Advances in Radio Science (2004) 2: 117–126© Copernicus GmbH 2004 Advances in

Radio Science

Geometry-based channel modelling of MIMO channels incomparison with channel sounder measurements

G. Del Galdo, M. Haardt, and C. Schneider

Communications Research Laboratory, Ilmenau University of Technology, P.O. Box 100565, 98684 Ilmenau, Germany

Abstract. In this paper we propose a flexible geometry-based propagation model for wireless communications de-veloped at Ilmenau University of Technology. The IlmPropcomprises a geometrical representation of the environmentsurrounding the experiment and a precise representation ofthe transmitting and receiving antennas. The IlmProp iscapable of simulating Multi-User MIMO scenarios and in-cludes a complete collection of tools to analyze the syntheticchannels. In order to assess the potentials as well as the limitsof our channel simulator we reconstruct the scenario encoun-tered in a recent measurement campaign at Ilmenau Univer-sity of Technology leading to synthetic data sets similar tothe ones actually measured. The measurements have beencollected with the RUSK MIMO multi-dimensional channelsounder. From the comparisons of the two channel matri-ces it is possible to derive useful information to improve themodel itself and to better understand the physical origins ofsmall-scale fading. In particular the effects of the differentparameters on the synthetic channel have been studied in or-der to assess the sensibility of the model. This analysis showsthat the correct positioning of a small number of scatterersis enough to achieve frequency selectiveness as well as spe-cific traits of the channel statistics. The size of the scatteringclusters, the number of scatterers per cluster, and the RicianK-factor can be modified in order to tune the channel statis-tics at will. To obtain higher levels of time variance, movingscatterers or time dependent reflection coefficients must beintroduced.

1 Introduction

The high complexity of the propagation phenomenon sets avast challenge in the effort of modelling the wireless channel.The most accurate approach would be to solve the Maxwellequations. However the computational effort required wouldbe prohibitive, especially if we considered multiple antennas

Correspondence to:G. Del Galdo(giovanni.delgaldo@tu-ilmenau.de)

at both ends of the link, i.e. a MIMO system (Multiple InputMultiple Output). In fact, ifMR andMT are the number ofantennas at the receiver and at the transmitter, respectively,the number of channels to be calculated grows toMR ·MT .Furthermore a precise physical and geometrical descriptionof the objects would be necessary.

In order to reduce the complexity of the model the socalled GTD (Geometric Theory of Diffraction) can be em-ployed (Correia, 2001). This approach postulates the exis-tence of direct, diffracted and reflected rays only. As thewavelengthλ approaches zero, this approximation becomesincreasingly accurate.

With a ray-tracing engine and a full 3D description of theenvironment it is then possible to calculate all possible rayslinking the antennas. However with a GTD approach a com-plete geometrical representation of the scenario is still re-quired. A further approximation is to simplify the geometrysurrounding the antennas with a discrete number of reflec-tors, calledscatterers; when only direct and reflected raysare considered the computational complexity drops signifi-cantly. In this paper we present the IlmProp, a simplifiedmodel which generates channels which mimic most charac-teristics of measured channels. To this aim it is extremely im-portant to determine which features of the synthetic channelare affected by the different parameters present in the model.A relatively low computational complexity allows us to gen-erate channels with all possible combinations of values forthe parameters. Then we analyze the data by finding strongtraits which link a particular parameter to a particular feature.At the same time we are able to assess whether a parameterhas no effect on a specific feature.

This extensive search and its results are reported in Sect. 3.Then we modeled the scenario found in one of the measure-ment campaigns undertaken at Ilmenau University of Tech-nology. Section 4 reports the direct comparison between thesynthetic and measured channels which led to a better under-standing on the origins of the fast and slow fading processes.Section 5 deals in particular with the comparison of the phaseresponses. Lastly, Sect. 6 concludes this paper.

118 G. Del Galdo et al.: Geometry-based channel modelling

Fig. 1. In this example 2 Mobiles (M’s) surrounded by scatterersmove in the proximity of one fixed Base Station (BS).

2 The IlmProp: a geometry-based channel model

The IlmProp developed at TU Ilmenau relies entirely on a3D geometric representation and has been designed to sim-ulate a Multi-User-MIMO scenario. It has been simplifiedto single-bounce reflections in order to reduce the compu-tational complexity. Clusters of scatterers can be arbitrarilypositioned in the 3D reconstruction of the environment. Eachscatterer corresponds to a single ray and is characterized by acomplex coefficient which, for simplicity, is independent ofthe Direction of Arrival (DoA) and the Direction of Depar-ture (DoD); but may vary in time with an arbitrary law. Thecoefficient determines the phase shift and power attenuationintroduced by the scatterer.

The receiver thus receives a number of rays equal to thenumber of scatterers plus the Line Of Sight (LOS), whichlinks the transmitter to the receiver directly. Figure 1 showsa simple example of this modelling approach. Such a sim-plified model still requires a fairly high computational com-plexity because the total number of rays to be computed is:MR ·MT ·Nscatt .

