small scale characterization of underwater acoustic channels
TRANSCRIPT
Small Scale Characterization of Underwater
Acoustic Channels
Parastoo QarabaqiNortheastern University
Milica StojanovicNortheastern University
ABSTRACT
Underwater acoustic (UWA) channel models provide a toolfor predicting the performance of communication systemsprior to system deployment and are thus essential for systemdesign. Small-scale modeling of the UWA channel targetsstatistical characterization of random effects such as scat-tering and motion-induced Doppler shifting. In this study,each transmission path is modeled as consisting of micro-multipath components that cumulatively result in a com-plex Gaussian multiplicative distortion, whose time- andfrequency-correlation are assessed analytically. Motion ofthe surface and transmitter/receiver displacements intro-duce additional variation whose temporal correlation is mod-eled via Bessel-type functions. Data obtained from threeunderwater experiments support the proposed model.
Categories and Subject Descriptors
C.2.1 [Computer-communication Networks]: NetworkArchitecture and Design—Wireless communication
General Terms
Theory
Keywords
underwater acoustic communications, channel modeling,small-scale fading
1. INTRODUCTIONUnderwater Acoustic (UWA) communication systems
have to be designed to operate in a variety of conditions thatdiffer from the nominal ones due to the changes in system ge-ometry and environmental parameters. Beam tracing tools,such as Bellhop [3], provide an accurate deterministic pic-ture of the UWA channel for a given geometry and signal fre-quency, but they do not take into account random variationof the channel. Numerous studies have been conducted tostochastically model the UWA channel, e.g. [1, 4–10]. These
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studies are usually based on the analyses of experimentalacoustic data collected in a particular location. Some au-thors find Ricean fading [4, 6, 7] or Rayleigh fading [1] toprovide a good match for their measurements, while othersfind log-normal distribution [5, 8] or the K-distribution [9,10], to be a better fit. The variety of the proposed statisti-cal models is due to experiment-specific properties, e.g. thedeployment site and the type of signals used for probing,as well as the time intervals during which the channel isobserved, i.e. small-scale versus large-scale phenomena.
We distinguish two types of channel variations: those thatare caused by displacements spanning many wavelengths,and those that are caused by displacements on the order ofone or a few wavelengths. The latter are referred to as thesmall-scale variations. Considering motion on the order of1 m/s, and frequencies on the order of 10 kHz, a wavelengthis traversed during a sub-second interval. Such short inter-vals of time incidentally correspond to typical communica-tion transactions (a packet or a frame of packets). Small-scale channel variations can thus be thought of as those vari-ations that occur over a communication transaction. Theyare to be distinguished from the variations that are causedby larger system displacements that span many wavelengthsand occur on correspondingly longer intervals of time.
This paper follows a previous study [5] in which a log-normal model was proposed for the large-scale variations ofthe acoustic loss. In the present work, small-scale phenom-ena such as micro-multipath formation due to scattering arethe target of analysis. These phenomena may cause addi-tional, possibly fast and comparatively large variations inthe instantaneous signal power.
The rest of the paper is organized as follows. In Sec. 2, atheoretical model is proposed for small-scale channel effects.Motion induced Doppler shifting is modeled in Sec. 3. Ex-perimental results that support the model are presented inSec. 4, and conclusions are summarized in Sec. 5.
2. SMALL-SCALE CHANNEL CHARACT-
ERIZATIONThe UWA channel is a multipath channel whose impulse
response can be represented as
h(τ ) =∑
p
hpq(τ − τp) (1)
where q(τ ) is the nominal impulse response of the referencepath, hp are the path gains, and τp are the path delays.These path gains and delays correspond to a particular large-scale system realization, i.e. a particular system geometry
and sound speed profile at a given time.A common way to model small-scale effects is to include
a multiplicative random factor into the overall path gain hp.While such a model is simple, it may be overly simplistic asit applies only to narrowband systems, while a more generalmodel calls for modifications to path filtering as well.
