Direct calculation of coherence bandwidth in urban microcells using a ray-tracing propagation model

An important parameter in characterizing radio communications channels is the coherence bandwidth. This paper presents an analysis of the coherence bandwidth in a urban microcell environment where the dynamic channel response is determined by a site-specific ray-tracing propagation model. Such an analytical model provides a direct calculation of signal fading envelope correlation as a function of frequency and location. The analysis here shows that coherence bandwidth is strongly dependent on location within a particular propagation environment and only weakly related to RMS delay spread. Typical results for frequency diversity gain for various frequency separations are also presented.


Introduction
Much of the effort in designing robust and reliable communications qstems focuses on choosing modulation, coding and receiver architecture schemes which mitigate the deleterious effects of the radio propagation channel.In free space, the propagation channel has a flat amplitude response (attenuation) and linear phase shift as a function frequency.When thc propagation environment is not free space but contains any other elements.including the atmosphere or a single reflecting surface, the frequency response of the channel is no longer flat over all frequencies.A single reflection results in the so-called "two-ray" model in which significant nulls in the amplitude response can occur at particular frequencies depending on the reflection coefficient and raj.geometry.
With highly complex propagation environments.signal energy arrives at the receiver along a \.aricty of paths \vith varying amplitudes, phases, and time dcla!s.The result is a channel frequent?. response which sarics from place to place.One measure of the varying frequency rcsponsc is the colierencc bandwidth ( A / ) , The colicrcnce bandwidth is the frequency separation bctwecu t u o frequenc!tones which results i n it giveii decorrciation iii signal envelope amplitudes.Thc decorrclation is usually defined as the point where the correlation coefficient p, bctn'een the fading envelopes at the two frequencies is reduce to 0.9 or 0.5.For the studies done here, a correlation coefficient of 0.9 is used to defined the correlation bandwidth.As explained in Section 2, a site-specific ray-tracing propagation model is used to find the fading envelopes and correlation coefficient.
The coherence bandwidth of the cliannel is particularly relevant to frequency-hopping spread spectrum (FHSS) systems[ 11, and to other multi-carrier systems, including OFDM.
In both cases robust transmission is achieved by choosing mu1 ti-carrier frequency separations, or frequency hop distalice.such that frequencies are sufficiently de-correlated that the probability of simultaneous fading inipairnicnts on multiple frequencies is low.This is the fundamental improvement which frequency diversity has to olfer [2].
For the hypothetical dense urban environment studied here.coherence bandwidths ranging from 30 kHz to 130 lcHz were found.The coherence bandwidth was found to be site-dependent and only weakly related to the inverse of the RMS delay spread of the power delay profile.

Ray-Tracing Propagation Model
A general model for the low-pass impulse response for an urban radio channel is: in which the impulse response h(t) is the sum of a set of S impulses arriving at delay times T,, with amplitudes -4n.phases On, and phase displacements AO,, .Thc phase displacements result from the niotioii of the receiver or other spatial change of the rccciver location relative to the rest of the propagation ewironmcnt bvhicli nia!; itself including moving objects (reflections i'roni cars and buses, etc.).For a mobile receiver the displaccincnt tcriii is given by AU,: = ( 2 m l ,' A) cos( Q,, 4 4,) .~l i c i c p, IS the arrival angle of the 11"' inipulsc.I' IS the speed of niolion, and q5,, is the direction of motion.
To USC the channel model in (1). it is ncccssav to Identify the amplitudes.timc dclays, and absolute phase shifts of the A' components of h(t).Tlic received @ IEEE coniponcnts consist of the line-of-sight signal froin tlie trarisniiltcr and a varicly of signals reaching the rcccivc anlcniia via rcflcctirig surfaccs.diffracting coriicrs and scattering surfaces.By using ray-tracing techniques.thc cncrgy cniittcd from thc sourcc transmitting anteiina is gconictrically traced to dctcrniinc thosc surf.xcs or coriicrs which arc illuminatcd.For the ray-tracing niodcl uscd here.cach illuniiiiatcd surface is rcplacc by an image transmitter or scattering source such that thc radiation from the image represents (in amplitude.phasc.and radiating directions) the energy rcflccted from thc source.Similarly, an illuminated corner is rcplaccd by an cquivalent wedge diffraction source.With thc first sct of images and illuminated corners in place, each of them is then considcred in turn by ray-tracing to dctcrniine the surfaces and corners they illuminate.This process is rcpeated for as many iterations as may be relevant to thc problem at hand, or which are practical from a coinputational point of view The ray interactions with the propagation environment are tracked for both HP and VP by taking into account the conducti\ity and pcrniittivih of the walls and corners.and the angle of incidence for the interaction at each u.all and corner.
Ray-tracing has become a widely used technique for analyzing propagation in outdoor microcells and indoor nirelcss LAN s j stems.The theoretical model used lierc is dcscribcd in detail 111 131.Rag-tracing niodcls along with comparisons to measurements can be found 111 141 and is].
A hpical ray-tracing study for a transmitter at point AA to a receiver at point R is shown in Figurc 1 along bvith the resulting power delay profile.As sho\vn in 1.31. the magnitude and phase of the reflection and diffraction coefficients will be a strong function of the angle of incidence on the reflecting surface.The magnitude and phase of thc reflection and diffraction cocfficicnts will also dcpcnd on the frequency A ray-tracing propagation model only provides thc ray amplitudes and phases to a single precise point.At this point it niay happen that the vector sum of thc rays rcsult in a null (fadc) or peak in the voltage cnvelope.However. in general the geonietq of the environnicnt is not kno\vn with sufficienl accuracy to predict the cnvclopc voltage so precisely.At the carrier frequcncics typicall!involved in PCS microcell systems (around 2000 MHz). the wavelength is on the order of 15 cni.I n a typical urban building database.the building wall locations may only be known u-ithin perhaps one nictcr.Bccausc absolutc phase can't be known.it is ncccssan.to dcterniinc the channel response over a range of positions around thc precise location where the ray-tracing malysis was pcrformed.
This can be donc b) considcring the ,/udi/?g e/7ve/ope over a raiigc of navclcngth displaccnicnts around this point.For a iypical analysis.the fading voltagc cnvclope is calculatcd at points spaced cvcry 0.125 wavelengths over ;i range of _f IO wavclciigths in four crossing dircctioiis around thc ray-tracilig analysis point.'This uniform pattcrii of four The linear correlation coefficient between the fading envelopes at any two frequencies is given by: where?, and 7, are the mean values of the voltage envelopes at frequencies I; and f2, respectively, and 0 ;, and CJ ; 2 are the correspondng variances of the envelope waveforms.both taken across N waveform samples as described in Section 2. Figure 3 shows an example of the correlation coefficient as a function of frequency separation for a