The advantage of a geometry-based model with respect toa stochastic one is indeed the possibility to easily generatea frequency selective time variant channel. In fact the fre-quency selectiveness is caused by the multi-path effect whilethe time variance is set off by the movement of the antennasand of the scatterers as well as the varying scattering coef-ficients with respect to time. In other words it well modelsa typical frequency selective time variant radio channel. Inorder to characterize such a channel we need to describe it inone of the four possible two-dimensional domains named theBello domains. (Bello, 1963). Either{t, f }, {t, τ }, {fD, f }

or {fD, τ } wheret , fD, f , andτ denote time, Doppler fre-quency, frequency, and delay time, respectively. The mostconvenient domain in which to compute the channel is cer-tainly {t, f }. Having a geometrical representation of all ob-jects in time makes the choice oft over fD quite obvious.The choice off over τ requires more explanation. Thanksto the geometrical representation it is simple to calculate thepath lengths for each ray present in the scenario. Assumingthat the velocity of light,c, is equal to the velocity of all rays,we can calculate the delay time associated with every ray, i.e.the time needed to reach the receiver. The delay times foundwill not necessarily match the sampling grid chosen. For this

Fig. 2. The geometrical representation of the modelMα . Twoscattering clusters and a strong LOS are visible. The transmitter (atthe top) moves on a linear path.

Fig. 3. The geometrical representation of the modelMβ . Themodel is characterized by three scattering clusters; one of them sur-rounds the transmitter.

reason more processing is required. On the contrary, in thefrequency domain we can calculate the channel response forthe specific frequency bins that we are interested in. Thechoice of a specific Bello domain is at the end a matter ofconvenience since it is possible to derive the other represen-tations via Fourier transforms. For instance, havingH(f, t),i.e. the channel expressed in delay time and time, we can cal-culateh(τ, t), the channel in frequency and time, as:

h(τ, t) =

∫H(f, t)ej2πf τ df . (1)

The amplitude and phase of each ray are calculated sepa-rately at the receiver. The phase of thei-th ray at a certaintime snapshotto and frequencyfo are calculated as follows:

ϕi(to, fo) = 6 (e−j2π foc Li )+ θi , (2)

whereLi is the ray’s path-length andθi is the phase shiftintroduced by the scatterer. The amplitude depends simplyon the scattering coefficient, antenna gain, and path-loss (VanTrees, 2002).

3 An exhaustive investigation

The first step in understanding how sensible the model is tothe different parameters and which features of the syntheticchannel are affected, is to model some typical scenarios andto generate channels with all possible values for the param-eters. We implemented two models, to which we will referasMα andMβ , seen in Figs. 2 and 3.Mα models a typi-cal scenario in which the areas around the transmitter and thereceiver are free of scatterers while some are present in the

G. Del Galdo et al.: Geometry-based channel modelling 119

middle of the link.Mβ , on the other hand, models a scatter-ing cluster in the proximity of the transmitter. We generatedeach channel for the following sampling grid:

– A bandwidthB of 400 MHz at a center frequency of 1GHz sampled with a1f = 1.3 MHz for a total of 301samples.

– A total of 200 time snapshots were taken, one every 10ms for a total duration of 2 s.

The choices of these parameters determine the samplinggrids in the coupled domains. In fact if the complex base-band signal is sampled in the range[−

B2 ,

B2 ], then1τ , the

resolution in delay time, is equal to1B

. In the same way thebandwidth in the Doppler frequency, 2· fD,max can be cal-culated from the resolution in time1t as:

fD,max =1

2 ·1t. (3)

– The resolution in delay time is1τ = 2.5 ns, whichcorresponds to a distance of less than a meter (75 cm).We can successfully resolve paths (i.e. without alias-ing), which are up to 220 meters long.

– Doppler shifts up tofD,max = ±50 Hz are visible with-out aliasing.

The transmitter moves with a uniform velocity of about 5m/s. Both transmitter and receiver employ a ULA (UniformLinear Array) withMR = MT = 4 antennas spaced byλ2 .

After several experiments with different geometries anddifferent configurations we narrowed down our extensivesearch to three dominant parameters:

– Cluster size

– Number of scatterers per cluster

– RicianK-factor

A fourth dominant parameter is certainly the modelling ofthe reflection coefficients, and more importantly their trendin time. This leaves, however, too many degrees of freedomsince we could implement any arbitrary function to describeit in time. For this reason we left this important parameterto future and more specific investigations. In our simulationsthe reflection coefficients are kept constant along all dimen-sions, i.e. frequency and time. The RicianK-factor is definedas

K =PLOS

Pscatt, (4)

i.e. the power attributed to the Line Of Sight divided by thepower of the scatterers. Clearly it is a function of the path-losses which on their side are determined by the geometry,the transmit power, and the reflection coefficients. We scalethe amplitudes of the reflection coefficients to obtain a de-siredK-factor.

We then generate many channels with different values forthese three parameters and then analyze the data to find anystrong dependency.

