8.3 Angular Dispersion and Correlation at the Base

Diversity at the base station is fundamental to good mobile communications design. But how far apart should the antennas be spaced to obtain adequate diversity? Here we see a striking difference between behaviour at the base station and the mobile. You recall from Section 5.1 that, at a typical mobile with a single isotropic antenna and surrounded by scatterers, decorrelation of the complex gain occurs over distances as short as half a wavelength. In contrast, you will see in this section that, at a typical base station, decorrelation requires separations of tens of wavelengths or more. Another difference is a strong dependence on direction of the separation with respect to the angular location of the mobile.

You may not have expected these differences between mobile and base, since the channel is reciprocal in the electromagnetic sense. However, they become clear if you consider the different geometries: the mobile is surrounded by scatterers and usually employs an isotropic antenna (though see Section 8.1 ), whereas the base is usually located where it is not troubled by local scatterers (e.g., on a tower) and it receives signals from the mobile over a narrow angular range that is determined by the size of the mobile's scattering neighbourhood and its distance from the base. In addition, the base frequently uses a directional antenna, such as a 120 degree or 60 degree sectoral antenna.

In this section, we will build a body of theory you need for diversity antenna design and for your study of smart, adaptive antenna arrays. We will start with a simple semi-quantitative argument that exposes the basic issues, then adapt the results of our analysis of directionality at the mobile in Section 8.1 to show the dependence of coherence distance on angular dispersion. Both discussions keep mathematics to a minimum and try to present the phenomena in an intuitive way.. Finally, we will obtain the autocorrelation function of complex gain - the fundamental issue in diversity design - as a function of spatial separation at the base, using an idealized model.

Simple Argument

Consider the simplified configuration in the sketch below. The mobile (seen from above) is a distance d from the base, and there are only two scatterers, separated by a distance s that is measured transverse to the axis, or line connecting base and mobile. Since decorrelation is a result of changing phase angles among the paths, we ask what is the differential distance and phase between the two paths if the base antenna is moved a distance x, also measured transverse to the axis.

A straightforward analysis based on right triangles gives the differential distance as

and series expansion of the square roots lets us approximate it as

so the differential phase is

We can observe the principal phenomena in this simple result:

• The quantity s/d is the angular dispersion, analogous to beamwidth W in Section 8.1 . We might have the scatterer diameter s as one or two hundred metres. In a large cell, where the distance d is large, we will therefore find very small values of angular dispersion - as low as 0.01 radian - so that x must be many wavelengths for a significant phase change, or significant decorrelation. Even in a microcellular environment, the angular dispersion is on the order of 0.1, and is unlikely to exceed 0.5 radian, so that the required antenna separation at the base is still an order of magnitude greater than at the mobile.
  

• From (8.1.15) and (8.1.18) , as well as (8.3.2) above, the coherence distance depends inversely on the angular dispersion (i.e., beamwidth W) for any orientation of mobile with respect to antenna displacement. Therefore we expect faster decorrelation, and consequently greater diversity, for mobiles that are closer to the base. Faraway mobiles appear as point sources, and are coherent in the displacement neighbourhood of the base station.
  

• Now generalize to the case of many scatterers and consider the impulse response. If the differential delay changes caused by base antenna displacement are small compared to the wavelength, they are negligible compared to the time scale of the modulation. Therefore the scatterers within each delay bin are fixed in individual amplitudes but sum to a different resultant as the differential delays cause phase changes of the carrier. We conclude from this that all base antenna locations in a neighbourhood of tens of wavelengths have the same power delay profile.

• We obtain the greatest change in differential distance with this transverse displacement of the base antenna. In contrast, displacements along the axis have no effect on differential delays, so that there is no possibility of diversity in that direction. In reality, of course, there are more than two scatterers and they are not confined to a straight line transverse to the axis, so the changes in differential path length are not strictly zero. However, they are very small, and there is very little decorrelation with antenna displacement along the axis. This means that if you have a linear array, as in the sketch below, don't expect much decorrelation in the endfire direction. Even broadside has slow decorrelation.

Coherence Distance

Our next step up in model sophistication is more quantitative: the coherence distance. We already obtained this quantity in Section 8.1 in connection with directionality at the mobile. We can use those results directly if we are willing to believe that the set of paths between the base and the scatterers associated with a particular mobile has power that is distributed uniformly over a small range of azimuth. From (8.1.15) and the approximations (8.1.16) and (8.1.17) , the coherence distance in wavelengths is then

This is consistent with our observations on the simple model above: the coherence distance is inversely proportional to the angular dispersion W and it is greatest at Q=0; that is, the direction of the mobile.

Autocorrelation Function

In the discussion so far, we have seen the principal effects of angular location and angular dispersion of the mobile's scatterers on coherence at the base station - and we found them without having to do detailed mathematics. If we intend to do any real analysis of diversity or adaptive beam forming, though, we need the autocorrelation of complex gain with respect to antenna displacements at the base. And, unfortunately, that means the holiday is over. It will take a little work to get this result.

The sketch below shows the situation. We have a reference location for the base antenna, and the mobile is at some angular position Q. As usual, the mobile is surrounded by a group of scatterers. We are interested in the correlation between the complex gain g(0) at the reference position and the complex gain g(x) at a distance x away.