Apart from experimental measurements, purely theoreti-cal arguments may be invoked to model the scattering pro-cess. Specifically, let us begin by considering a narrowbandcase, in which the transmitted signal
s(t) = Re{u(t)ej2πft}has bandwidth ∆f � f . We will later extend this conceptto the wideband case by considering different frequencies fwithin a signal bandwidth B. Let us also focus on a singlepath, say path p. So far, we have modeled this path ashaving the gain hp and delay τp. However, if scatteringoccurs along this path, the arriving signal can be modeledinstead as
rp(t) = Q(f)∑
i≥0
hp,is(t− τp − δτp,i) = Re{vp(t)ej2πft}
where Q(f) = F {q(τ )}, vp(t) is the equivalent baseband re-ceived signal, and hp,i and δτp,i describe the intra-path gainsand relative delays of the micro-multipath components, re-spectively. Defining hp,i = hpγp,i, the equivalent basebandsignal is given by
vp(t) = Q(f)hpe−j2πfτp
∑
i≥0
γp,ie−j2πfδτp,iu(t− τp − δτp,i)
The delays δτp,i correspond to micro-multipath length dif-ferences that can be on the order of the signal wavelength,λ = c/f . Since ∆f � f , the signal u(t) will remain practi-cally insensitive to such delays, i.e. u(t− δτp,i) ≈ u(t). Wecan thus express the arriving signal as
vp(t) ≈ Q(f)hpe−j2πfτpγp(f)u(t− τp) (2)
where
γp(f) =∑
i≥0
γp,ie−j2πfδτp,i (3)
This expression implies that the effect of scattering in a nar-rowband system is that of a multiplicative distortion only.Collecting all the frequency components of a wideband sig-nal, and allowing for the possibility of time-varying pathcoefficients and delays, the overall channel transfer functionis given by
H(f) = Q(f)∑
p
hpγp(f)e−j2πfτp (4)
Note that although we have dropped the time from nota-tion, both the path gain hp and the distortion coefficient γpvary with time, so strictly speaking we should write hp(t)and γp(f, t) in the above expression. Allowing for the time-varying delays τp as well, we have a general model
H(f, t) = Q(f)∑
p
hp(t)γp(f, t)e−j2πfτp(t) (5)
The fundamental difference between the two multiplicativefactors is that hp(t) varies slowly while γp(f, t) may varyrapidly . Depending upon the system functions on whichone is focusing, the appropriate time-dependence should ex-plicitly be taken into account. For example, if one assumes
that large-scale power control is in place at the transmitter,then it suffices to focus on γp(t) in order to evaluate theaverage probability of bit error. Conversely, if one is focus-ing on large-scale power control, then the notion of averagepower is used to signify the fact that small-scale fading is notconsidered, i.e. averaging is performed over γp. Specifically,one can define the instantaneous power gain correspondingto the p-th path as
G̃p(f, t) = Q2(f)h2p(t)|γp(f, t)|2 (6)
and the average power gain as
Gp(f, t) = Eγ
{
G̃p(f, t)}
= Q2(f)h2p(t)E{γ2
p(f, t)} (7)
2.1 Probability density functionThe constituent terms of the distortion coefficient γp(f) in
expression (3) are treated as random variables, whose behav-ior is dictated by the amplitudes γp,i and delays δτp,i. Whilethe amplitudes and delays may exhibit little variation, thephases φp,i = −2πfδτp,i|mod2π
will exhibit significant vari-ation whenever the delays are on the order of a wavelength,causing in turn a significant variation in γp(f).
Assuming that the constituent terms are independent andidentically distributed, the Central Limit Theorem implies acomplex Gaussian distribution for γp(f). It is also possibleto consider a situation in which there is a component whosedelay is stable. In that case, the distortion is modeled by
γp(f) = γp,0 +∑
i≥1
γp,ie−j2πfδτp,i (8)
where γp,0 represents the relative coefficient of the stablepath (whose δτp,0 = 0). In general, the coefficient γp iscomplex Gaussian with mean γ̄p(f) and variance σ2
p(f) percomplex dimension, i.e. its magnitude |γp| is Ricean dis-tributed. The corresponding Ricean factor observed on thep’th path is Kp = γ̄p/2σ
2p. The second order statistics γ̄p
and σ2p will be influenced by the environmental conditions,
e.g. the bottom and surface roughness which are in turnrelated to wave activity.