3.1 K-factor

TheK-factor affects several features of the channel. Firstlyit influences its variance with respect to all four domains. Itis thus very interesting to assess this information quantita-tively computing the RMS delay spreadτRMS, the coherencebandwidth(1f )c, the RMS Doppler spreadfD,RMS and thecoherence time(1t)c.

The RMS delay spread of the channel,τRMS, is derivedfrom the multipath intensity profileψDe(τ ) which representsthe average power of the channel output as a function of de-lay time (Paulraj et al., 2003; Stuber, 1996).

τRMS =

√√√√∫ τmax0 (τ − τ )2ψDe(τ )dτ∫ τmax

0 ψDe(τ )dτ. (5)

τmax is the maximum delay spread whileτ is the averagedelay spread given by

τ =

∫ τmax0 τψDe(τ )dτ∫ τmax0 ψDe(τ )dτ

. (6)

The coherence bandwidth(1f )c corresponds to the fre-quency lag in which the channel’s autocorrelation functionreduces to 0.7 (Paulraj et al., 2003). As previously described,the channel representations in the four domains are coupledthrough Fourier transforms. For this reasons(1f )c andτRMSare connected as well. In fact the coherence bandwidth is in-versely proportional to the RMS delay spread so that

(1f )c =const1

τRMS. (7)

The same concepts are applied on the{fD, t} pair. The RMSDoppler spread,fD,RMS, is derived in a similar way fromψDo, the Doppler power spectrum, which corresponds to theaverage power as a function of Doppler frequencyfD.

fD,RMS =

√∫F (fD − fD)

2ψDo(fD)dfD∫F ψDo(fD)dfD

fD =

∫F fDψDo(fD)dfD∫F ψDo(fD)dfD

(8)

The integration domain,F , corresponds to the Doppler fre-quency interval where the Doppler power spectrum is nonzero. The coherence time(1t)c is calculated in the sameway as the coherence bandwidth. It corresponds to the timelag at which the autocorrelation function drops to 0.7 (Paulrajet al., 2003).(1t)c indicates how time variant the channel isand similarly to the{f, τ } pair

(1t)c =const2

fD,RMS(9)

Figure 4 shows typical trends forτRMS, fD,RMS, (1f )c, and(1t)c for differentK-factors. Similar plots can be observedfor both modelsMα andMβ for almost any cluster size andnumber of scatterers. AsK grows τRMS decreases, mean-ing that the channel becomes rapidly less frequency selec-tive. The same observation can be derived from the trend of(1f )c. fD,RMS and(1t)c show the same trends. For biggerK’s the channel becomes increasingly less time variant. The

120 G. Del Galdo et al.: Geometry-based channel modelling

0 10 20 30 400.5

1

1.5

2

2.5x 10

−8

K factor

τ RM

S

0 10 20 30 402

4

6

8

K factor

f D,R

MS

0 10 20 30 400.02

0.04

0.06

0.08

0.1

K factor

(∆t)

c

0 10 20 30 402

3

4

5x 10

7

K factor

(∆f)

c

0 10 20 30 400.1

0.2

0.3

0.4

0.5

K factor

con

st1 =

(∆t

) c ⋅ f D

,RM

S

0 10 20 30 400.2

0.4

0.6

0.8

1

K factor

con

st2 =

(∆f

) c ⋅ τ R

MS

Fig. 4. τRMS, fD,RMS, (1f )c and(1t)c plotted for differentK-factors.

0.8 0.9 1 1.1 1.2

x 109

0.2

0.3

0.4

0.5

0.6

0.7

0.8

K factor=200.8 0.9 1 1.1 1.2

x 109

0.2

0.3

0.4

0.5

0.6

0.7

0.8

K factor=10

Coe

ffici

ent o

f Var

iatio

n

0.8 0.9 1 1.1 1.2

x 109

0.2

0.3

0.4

0.5

0.6

0.7

0.8

K factor=50.8 0.9 1 1.1 1.2

x 109

0.2

0.3

0.4

0.5

0.6

0.7

0.8

K factor=0.5

Coe

ffici

ent o

f Var

iatio

n

Fig. 5. The coefficient of variationξ plotted against frequencyffor differentK-factors.

two constantsconst1 andconst2 are plotted as well to showthat the inverse proportionality expressed in Eqs. (7) and (9)is a good approximation.

Another interesting measure of the variation of the chan-nel is to look at the fluctuation of‖H‖

2F . A good index is

the so calledcoefficient of variation(Nabar, 2003) which iscommonly used as a quantitative measure of the spread (fluc-tuation) of a random variable. The coefficient of variationξis defined as:

ξ =

√E{(‖H‖

2F)2} − (E{‖H‖

2F

})2

E{‖H‖2F

}. (10)

We calculatedξ for all frequency bins to discover that forgrowingK’s the coefficient of variation becomes increasinglysmaller (Fig. 5). This is actually expected if we associate

−140 −130 −120 −110 −1000

10

20

30

40

50

60

K factor=15−140 −130 −120 −110 −100

0

10

20

30

40

50

60

K factor=5

−140 −130 −120 −110 −1000

10

20

30

40

50

60

K factor=2.5−140 −130 −120 −110 −100

0

10

20

30

40

50

60

K factor=1

firstsecondthirdfourth

Fig. 6. Eigenvalue distributions in dB asK grows forMα .

