sp). If the Doppler fators orresponding to eah of these types of motion are fixed, the orresponding Doppler shift is always the same. On the ontrary, if the Doppler fators hange randomly from one realization to another, so do the shifts. The resulting effet is that of random Doppler shifting or Doppler spreading. To haraterize Doppler spreading, we fous on the autoorrelation funtion of the random proess e j2πapft, i.e. E{e j2πapf t }. Assuming independene between various fators ontributing to motion, we have that R p( t) = E{e j2πfa dp t }E{e j2πfavp t }E{e j2πfasp t } Should any one of these omponents be regarded as deterministi, its expetation is dropped. To haraterize the drifting omponent, we assume that the transmitter and reeiver drift at veloities v td, v rd in diretions θ td, θ rd with respet to horizontal pointed toward 2 It is also possible to aount for a path-dependent propagation speed p.

4 eah other. The relative speed, projeted onto the p-th path, is v dp = v td os(θ p θ td ) v rd os(θ p +θ rd ) and the orresponding Doppler fator is a dp = v dp /. If the transmitter/reeiver drift in random diretions, then E{e j2πfa dp t } involves averaging over θ td, θ rd. Assuming that drifting is equally likely in any diretion, and that it ours independently for the transmitter and reeiver, E{e j2πfadp t } = J v td (2π f t)j(2πv rd f t) (11) where J (.) is the Bessel funtion of the first kind and order zero. Assuming v td = v rd = v d, and a d = v d /, the above expression redues to J(2πa 2 d f t). Vehiular omponent of the Doppler effet is obtained similarly, exept that those omponents of motion that an be estimated and ompensated by synhronization are not to be regarded as part of the hannel distortion. Hene, we regard a vp as the residual Doppler fator after initial synhronization. For example, if synhronization ompensates for the pre-dominant Doppler fator orresponding to the projetion of the transmitter/reeiver intentional veloity onto the referene path p=, the effetive Doppler fator is a vp = 1 [ [v tvos(θ p θ tv) v rvos(θ p +θ rv)] ] [v tvos(θ θ tv) v rvos(θ +θ rv)] = 1 [ v (θp +θ 2θ tv ) (θ p θ ) tv[ 2sin sin ] 2 2 +v (θp +θ ] +2θ rv ) (θ p θ ) rv[2sin sin ] 2 2 If we assume the transmitter/reeiver motion to be equally likely in any diretion θ tv/rv, the auto-orrelation funtion orresponding to vehiular motion is E{e j2πfavp t 2vtv sin((θp θ)/2) } =J (2π f t) 2vrv sin((θp θ)/2) J (2π f t) (12) Finally, to assess the surfae omponent, let us fous on waves thatause apointon thesurfae tomove upanddown reating a displaement that varies sinusoidally in time, with amplitude A w and frequeny ω w/2π. A signal impinging upon the j-th refletion point along the p-th path athes it in a random phase, i.e. at vertial veloity v wsin(ψ p,j+ω wt) where ψ p,j U[ π,π], and v w = A wω w. Projetions of this veloity onto the p-th path, summed over all surfae refletion points, yield v sp = 2v w sinθ p K s(p) j=1 sin(ψ p,j +ω wt) where K s(p) is the number of surfae refletions along the p-th path. Assuming that refletion points are suffiiently far apart suh that ψ p,j are independent, time-orrelation is obtained by taking the expetation over a sp = v sp/ with the angles ψ p,j uniformly distributed over 2π. The result is E{e j2πfasp t } = [J (2πa wpf t)] Ks(p) (13) where a wp = 2v w sinθ p/. It is worth noting that the main lobe of this Bessel-like (or sin-like) funtion, whose behavior Table 1: Nominal parameters of the experimental hannels. BW[kHz] d[km] h w[m] h tx[m] h rx[m] SPACE MACE KAM indiates the oherene time and is ditated by a wp, narrows with eah additional surfae enounter. Putting together Eqs. (1) (13), the omplete autoorrelation funtion of the overall sattering oeffiient γ p(f,t) = γ p(f,t)e 2πapft is obtained as [ R γp ( t) = γ p(f)+2σ 2 p(f)e 2 πbp(f) t] J(2πa 2 d f t) 2vtv sin((θp θ)/2) J (2π f t) 2vrv sin((θp θ)/2) J (2π f t) [ ] Ks(p) J (2π 2vwsinθp f t) (14) This funtion exhibits an overall Bessel-like behavior, dampened by the exponentially deaying orrelation of the γ p oeffiient. As suh, it is helpful in explaining experimental measurements, whih exhibit similar behavior, and whih we present in the next setion. 4. EXPERIMENTAL RESULTS We present experimental data olleted during three underwater experiments. The transmitted signals were PN sequenes, BPSK modulated, and transmitted repeatedly over the ourse of one minute. The first experiment, alled the Surfae Proesses Aousti Communiation Experiment (SPACE) was onduted south of the oast of Martha s Vineyard in Massahusetts, in the fall of 28. The arrier frequenywas 13 khz and the transmission rate was 6.5 kbps. The water depth was 1 m, the transmitter and reeiver were deployed at 4 m and 2 m above the sea-floor respetively, and separated by 1 m. The seond experiment, Mobile Aousti Communiation Experiment (MACE), was onduted in the Atlanti Oean about 1 miles south of Cape Cod in July 21. The reeiver was suspended at the depth of 4 m and the transmitter was towed at the depth of 5 6 m. The water depth was approximately 1 m and the distane varied between 5 m and 4 km. The signals were transmitted ontinuously at 5 kbps. Finally, the third experiment, known as the Kauai Aomms MURI (KAM), was onduted in July 211 off the oast of Kauai Island, Hawaii. The transmitter and the reeiver were deployed at mid-water olumn and were 3 km apart in approximately 1 m deep water. The signals were modulated onto the arrier of 13 khz and transmitted at the rate of 6.5 kbps. Table 1 shows the nominal parameters of the experimental hannel and the signal frequeny range for different experiments. The following approah was taken to prepare the signals for analysis. First, the reeived signal was resampled based on the reeived paket length to ompensate for the motionindued time saling and frequeny shifting whih an be