The path statistics γ̄p(f) and σ2p(f) can be determined
experimentally, or analytically if the distribution of the con-stituent terms is known. To illustrate an analytical ap-proach, let us assume that the micro-path amplitudes arefixed at γp,0 and γp,i = µp, i ≥ 1, and that the delays δτp,iare zero-mean Gaussian, with variance σ2
δp .1 The mean and
variance are then obtained as
γ̄p(f) = γp,0 + µpSpρp(f)
2σ2p(f) = µ2
pSp[1− ρ2p(f)]
where Sp is the number of intra-paths and ρp(f) is defined
as ρp(f) = E{e−j2πfδτp,i} = e−(2πf)2σ2
δp/2. In this model,
we note that the second-order statistics are determined bythe function ρp(f), whose value depends on the parameterfσδp , i.e. the standard deviation of the micro-path delaynormalized by the signal wavelength.
Note that regardless of the distribution of δτp,i, for highfrequencies, such that fσδp � 1, the model reduces to thestandard one in which the phases φp,i are uniformly dis-tributed.
1Gaussian-distributed intra-path delays correspond toGaussian-distributed displacements of scattering points.
0 1 2 3 4 50.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
∆f [kHz]
ρp(∆
f)
σδ p
=0.02ms
σδ p
=0.05ms
Figure 1: Frequency correlation term ρp(∆f) for dif-ferent standard deviations of the micro-path delays.
2.2 Correlation in the frequency domainFrequency-correlation of the small-scale path coefficients
is described by the function E{
γp(f +∆f, t)γ∗q (f, t)
}
. As-suming that the scattering process is independent betweendifferent paths, i.e. that reflection points of different pathsare sufficiently far apart, we have that
E{γp(f +∆f)γ∗q (f)} = γ̄p(f +∆f)γ̄q(f)
+ δp,qE{[γp(f +∆f)− γ̄p(f +∆f)][γq(f)− γ̄q(f)]∗}
Note that although the paths exhibit uncorrelated scatter-ing, the above function is not zero in general, due to thenon-zero mean values.
In order to evaluate the frequency-correlation, the distri-bution of the intra-path delays δτp,i has to be known. Forthe Gaussian distributed delays with zero mean and varianceσ2δp , frequency-correlation is obtained as
E {[γp(f +∆f)− γ̄p(f +∆f)][γp(f)− γ̄p(f)]∗}
= µ2pSpρp(∆f)[1− ρp(
√
2f(f +∆f))]
≈ µ2pSpρp(∆f) (9)
where the approximation holds well for σδp > 1/√2max
{f, f +∆f}. Fig. 1 illustrates the function ρp(∆f). Wenote that depending upon the standard deviation of the pathdelays, there may be more or less correlation between thesmall-scale coefficients γp within the signal bandwidth. Inour analysis of experimental data in Sec. 4, we will assumethat full correlation exists across the signal bandwidth, i.e.that a single value γp suffices to describe small-scale effects.However, we note that this assumption may not hold always,and, hence, our results should be regarded as preliminary.Note also that while the particular function of Fig. 1 pertainsto Gaussian distributed delays, one can expect a similar con-clusion to hold for a different distribution as well.