ξ with the effective diversity order, as described in (Nabar,2003).

In fact the effective diversity order,Ndiv, can be written as:

Ndiv =1

ξ2, (11)

and it approaches infinity for a channel with only the LOScomponent (AWGN channel) whileNdiv = MR · MT forRayleigh channels. The trend inξ and consequently inNdiv

can be observed for both models proposed and independentlyfrom other parameters.

Another interesting analysis can be performed on thestatistics of the eigenvalues. Figure 6 shows their distribu-tions for differentK-factors for the modelMα. Eigenvaluedecompositions have been applied on the spatial correlationmatrices at the receiverRR calculated for each time snap-shot. Then we generated the histograms to derive the am-plitudes distribution on a dB scale. The spatial correlationmatrixRR at the receiver for the specific timeto is:

RR(to) =1

F

F∑i=1

H (:, :, i, to) · H (:, :, i, to)H , (12)

whereF is the total number of frequency bins,HH is theHermitian transpose ofH and the channelH has dimensionsMR×MT ×F ×T . The MATLAB like notationH (:, :, i, to)

denotes the two-dimensional channel matrixH ∈ CMR×MT

for a specific frequencyf = i and timet = to. Figure 6shows the distributions of the eigenvalues plotted in dB. Thechannels have been generated without noise and without anynormalization, i.e. the absolute amplitudes of the eigenval-ues appear very low (around−120 dB). This has no physicalmeaning and only the relative distances between the eigen-values should be taken into account. As theK-factor grows,the Line Of Sight becomes increasingly stronger and thechannel displays low rank traits. In other words the strongesteigenvalue separates itself from the weaker ones asK in-creases.

G. Del Galdo et al.: Geometry-based channel modelling 121

0 2 4

x 10−6

0

500

1000

1500

2000

2500

3000

3500

K factor=150 2 4

x 10−6

0

500

1000

1500

2000

2500

3000

3500

K factor=50 2 4

x 10−6

0

500

1000

1500

2000

2500

3000

3500

K factor=3

0 2 4

x 10−6

0

500

1000

1500

2000

2500

3000

3500

K factor=20 2 4

x 10−6

0

500

1000

1500

2000

2500

3000

3500

K factor=10 2 4

x 10−6

0

500

1000

1500

2000

2500

3000

3500

K factor=0

Fig. 7. Amplitude distributions of channels considering all frequen-cies and time snapshots for different RicianK-factors forMα .

A higherK-factor inhibits the effects of the fast-fading.This can be easily seen by plotting the distributions of theamplitudes of the channel considering all frequencies andtime snapshots (Fig. 7).

3.2 Cluster size and number of scatterers

While there exists a strong relationship between the differentchannel statistics and the RicianK-factor, there is no strictrelation with the parameters which characterize a scatteringcluster, namely the number of scatterers, the cluster size andthe type of distribution of the scatterers inside the cluster. Inthe first simulations we investigated whether the geometricaldistribution affected the channel. We generated clusters withscatterers distributed uniformly on a sphere, on a circle onthe horizontal plane (as seen in Figs. 2 and 3), and on thesurface of a cylinder or a sphere. All channels showed thesame trends and for this reason we continued with more de-tailed investigations fixing the distribution of the scattererson a circle. We generated models characterized by clustersizes with diameters ranging fromλ2 to 20λ and with num-bers of scatterers ranging from 1 up to 100 reflection pointsper cluster.

We then investigated trends in the different channel fea-tures fixing all parameters as either the cluster size or thenumber of scatterers changed. All features but the effectivediversity order did not show any strong dependency eventhough they were all affected. For instance, increasing thesize of the scattering clusters affects the frequency selective-ness of the channel, even though there is neither a direct noran inverse proportionality between the two. Figure 8 showsthe effective diversity orderNdiv plotted for different frequen-ciesf and for different cluster diameters expressed inλ’sfor the first model,Mα. AnalyzingNdiv along different fre-quencies we can recognize a slow-moving trend corruptedby faster variations. For instance, in the first plot of Fig. 8

0.8 1 1.2

x 109

2

4

6

8

10

cluster size=200.8 1 1.2

x 109

2

4

6

8

10

cluster size=150.8 1 1.2

x 109

2

4

6

8

10

cluster size=10

Effe

ctiv

e D

iver

sity

Ord

er

0.8 1 1.2

x 109

2

4

6

8

10

cluster size=50.8 1 1.2

x 109

2

4

6

8

10

cluster size=2.50.8 1 1.2

x 109

2

4

6

8

10

cluster size=0.5

Effe

ctiv

e D

iver

sity

Ord

er

Fig. 8. Theeffective diversity order Ndiv plotted against frequencyf for different cluster sizes (expressed inλ’s) forMβ .