5 modeled by a rough Doppler fator. The resampling was notably neessary for the MACE data, where the transmitter moved at about 1 m/s reating a Doppler rate on the order of 1 3. Fine Doppler ompensation was then arried out using a ombined RLS estimator and seond order Phase Loked Loop. One these steps were ompleted, the Orthogonal Mathing Pursuit algorithm was used for hannel estimation. This algorithm is ommonly used for estimation of UWA hannels (see for example [2]) as it outperforms onventional least-squares methods when the hannel is sparse. The resulting estimate of the baseband hannel response was then used to extrat the s. Figs. 2 4 show the results obtained for the three experiments. Eah figure shows an ensemble of estimated baseband impulse responses (magnitude) over the duration of 1 minute at resolution of one bit. Several different loal maxima over the delay axis are highlighted in the figures. The loal maxima indiate hannel taps over whih the impulse response is the strongest, and they orrespond to the delays at whih physial paths our. Physial path delays orresponding to the nominal hannel geometry are marked by arrows and labeled as s, b, sb, et. referring to surfae, bottom, surfae-bottom, et. refletions. The observed path delays deviate from the nominal beause of loation unertainty and small-sale motion. Also, eah loal maximum may represent more than one physial path if the path delays are too lose to be distinguished given the delay resolution (finite bandwidth) of the system. Finally, the time variability of the hannel may ause eah path to (i) spread over several adjaent taps, and (ii) enompass a slowly varying mean. Path spreading, whih is a onsequene of both the miro-path dispersion and bandwidth limitation, may our uniformly at a tratable rate, e.g. the path labeled p in the KAM experiment, or an intratable one, e.g. the path labeled p bs. In order to take into aount the effet of all ontributing taps, whenever the phase information is not needed, several taps an be ombined in a root mean square fashion to form the absolute value of eah. Slow variation of the mean is observed if the oneminute window is longer than the stationarity interval of the proess h p(t). For example, suh slow variation is evident in the path of the MACE experiment. In this ase, in order to extrat the absolute value of the small-sale fator γ p(t), the slowly varying average of the proess h p(t)γ p(t) is removed. When the phase information is not to be negleted, the signal statistis are estimated over shorter time intervals, i.e. a fration of a minute, and ensemble averaged. In addition to the time-evolution of the hannel response, shown in the figures are also the histograms of the estimated γ p(t) for several different paths, along with a theoretial Riean urve. The onditional Riean distribution (onditioned on the slowly-varying mean) appears to provide a good fit. In all three experiments, the Riean K-fator orresponding to the diret path (p=) is greater than that of the other paths, indiating a stable arrival. Finally, the time domain auto-orrelation funtions of the path oeffiients are plotted. For all three experiments, the auto-orrelations orresponding to the diret path show less flutuation and a higher oherene time. All other paths show Bessel-type autoorrelation, and, as noted in Setion 2, Doppler bandwidth that inreases with the number of surfae enounters. 5. CONCLUSIONS Inthispaper, amodel for small-sale fadingof UWAhannels was developed. Miro-multipath omponents in a sattering field were modeled by omplex Gaussian multipliative oeffiients, whose orrelation properties in frequeny and time were assessed. Speifially, it was shown that an AR-1 Gaussian displaement of sattering points leads to an exponential auto-orrelation funtion of the small-sale fading oeffiients, while motion-indued random Doppler shifting, resulting from surfae waves or transmitter/reeiver drifting, was shown to lead to Bessel-type auto-orrelation funtions. Preliminary results from three UWA experiments showed a good math with the proposed model. Future work will fous on examining frequeny-orrelation properties of the small-sale fading. 6. ACKNOWLEDGMENTS This work was supported by the ONR MURI grant #N and the ONR grant#n REFERENCES [1] R. Galvin and R. Coats. A stohasti underwater aousti hannel model. In OCEANS 96. MTS/IEEE. Prospets for the 21st Century. Conferene Proeedings, volume 1, pages 23 21, Sep [2] W. Li and J. C. Preisig. Estimation of rapidly time-varying sparse hannels. IEEE J. Oean. Eng., 32(4): , Ot 27. [3] M. B. Porter. Bellhop ode, Available: [4] P. Qarabaqi and M. Stojanovi. Statistial modeling of a shallow water aousti ommuniation hannel. In Underwater Aousti Measurements Conferene. Nafplion, Greee, June 29. [5] P. Qarabaqi and M. Stojanovi. Modeling the large sale transmission loss in underwater aousti hannels. In the 49th Annual Allerton Conferene on Communiation, Control, and Computing, Sep 211. [6] A. Radosevi, J. Proakis,, and M. Stojanovi. Statistial haraterization and apaity of shallow water aousti hannels. In IEEE Oeans Europe Conferene, 29. [7] F. Soheleau, J. Passerieux, and C. Laot. Charaterisation of time-varying underwater aousti ommuniation hannel with appliation to hannel apaity. In Underwater Aousti Measurements Conferene. Nafplion, Greee, June 29. [8] B. Tomasi, P. Casari, L. Badia, and M. Zorzi. A study of inremental redundany hybrid ARQ over Markov hannel models derived from experimental data. In WUWNet. Woods Hole, USA, Sep 21. [9] W. B. Yang and T. C. Yang. High-frequeny hannel haraterization for M-ary frequeny-shift-keying underwater aousti ommuniations. J. Aoust. So. Am., 12(5): , Nov 26. [1] J. Zhang, J. Cross, and Y. R. Zheng. Statistial hannel modeling of wireless shallow water aousti ommuniations from experiment data. In Military Communiations Conferene, 21.