2.3 Correlation in the time domainTime-correlation of the scattering coefficients is described
by E{γp(f, t +∆t)γ∗p(t)}. This function captures the effect
of motion within the scattering field which influences thecoefficients γp(f, t) through the time-varying micro-path de-lays. To assess the time correlation function, power spectral
density of δτp,i(t) has to be known. Without loss of gen-erality, let us assume that the Gaussian-distributed delaysδτp,i(t) obey a first-order auto-regressive process (AR-1):
δτp,i(t+∆t) = αδpδτp,i(t) +wδp,i(t)
where wδp,i(t) ∼ N (0, σ2δp(1 − α2
δp)), αδp = e−πBδp∆t andBδp is the 3 dB width of the psd of δτp,i(t). The time-correlation function is obtained as
E{[γp(f, t+∆t)− γ̄p(f)][γp(f, t)− γ̄p(f)]∗}
= µ2pSpe
−(1−αδp)(2πf)2σ2
δp [1− e−αδp
(2πf)2σ2
δp ]
≈ 2σ2p(f)e
−πBp(f)∆t (10)
where we have defined
Bp(f) = (2πfσδp)2Bδp
and the approximation holds under the assumption that∆t � 1/Bδp. Note that the expression (10) implies thatγp(f, t) can equivalently be modeled as a Gauss–MarkovAR-1 process.
3.MOTION-INDUCED DOPPLER SHIFTINGMotion of the transmitter/receiver or any reflection points
in the channel leads to time-varying path delays τp. Focus-ing on the small-scale phenomena, we are interested in thevariations that occur over short intervals of time (e.g. sub-second intervals). During such intervals, it is reasonable toassume that any motion occurs at a constant velocity, i.e.that it is only the velocity and not acceleration that mat-ters. When that is the case, the path delays are modeled asτp(t) = τp − apt where ap = vp/c is the Doppler factor cor-responding to velocity vp.
2 Note that the Doppler factorsap may vary over longer periods of time and a more generalquasi-stationary model can be introduced to address thisfact. We will keep this fact in mind, but drop time indexingfor simplicity.
At least three types of motion influence the Doppler fac-tor, (i) unintentional transmitter/receiver motion (drifting,which gives rise to a Doppler scaling factor adp), (ii) inten-tional transmitter/receiver motion (vehicular motion, whichgives rise to avp) and (iii) waves (surface motion, which givesrise to asp). If the Doppler factors corresponding to each ofthese types of motion are fixed, the corresponding Dopplershift is always the same. On the contrary, if the Dopplerfactors change randomly from one realization to another, sodo the shifts. The resulting effect is that of random Dopplershifting or Doppler spreading.
To characterize Doppler spreading, we focus on the auto-correlation function of the random process ej2πapft, i.e.E{ej2πapf∆t}. Assuming independence between various fac-tors contributing to motion, we have that
Rp(∆t) = E{ej2πfadp∆t}E{ej2πfavp∆t}E{ej2πfasp∆t}
Should any one of these components be regarded as deter-ministic, its expectation is dropped.
To characterize the drifting component, we assume thatthe transmitter and receiver drift at velocities vtd, vrd indirections θtd, θrd with respect to horizontal pointed toward
2It is also possible to account for a path-dependent propa-gation speed cp.
each other. The relative speed, projected onto the p-th path,is
vdp = vtdcos(θp − θtd)− vrdcos(θp + θrd)
and the corresponding Doppler factor is adp = vdp/c. Ifthe transmitter/receiver drift in random directions, thenE{ej2πfadp∆t} involves averaging over θtd, θrd. Assumingthat drifting is equally likely in any direction, and that itoccurs independently for the transmitter and receiver,
E{ej2πfadp∆t} = J0(2πvtdcf∆t)J0(2π
vrdcf∆t) (11)
where J0(.) is the Bessel function of the first kind and orderzero. Assuming vtd = vrd = vd, and ad = vd/c, the aboveexpression reduces to J2
0 (2πadf∆t).Vehicular component of the Doppler effect is obtained sim-