(cluster size = 0.5 λ), we see a slow moving trend whichlooks like the sum of a concave semicircle and a zero meannoise-like process which introduces fast variations. Aftermany simulations we observed that the magnitude of the fastmovingNdiv changed at every realization while the varianceof the slow movingNdiv changed with the cluster diameterwith an undoubtedly direct proportionality. In other words,for bigger cluster sizes,Ndiv changes more rapidly along fre-quency while this does not necessarily correspond to higherfrequency selectiveness nor to greater time variance. Thenumber of scatterers affects the channel in a similar way butwith a much weaker dependency.

4 A comparison with measurements

In order to assess the results derived in Sect. 3 with an evenmore realistic scenario we modeled the surroundings of ameasurement campaign undertaken at TU Ilmenau. Addi-tionally the simulation allowed us to investigate the impor-tance of other critical parameters of the model: the positionsof the scatterers and the modelling of the reflection coeffi-cients. The first step was to determine whether a rough po-sitioning of the scatterers was enough to recreate the mainfeatures which characterize the measured channel. To thisend, we chose, among the many measurements gathered, achannel with a strong LOS and characterized by a simplegeometry so that the position of the scatterers could easilybe estimated. Figure 9 shows the bird’s eye view of the re-constructed scenario. The receiver was a 4× 8 polarizationsensitive patch-array URA (Uniform Rectangular Antenna)and it was positioned by a window on the second floor ofa building. The transmitter was a single 2 GHz omnidirec-tional antenna mounted on a wheeled cart pulled by hand upa road in the direction of a second building (Fig. 10). Thetransmitter path is represented by a continuous line in Fig. 9.

122 G. Del Galdo et al.: Geometry-based channel modelling

Fig. 9. Bird’s eye view of the synthetic reconstruction of the sce-nario where the measurements were taken.

In the proximity of the building the cart was turned aroundand pulled in the opposite direction (towards the receiver)approximately on the same path. The whole measurementtook approximately 1 min at a speed of 3 m/s. The measure-ment device employed was the broadband real-time channelsounder RUSK MIMO from MEDAV (Thoma et al., 2001).The following parameters characterize the measurements:

– A bandwidthB of 120 MHz at a center frequency of1.95 GHz sampled with1f = 300 kHz for a total of385 samples.

– A total of 6313 time snapshots were taken, one every9.2 ms for a total of 58 s.

The same parameters were chosen for the synthetic IlmPropmodel so that the two data sets had same dimensions andsampling grids. However, in order to compare the measure-ments with the synthetic channels, adding the noise and per-forming a proper normalization must be taken into account.

The need for a normalization is due to the fact that theabsolute powers sent by the antennas have not been consid-ered. Only the relative difference in power between the LOSand the scattering components have been modeled throughthe RicianK-factor which can be set at will. This simpli-fied the modelling effort even though the results need post-processing in order to be finally compared with the measure-ments. The normalization is performed in such a way thatthe total power received for the synthetic channel matchesthe one received in the measurements. Clearly both datasets must have the same number of samples, dimensions andmore in general, the same sampling grids. IfHM denotes thechannel for the measurements andH I the one generated with

Fig. 10. The receiver (top circle) and the transmitting antenna (bot-tom circle) mounted on a cart.

the IlmProp then the whole channel matrix must be scaled sothat∑2

‖HM‖2

=

∑2

‖H I ‖2 (13)

where2 spans the complete domain of all dimensions. Inour simulationsH I has been generated without noise. Thenormalization and the adding of the noise must then be doneconcurrently. First we have to estimate the power of the noisefrom the measurements. We assume constant AWGN (Addi-tive White Gaussian Noise) over all dimensions (delay time,time, antennas, etc.). An effective way to measure the noisefloor in the measurements is to observe the channel matrix inthe time and delay time domain. If the path lengths are shortenough, so that the last echoes extinguish before the maxi-mum delay resolvable, we have a measurement of the noisewithout signal. If the number of samples is sufficient we canuse this data to estimate the power of the noise. The noisysynthetic channel will then be

H I ,noisy = H I ,noiseless+ Hw, (14)

whereHw is a matrix whose elements are ZMCSCG (ZeroMean Circular Symmetric Complex Gaussian) random num-bers with varianceσ 2 (Paulraj et al., 2003).

Equation (13) then becomes:∑2

‖HM‖2

− γ · σ2=

∑2

‖H I ‖2, (15)

whereγ is the total number of samples present inH , i.e.γ = MR ·MT ·T ·F , whereT andF are the number of timesnapshots and frequency bins, respectively.

The geometry has been roughly reconstructed with thehelp of blueprints of the area and through visual inspection.Cars, buildings, light poles, and trash bins have been identi-fied as the probable sources of scattering.