6 p... 2b p b3s3 p b4s delay [ms] (a) p 2b p b3s3 p b4s k= k= k= k= x 1 3 (b) x 1 3 () x 1 3 (d) (e) p 2b p b3s3 p b4s3 auto orrelation auto orrelation auto orrelation auto orrelation 5 5 (f) 5 5 (g) 5 5 (h) 5 5 (i) Figure 2: SPACE experiment: Time-evolution of the baseband impulse response (magnitude), histograms of signifiant path magnitudes, and time-orrelation funtions.

7 5 1 p p b b p bs x delay [ms] (a) p p b b k= (b) 8 6 k= x 1 3 () 6 k= x 1 3 (d) 5 4 k= x 1 3 (e) p p b b auto orrelation auto orrelation auto orrelation auto orrelation 5 5 (f) 5 5 (g) 5 5 (h) 5 5 (i) Figure 3: MACE experiment: Time-evolution of the baseband impulse response (magnitude), histograms of signifiant path magnitudes, and time-orrelation funtions.

8 p p b b p bs 2b p b2s x delay [ms] 1 (a) p b p bs k= k= k= k= x 1 3 (b) x 1 3 () x 1 3 (d) (e) p b p bs auto orrelation auto orrelation auto orrelation auto orrelation 5 5 (f) 5 5 (g) 5 5 (h) 5 5 (i) Figure 4: KAM experiment: Time-evolution of the baseband impulse response (magnitude), histograms of signifiant path magnitudes, and time-orrelation funtions.