Coherence Bandwidth In an Urban M icrocel I Environment
The corrclation coefficient p \!as calculated at a set of points along the study route shown in Figure 4.This route includes 114 points spaced at 5 meter intervals.soinc of uhich are line-o€-sight n ith the transmitter at point AA, and some of which are shadowed are in the ''plaza'' area whcrc the RMS dela) spread is higher due to the t\idely spaced opposing reflecting surfaces The resulting Coherence bandwidth (Af), at each point along 000110000 Fig. 4. Map of study route.
the route is plotted in Figure 5. Figure 5 shows that (w), varies considerably as the receiver is moved along the route, with a maximum value of 130 kHz and a minimum value of 30 kHz.The average coherence bandwidth over this route is 66 kHz.
The relationship between coherence bandwidth and RMS delay spread is shown by the scatter plot in Figure 6.A line has been fitted through this data using the ordinary least square error (OLSE) linear curve fitting techniques.The resulting equation relating FWS delay spread and coherence bandwidth is: where oD is the RMS delay spread in nanoseconds over the range of 100 to 500 nanoseconds.The RMS delay spread \vas found from the power delay profile in the usuall) way (see [3] for typical equations).The RMS delay spread also varies along the route but doesn't closely track the variations in the coherence bandwidth.This is not surprising since the RMS delay spread found from the power delaj profile does not take into account the detailed phase and arrival angle information \vhich is inherently a part of the fading envelopes used to develop the frequency correlation coefficients.Also. the RMS delay spread may be greatly affected by long-delayed Ion-amplitude echoes.Even though such low amplitude echoes have an impact on the RMS delay spread value.the>-have very little impact on the actual fading voltage envelope.
This equation was deri\.edfor the spccific hypothetical urban environment studied here.It may not necessarily apply to other environments.The significant point here is that R M S dela\-spread is only an approsiniate indicator of coherence bandwidth.
The treatment of coherence bandwidth in [6] showed coherence bandlvidth to be a niuch stronger function of RMS delay spread.However. the analysis in [6] assumed a hypothetical exponential dela! profile rather than the more realistic profiles derived from the ray-tracing propagation model.The raytracing model as applied to the particular environment here does exhibit a weakness in that due to its limited overall dimensions, RMS delay spreads greater than 500 nSec are rarely produced.Higher RMS delay spreads are common in the environments treated in 161.

Frequency Diversity Gain
Because of the lack of correlation between the fading cn\ clopcs at differcnt frequencies, t\i o scparatcd frequencics can be combmcd to achieve diversity gain Common divcrsit!combining technique include su itchcd (sclcction).equal gain.or nia.rima1 ratio For the In simple switched diversity the amplitude of the signals on the diversity branches are continuously compared and a the branch with the higher amplitudc signal selected.Diversity improvement is usually assessed in terms of diversity gain; i.e., the equivalent transmitter power increase in dB that would be needed to achieve the same system performance improvement as the diversity scheme provides.The diversity gain can usually be determined from the cumulative distribution hnction (CDF's) of the envelopes before and after diversity combining.As an example, the CDF's of the fading envelopes distributions for all the envelopes for all the points on the study route in Figure 4 are plotted in Figure 7.In this figure the leftmost line is the CDF distribution of the fading envelope for a frequency separation of 0 1<Hz.The other lines show the CDF's of the resulting envelope when the two frequencies are separated by amounts ranging from 10 to 250 Wz.This figure clearly shows that diversity gain increases with increasing frequency separation.For a 250 kHz frequency separation in this particular propagation environment, the diversity gain achieved is essentially the same as that achieved t\ ith two independent Fbyleigh-fading diversity branches[ 71.The fading envelope CDF's in Figure 7 can be further used to estimate the diversity improvement in bit error rate following the approach found in 181.

Conclusions
A method for finding the coherence bandwidth 111 an dense inicrocell cnvironnient using a ra! -tracing propagation model has been preseiited Using this model it IS possible to develop realistic fading envclopc pattcrns for different frequencies By comparing the envelopes.the degree of correlation between them and hcnce.the coherence bandwidth can be cstimatcd

Fig. 3 .
Fig. 3. Correlation coefficient vs. tone frequency separation for one point on the study route in Figure 1