Figure 11 shows the channel impulse responses in timet

and delay timeτ . The channel has been gathered originallyin the{t, f } domain; the CIRs in delay time have been com-puted via the Fourier transform described in Eq. (1). The

G. Del Galdo et al.: Geometry-based channel modelling 123

delay time [µs]

time

[s]

0 0.5 1 1.5 2 2.5 3

0

10

20

30

40

50

−90

−80

−70

−60

−50

−40

−30

LOS

double−bounceecho

single−bounceecho

Fig. 11. Channel Impulse Responses from the measurements intime t and delay timeτ . The Line Of Sight LOS and echos areclearly visible. The grayscale represents amplitudes in a dB scale

LOS is clearly visible and as expected its delay lag grows asthe transmitter moves further away from the receiver. Whenthe cart is turned around and pulled back towards the re-ceiver, the arrival delay time decreases. Many echoes arevisible as well. Note that as the transmitter moves mostechoes arrive earlier, meaning that their path length becomessmaller. When the curve corresponding to one echo changesin time with the same slope characterizing the Line Of Sight,then we can deduce that the path length of the echo changeswith the same rate of the path length of the LOS. The strongecho labeleddouble-bounce echohas the same trend as theLOS just delayed by approximately 0.8 µs, which corre-sponds to 240 m, the distance between the top and bottombuildings doubled. For this reason it is very likely that this ar-rival corresponds to the signal reflected by the bottom build-ing first, and then by the top building, thus a doble-bounceecho. Figure 12 shows the channel generated by our model.The trend of the LOS is extremely well reproduced. Thiswas, however, expected since the LOS is the only compo-nent easily predictable once the position of the transmitterin known. Most of the scattering contributions can be ob-served in the synthetic data set as well, with the exceptionof the double-bounce paths which have been omitted. An-alyzing the two data sets in the{t, τ } domain suggests im-provements in the modelling approach: in the measurementswe can observe that certain echo traces are stronger than oth-ers. In the modeled channel this does not happen. This is aresult of the modelling of the reflection coefficients. In factwhile their phase has been generated uniformly distributed inthe[0, 2π] range, the amplitude has been taken equal for all.The comparison suggests that the reflection coefficients needa more sophisticated modelling. Furthermore, a closer lookon the measurements reveals that some echoes exist only incertain time periods. This suggests that the amplitude of thereflection coefficients (perhaps their phases too) should be

delay time [µs]

time

[s]

0 0.5 1 1.5 2 2.5 3

0

10

20

30

40

50

−90

−80

−70

−60

−50

−40

−30

Fig. 12.The synthetically generated Channel Impulse Responses intime t and delay timeτ .

0 5 10 15 20 25−50

−45

−40

−35

−30

−25

Am

plitu

de [d

B]

0 5 10 15 20 25−50

−45

−40

−35

−30

−25

Am

plitu

de [d

B]

time [s]

Fig. 13. Amplitudes of the CIRs for a specific frequency bin plot-ted in dB against time for the measurements (top) and the IlmPropmodel (bottom).

dependent on the Direction of Arrival (DoA) and Directionof Departure (DoD). Figure 13 shows the fluctuations of theamplitudes of the channel for a specific frequencyfo plot-ted against time for the first 25 s. Fast-fading characterizesboth signals even though the measurements appear somehowmore variant. In order to assess this feature quantitatively wecompute the multipath intensity profileψDe(τ ), the Dopplerpower spectrumψDo(fD) and the autocorrelation functionsin frequency acf(1f ) and time acf(1t) for both data sets.From their analysis we can obtain the RMS delay spreadτRMS, the coherence bandwidth(1f )c, the RMS DopplerspreadfD,RMS and the coherence time(1t)c (Figs. 14 and15). The RMS delay spread shows that the energy is simi-larly distributed in delay timeτ . In the measurements two

124 G. Del Galdo et al.: Geometry-based channel modelling

0 1 2 3 4 5

x 10−7

0

1

2x 10

−10

delay time

ψ D

e(τ)

−40 −20 0 20 400

0.2

0.4

0.6

0.8

1x 10

−10

Doppler Frequency

ψ D

o(fD

)

−4 −2 0 2 4

x 106

0

0.2

0.4

0.6

0.8

1

frequency

acf(

∆f)

−0.1 −0.05 0 0.05 0.10

0.2

0.4

0.6

0.8

1

time

acf(

∆t)

Fig. 14. The multipath intensity profileψDe(τ ), Doppler powerspectrumψDo(fD) and the autocorrelation functions in frequencyacf(1f ) and time acf(1t) for the measurements

more spikes can be observed. This is due to the fact that twospecific echoes have a significantly stronger magnitude com-pared to other echoes. The RMS delay spreadτRMS is any-how very similar suggesting that the two channels will have asimilar frequency selectiveness. The RMS delay spread cor-responds to the distance between the two vertical lines plot-ted in Figs. 14 and 15. From the autocorrelation function inthe frequency domain acf(1f ) we can derive the coherencebandwidth(1f )c which confirms that the two channels haveapproximately the same coherence bandwidth. In the{t, fD}

pair the two channels show more differences. While theDoppler power spectrumψDo(fD) has very similar trends,the autocorrelation function in the time domain shows thatthe measured channel is much more time variant than thesynthetic one. Furthermore this analysis proves that Eq. (9)only represents a good empirical law, valid in most but not allcases. In Fig. 13 we can in fact recognize that the measure-ments have a faster moving small-fading effect, especiallyduring the first 10 s. We then investigated how we could gen-erate more time variance changing only positions, the size ofthe scattering clusters, and the number of scatterers per clus-ter, discovering that this was not possible. Our simulationsshowed that in order to achieve higher order of time varianceit is absolutely necessary to model time variant reflection co-efficients or to introduce moving scatterers along the timedimension.