ilarly, except that those components of motion that can beestimated and compensated by synchronization are not tobe regarded as part of the channel distortion. Hence, we re-gard avp as the residual Doppler factor after initial synchro-nization. For example, if synchronization compensates forthe pre-dominant Doppler factor corresponding to the pro-jection of the transmitter/receiver intentional velocity ontothe reference path p=0, the effective Doppler factor is
avp =1
c
[
[vtvcos(θp − θtv) − vrvcos(θp + θrv)]
− [vtvcos(θ0 − θtv)− vrvcos(θ0 + θrv)]
]
=1
c
[
vtv[−2 sin(θp + θ0 − 2θtv
2
)
sin(θp − θ0
2
)
]
+ vrv[2 sin(θp + θ0 + 2θrv
2
)
sin( θp − θ0
2
)
]
]
If we assume the transmitter/receiver motion to be equallylikely in any direction θtv/rv, the auto-correlation functioncorresponding to vehicular motion is
E{ej2πfavp∆t} =J0(2π2vtv sin((θp − θ0)/2)
cf∆t)
×J0(2π 2vrv sin((θp − θ0)/2)
cf∆t) (12)
Finally, to assess the surface component, let us focus onwaves that cause a point on the surface to move up and downcreating a displacement that varies sinusoidally in time, withamplitude Aw and frequency ωw/2π. A signal impingingupon the j-th reflection point along the p-th path catches itin a random phase, i.e. at vertical velocity vw sin(ψp,j+ωwt)where ψp,j ∼ U [−π, π], and vw = Awωw. Projections ofthis velocity onto the p-th path, summed over all surfacereflection points, yield
vsp = 2vw sin θp
Ks(p)∑
j=1
sin(ψp,j + ωwt)
where Ks(p) is the number of surface reflections along thep-th path. Assuming that reflection points are sufficientlyfar apart such that ψp,j are independent, time-correlation isobtained by taking the expectation over asp = vsp/c withthe angles ψp,j uniformly distributed over 2π. The result is
E{e−j2πfasp∆t} = [J0(2πawpf∆t)]Ks(p) (13)
where awp = 2vw sin θp/c. It is worth noting that the mainlobe of this Bessel-like (or sinc-like) function, whose behavior
Table 1: Nominal parameters of the experimentalchannels.
BW[kHz] d[km] hw[m] htx[m] hrx[m]
SPACE 8–17 1 10 4 2
MACE 10.5–15.5 0.5–4 100 45 60
KAM 8.5–17.5 2.9 103 58 59
indicates the coherence time and is dictated by awp, narrowswith each additional surface encounter.
Putting together Eqs. (10)–(13), the complete auto-correlation function of the overall scattering coefficientγ̃p(f, t) = γp(f, t)e
2πapft is obtained as
Rγ̃p(∆t) =[
γ̄2p(f) + 2σ2
p(f)e−πBp(f)∆t
]
J20 (2πadf∆t)
×J0(2π 2vtv sin((θp − θ0)/2)
cf∆t)
×J0(2π 2vrv sin((θp − θ0)/2)
cf∆t)
×[
J0(2π2vw sin θp
cf∆t)
]Ks(p)
(14)
This function exhibits an overall Bessel-like behavior, damp-ened by the exponentially decaying correlation of the γp co-efficient. As such, it is helpful in explaining experimentalmeasurements, which exhibit similar behavior, and whichwe present in the next section.
4. EXPERIMENTAL RESULTSWe present experimental data collected during three un-
derwater experiments. The transmitted signals were PN se-quences, BPSK modulated, and transmitted repeatedly overthe course of one minute.
The first experiment, called the Surface Processes Acous-tic Communication Experiment (SPACE) was conductedsouth of the coast of Martha’s Vineyard in Massachusetts,in the fall of 2008. The carrier frequency was 13 kHz and thetransmission rate was 6.5 kbps. The water depth was 10 m,the transmitter and receiver were deployed at 4 m and 2 mabove the sea-floor respectively, and separated by 1000 m.
The second experiment, Mobile Acoustic CommunicationExperiment (MACE), was conducted in the Atlantic Oceanabout 100 miles south of Cape Cod in July 2010. The re-ceiver was suspended at the depth of 40 m and the trans-mitter was towed at the depth of 50–60 m. The water depthwas approximately 100 m and the distance varied between500 m and 4 km. The signals were transmitted continuouslyat 5 kbps.