5 A deeper look at the phase response

Looking more closely at the phase of the received signalleads to interesting comparisons and remarks. Figure 16shows how the phase of the received signal (in its complexbase-band representation) is constructed in a flat-fading fre-quency selective channel. The resulting phaseϕR will be

0 1 2 3 4

x 10−7

0

0.5

1

1.5

x 10−10

delay time

ψ D

e(τ)

−40 −20 0 20 400

1

2x 10

−10

Doppler Frequency

ψ D

o(fD

)

−4 −2 0 2 4

x 106

0

0.2

0.4

0.6

0.8

1

frequency

acf(

∆f)

−0.2 −0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

time

acf(

∆t)

Fig. 15. The multipath intensity profileψDe(τ ), Doppler powerspectrumψDo(fD) and the autocorrelation functions in frequencyacf(1f ) and time acf(1t) for the IlmProp model

8 Del Galdo and Haardt: Geometry-based channel modelling

0 1 2 3 4 5

x 10−7

0

1

2x 10

−10

delay time

ψ D

e(τ)

−40 −20 0 20 400

0.2

0.4

0.6

0.8

1x 10

−10

Doppler Frequencyψ

Do(f

D)

−4 −2 0 2 4

x 106

0

0.2

0.4

0.6

0.8

1

frequency

acf(

∆f)

−0.1 −0.05 0 0.05 0.10

0.2

0.4

0.6

0.8

1

time

acf(

∆t)

Fig. 14. The multipath intensity profileψDe(τ), Doppler powerspectrumψDo(fD) and the autocorrelation functions in frequencyacf(∆f) and timeacf(∆t) for the measurements

derive the coherence bandwidth(∆f)c which confirms thatthe two channels have approximately the same coherencebandwidth. In the{t, fD} pair the two channels show moredifferences. While the Doppler power spectrumψDo(fD)has very similar trends, the autocorrelation function in thetime domain shows that the measured channel is much moretime variant than the synthetic one. Furthermore this analy-sis proves that equation (9) only represents a good empiricallaw, valid in most but not all cases. In Figure 13 we can infact recognize that the measurements have a faster movingsmall-fading effect, especially during the first10 seconds.We then investigated how we could generate more time vari-ance changing only positions, the size of the scattering clus-ters, and the number of scatterers per cluster, discoveringthatthis was not possible. Our simulations showed that in or-der to achieve higher order of time variance it is absolutelynecessary to model time variant reflection coefficients or tointroduce moving scatterers along the time dimension.

5 A Deeper Look at the Phase Response

Looking more closely at the phase of the received signalleads to interesting comparisons and remarks. Figure 16shows how the phase of the received signal (in its complexbase-band representation) is constructed in a flat-fading fre-quency selective channel. The resulting phaseϕR will bethe sum of the phase introduced by the LOSϕLOS plus theadditional phase introduced by the echoes. The smaller theamplitude of the echoes is compared to the LOS component,the smaller will be the differenceϕR − ϕLOS . As the radialdistance between transmitter and receiver changes, the LOScomponent,ϕLOS , will change accordingly. If the move-ment is slow enough and the phase-noise introduced by theechoes is negligible, it is possible to derive the displacementfrom the phase trend of the received signal. In fact, a move-

0 1 2 3 4

x 10−7

0

0.5

1

1.5

x 10−10

delay time

ψ D

e(τ)

−40 −20 0 20 400

1

2x 10

−10

Doppler Frequency

ψ D

o(fD

)

−4 −2 0 2 4

x 106

0

0.2

0.4

0.6

0.8

1

frequency

acf(

∆f)

−0.2 −0.1 0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

time

acf(

∆t)

Fig. 15. The multipath intensity profileψDe(τ), Doppler powerspectrumψDo(fD) and the autocorrelation functions in frequencyacf(∆f) and timeacf(∆t) for the IlmProp model

I

Qechoes

LOS

received

ϕLOS ϕR

Fig. 16. When the amplitude of the echoes is small the phase of thereceived signalϕR is a good approximation of the phase of the LOScomponent onlyϕLOS .

ment ofλ in the radial direction corresponds to a full rotationof 2π radians in the phase. If this wavelength has been sam-pled with more than two samples per wavelength, thus ful-filling Shannon’s theorem, the phase can be efficiently ’un-wrapped’, or in other words, absolute jumps greater thanπ

are converted to their2π complement. The spatial displace-ment can be then derived from the phase simply multiplyingby the factor: λ

2π. Figure 17 shows the unwrapped phase

for every frequency bin derived directly from the measuredchannel analyzed in Section 4. Even though a different fluc-tuation perturbs each frequency bin, the overall trend is con-stant and shows a descending phase for half of the time anda symmetrical ascending phase for the other half, in accor-dance with the experiment’s geometry. The different slopeis due to the different velocity. In fact the transmitter movedfaster in the second half of the experiment. If the channelwere flat-fading we would see the slow moving LOS com-ponent only, absolutely equal for every frequency. It is in-teresting to compare the overall phase trend with the LOScomponent only. In case of the synthetic channel this is veryeasy to obtain. More complicated is the case of the measure-ments. Once the channel has been transformed in the{t, τ}

Fig. 16. When the amplitude of the echoes is small the phase of thereceived signalϕR is a good approximation of the phase of the LOScomponent onlyϕLOS .

the sum of the phase introduced by the LOSϕLOS plus theadditional phase introduced by the echoes. The smaller theamplitude of the echoes is compared to the LOS component,the smaller will be the differenceϕR − ϕLOS . As the radialdistance between transmitter and receiver changes, the LOScomponent,ϕLOS , will change accordingly. If the movementis slow enough and the phase-noise introduced by the echoesis negligible, it is possible to derive the displacement fromthe phase trend of the received signal. In fact, a movementof λ in the radial direction corresponds to a full rotation of2π radians in the phase. If this wavelength has been sam-pled with more than two samples per wavelength, thus ful-filling Shannon’s theorem, the phase can be efficiently ’un-wrapped’, or in other words, absolute jumps greater thanπ

are converted to their 2π complement. The spatial displace-ment can be then derived from the phase simply multiplyingby the factor: λ

2π . Figure 17 shows the unwrapped phasefor every frequency bin derived directly from the measured

G. Del Galdo et al.: Geometry-based channel modelling 125

Fig. 17. Unwrapped phase trend derived from measurements plot-ted in radians against frequencyf and timet .

channel analyzed in Sect. 4. Even though a different fluctua-tion perturbs each frequency bin, the overall trend is constantand shows a descending phase for half of the time and a sym-metrical ascending phase for the other half, in accordancewith the experiment’s geometry. The different slope is dueto the different velocity. In fact the transmitter moved fasterin the second half of the experiment. If the channel wereflat-fading we would see the slow moving LOS componentonly, absolutely equal for every frequency. It is interestingto compare the overall phase trend with the LOS componentonly. In case of the synthetic channel this is very easy toobtain. More complicated is the case of the measurements.Once the channel has been transformed in the{t, τ } domain,it is possible to identify the samples in delay timeτ withhighest energy for every time snapshot. The changing phaseof these samples against time approximates the phase trendintroduced by the LOS component only. The comparison ofthe unwrapped LOS phase with the overall phase is shown inFig. 18. The model and the measurements show very sim-ilar traits. The upper curves show the phase trend for onefrequency bin derived directly from the unwrapping of thephase. The lower curves are derived, respectively, with theLOS approximation for the measurements, and from a directcalculation of the displacement for the model. In fact, forthe synthetic data, the exact path of the transmitter is known.From these coordinates it is possible to derive the radial dis-placement and consequentially the phase. The phase trendsmatch for the first 10–15 s, while they start to diverge signifi-cantly around 25 s. This can be simply explained consideringthat the RicianK-factor drops in the middle part of the exper-iment. When the amplitude of the echoes is comparable withthe LOS, then the received vector will have a phase uniformlydistributed distributed in the interval[0, 2π ]. The unwrapperwill not find any trend and will jump from ascending to de-scending phases randomly. The resulting unwrapped signalwill have a slope close to 0, that is horizontal. For this rea-son, as we pass from a higherK-factor to a smaller one we

0 5 10 15 20 25 30 35 40 45 50−3500

−3000

−2500

−2000

−1500

−1000

−500

0

phas

e [r

ad]

time [sec]

realmodelreal LOSmodel LOS

Fig. 18. Phase comparison between the measurements and its syn-thetic reconstruction. The upper curves are obtained simply un-wrapping the signal received at one specific frequency. The lowercurves represent the LOS component only.

find increasingly less steep slopes. In other words the phase-noise introduced by the echoes is uniformly distributed, butdue to the non-linearity of the unwrapper, the resulting errorleads always to more moderate slopes.

6 Conclusions

Our simulations show that for certain modelling applicationsa simple geometry-based single-bounce model can mimicmost channel features with sufficient precision, among whichthe frequency selectiveness and the time variance are themost characterizing. A rough positioning of the scatteringclusters allows us to achieve the desired frequency selective-ness as well as other specific traits of the channel statisticsby tuning three basic parameters, i.e, the size of the scatter-ing clusters, the number of scattering coefficients, and theRicianK-factor. To obtain higher levels of time variance andmore realistic synthetic channels, moving scatterers and timedependent reflection coefficients must be modeled as well.

Acknowledgements.The authors gratefully acknowledge the sup-port of MEDAV (www.channelsounder.de) and of the ElectronicMeasurements Engineering Laboratory at Ilmenau University ofTechnology in performing the channel measurements.

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