Finally, the third experiment, known as the KauaiAcomms MURI (KAM), was conducted in July 2011 off thecoast of Kauai Island, Hawaii. The transmitter and the re-ceiver were deployed at mid-water column and were 3 kmapart in approximately 100 m deep water. The signals weremodulated onto the carrier of 13 kHz and transmitted at therate of 6.5 kbps. Table 1 shows the nominal parameters ofthe experimental channel and the signal frequency range fordifferent experiments.
The following approach was taken to prepare the signalsfor analysis. First, the received signal was resampled basedon the received packet length to compensate for the motion-induced time scaling and frequency shifting which can be
modeled by a rough Doppler factor. The resampling wasnotably necessary for the MACE data, where the transmit-ter moved at about 1 m/s creating a Doppler rate on theorder of 10−3. Fine Doppler compensation was then carriedout using a combined RLS estimator and second order PhaseLocked Loop. Once these steps were completed, the Orthog-onal Matching Pursuit algorithm was used for channel esti-mation. This algorithm is commonly used for estimation ofUWA channels (see for example [2]) as it outperforms con-ventional least-squares methods when the channel is sparse.The resulting estimate of the baseband channel response wasthen used to extract the path gains.
Figs. 2–4 show the results obtained for the three exper-iments. Each figure shows an ensemble of estimated base-band impulse responses (magnitude) over the duration of1 minute at resolution of one bit. Several different localmaxima over the delay axis are highlighted in the figures.The local maxima indicate channel taps over which the im-pulse response is the strongest, and they correspond to thedelays at which physical paths occur. Physical path delayscorresponding to the nominal channel geometry are markedby arrows and labeled as s, b, sb, etc. referring to surface,bottom, surface-bottom, etc. reflections.
The observed path delays deviate from the nominal be-cause of location uncertainty and small-scale motion. Also,each local maximum may represent more than one physi-cal path if the path delays are too close to be distinguishedgiven the delay resolution (finite bandwidth) of the system.Finally, the time variability of the channel may cause eachpath to (i) spread over several adjacent taps, and (ii) encom-pass a slowly varying mean. Path spreading, which is a con-sequence of both the micro-path dispersion and bandwidthlimitation, may occur uniformly at a tractable rate, e.g. thepath labeled p0 in the KAM experiment, or an intractableone, e.g. the path labeled pbs. In order to take into accountthe effect of all contributing taps, whenever the phase in-formation is not needed, several taps can be combined in aroot mean square fashion to form the absolute value of eachpath gain. Slow variation of the mean is observed if the one-minute window is longer than the stationarity interval of theprocess hp(t). For example, such slow variation is evidentin the path ps of the MACE experiment. In this case, inorder to extract the absolute value of the small-scale factorγp(t), the slowly varying average of the process |hp(t)γp(t)|is removed. When the phase information is not to be ne-glected, the signal statistics are estimated over shorter timeintervals, i.e. a fraction of a minute, and ensemble averaged.
In addition to the time-evolution of the channel response,shown in the figures are also the histograms of the estimated|γp(t)| for several different paths, along with a theoreticalRicean curve. The conditional Ricean distribution (condi-tioned on the slowly-varying mean) appears to provide agood fit. In all three experiments, the Ricean K-factor cor-responding to the direct path (p=0) is greater than that ofthe other paths, indicating a stable arrival.
Finally, the time domain auto-correlation functions of thepath coefficients are plotted. For all three experiments, theauto-correlations corresponding to the direct path show lessfluctuation and a higher coherence time. All other pathsshow Bessel-type autocorrelation, and, as noted in Section 2,Doppler bandwidth that increases with the number of sur-face encounters.
5. CONCLUSIONSIn this paper, a model for small-scale fading of UWA chan-
nels was developed. Micro-multipath components in a scat-tering field were modeled by complex Gaussian multiplica-tive coefficients, whose correlation properties in frequencyand time were assessed. Specifically, it was shown that anAR-1 Gaussian displacement of scattering points leads toan exponential auto-correlation function of the small-scalefading coefficients, while motion-induced random Dopplershifting, resulting from surface waves or transmitter/receiverdrifting, was shown to lead to Bessel-type auto-correlationfunctions. Preliminary results from three UWA experimentsshowed a good match with the proposed model. Future workwill focus on examining frequency-correlation properties ofthe small-scale fading.
6. ACKNOWLEDGMENTSThis work was supported by the ONR MURI grant
#N00014-07-1-0738 and the ONR grant #N00014-09-1-0700.
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delay [ms]
tim
e [s]
−3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
10
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0
0.002
0.004
0.006
0.008
0.01
0.012
ps2b
pb3s3
pb4s3
p0... ... ...
(a)
6 8 10 12
x 10−3
0
200
400
600
800
1000
1200
k=304.8148
path gain
p0
(b)
1 2 3 4 5 6 7
x 10−3
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800
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k=63.5124
path gain
ps2b
(c)
2 4 6 8 10 12 14
x 10−3
0
100
200
300
400
500
k=34.266
path gain
pb3s3
(d)
0 0.002 0.004 0.006 0.008 0.010
100
200
300
400
500
k=10.1009
path gain
pb4s3
(e)
−5 0 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
time [s]
auto
−corr
ela
tion
p0
(f)
−5 0 5−0.4
−0.2
0
0.2
0.4
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time [s]
auto
−corr
ela
tion
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(g)
−5 0 5−0.4
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time [s]
auto
−corr
ela
tion
pb3s3
(h)
−5 0 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
time [s]
auto
−corr
ela
tion
pb4s3
(i)
Figure 2: SPACE experiment: Time-evolution of the baseband impulse response (magnitude), histograms ofsignificant path magnitudes, and time-correlation functions.
delay [ms]
tim
e [s]
−4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
0
5
10
15
20
25
30
35
40
45
50
0
2
4
6
8
10
12x 10
−3
p0
ps
pb
psb
pbs
(a)
0.006 0.008 0.01 0.012 0.014 0.0160
100
200
300
400
500
600
700
k=128.1378
path gain
p0
(b)
2 4 6 8
x 10−3
0
200
400
600
800
1000
k=74.5349
path gain
ps
(c)
4 6 8 10
x 10−3
0
200
400
600
800
k=93.2281
path gain
pb
(d)
0 2 4 6 8
x 10−3
0
100
200
300
400
500
600
k=16.236
path gain
psb
(e)
−5 0 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
time [s]
auto
−corr
ela
tion
p0
(f)
−5 0 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
time [s]
auto
−corr
ela
tion
ps
(g)
−5 0 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
time [s]
auto
−corr
ela
tion
pb
(h)
−5 0 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
time [s]
auto
−corr
ela
tion
psb
(i)
Figure 3: MACE experiment: Time-evolution of the baseband impulse response (magnitude), histograms ofsignificant path magnitudes, and time-correlation functions.
delay [ms]
tim
e [s]
−3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
0
5
10
15
20
25
30
35
40
45
50
0
1
2
3
4
5
6
7
x 10−3
p0
ps
pb
pbs
psb
ps2b
pb2s
(a)
5 6 7 8
x 10−3
0
500
1000
1500
k=359.98
path gain
p0
(b)
2 3 4 5 6 7
x 10−3
0
200
400
600
800
1000
k=68.0447
path gain
ps
(c)
0 2 4 6
x 10−3
0
200
400
600
800
1000
1200
k=33.7406
path gain
psb
(d)
0 0.002 0.004 0.006 0.008 0.010
100
200
300
400
500
k=18.5864
path gain
pbs
(e)
−5 0 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
time [s]
auto
−corr
ela
tion
p0
(f)
−5 0 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
time [s]
auto
−corr
ela
tion
ps
(g)
−5 0 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
time [s]
auto
−corr
ela
tion
psb
(h)
−5 0 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
time [s]
auto
−corr
ela
tion
pbs
(i)
Figure 4: KAM experiment: Time-evolution of the baseband impulse response (magnitude), histograms ofsignificant path magnitudes, and time-correlation functions.