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.

Assume that the mobile transmits unmodulated carrier, with a complex envelope s(t)=1. Then we can interpret the arriving signal as the complex gain of the channel. At any angle q, the arrivals in a differential angle dq sum to the complex amplitude A(q). Then the complex gain at the reference position (the original base antenna) is just the sum of all scatterers

At the new position, the path length for scatterers at azimuth q has decreased by x sin(q), which introduces a corresponding phase change. In reality, this expression for path length change is an approximation that holds for displacements that are much smaller than the distance to the mobile (not the same thing as small angular dispersion). Jakes used it for both mobile and base autocorrelations [ Jake74 ], where it appears as a Doppler shift in his derivations. In any case, the resultant complex gain becomes

where b is the wave number 2p/l. The autocorrelation function as a function of position is

Since the scatterers at different angles q and a are independent (they are different scatterers), we have

where P(q) is the angular power density

In one sense, (8.3.7) is simple, because it shows that the autocorrelation function is determined by the angular power density. On the other hand, we can make things arbitrarily difficult for ourselves by trying to solve it exactly. Recall that, even in the easy case of a uniform P(q) over [-p,p), the autocorrelation function (8.3.7) becomes the Bessel function J₀(bx), and things can only get worse from there. In reality, though, there is little need for an exact solution, since even our starting point was an approximate model of the true propagation environment (and if we really cared, we could evaluate (8.3.7) numerically, at the cost of insight).

So we'll take a simpler tack. Represent the azimuth q of the scatterers as a small increment f on the angular location Q of the mobile, so that q=f+Q (see the sketch above). Next, expand the sine in the exponential in (8.3.7) trigonometrically, then in a series in f

The angular dispersion W at a base is usually small: W=0.4 would be a very large value, seen only in a microcellular system, and an order of magnitude smaller is more typical of a macrocell. Consequently, we can truncate the series (8.3.9) to the linear term in f. Substitution of (8.3.9) into (8.3.7) gives

(approximately)

Fascinating! The autocorrelation function is proportional to the inverse Fourier transform p of the angular power density P, dilated by the factor 1/cos(Q), provided W is small.

Here's a quick inference from (8.3.10). Represent the displacement of the base station antenna as a component xsin(Q) along the axis between base and mobile (the endfire direction) and a component xcos(Q) transverse to the axis (the broadside direction). The endfire component

then affects only the phase of the autocorrelation, not the magnitude, so that it has no effect on diversity. It is just the broadside component that provides diversity. Of course, these statements are valid only to the validity of ignoring second and higher order terms in the series (8.3.9). If W is too large for this, use the original form (8.3.7).

Autocorrelation Functions for Three Scatterer Models

We'll look at three candidates for angular power density P(q): a uniform distribution, a ring of scatterers and a disc of scatterers. They and their corresponding autocorrelation functions are shown together in a graph following the discussion.

To simplify things, we'll express distance in wavelengths. That's equivalent to setting . Also, we'll normalize to unit power, or . Finally, we define

The first candidate for angular power density is a uniform distribution over the beamwidth W. We used it in Section 8.1 as a simple model for the signal environment at a mobile with a directional antenna in isotropic scattering, and it was also employed in [ Salz94 ] as a base station model. This gives

so that Fourier inversion produces the familiar sinc function:

As a second candidate, you may feel that a more realistic model for angular power density would account for the clustering of scatterers around the mobile. Referring to the sketch above, let us assume that the scatterers are distributed uniformly on the circumference of a circle of radius W/2 centred on Q. This is the well-known "ring of scatterers" model, originally proposed by Jakes [ Jake74 ]. A little geometry gives

for

It has the same shape as the "U-shaped spectrum" of Section 5.1 , so its transform is the Bessel function. From (8.3.10), its autocorrelation function is

For our third candidate, we take the scatterers to be uniformly distributed with equal power on a disc of radius W/2 centred on Q. To obtain the angular power density from the disc, we need the area in the disc that lies in dq at q. As you can imagine from the first part of the sketch below, this is a challenging computation in itself.

But why bother with this computation? We are assuming relatively small angular dispersion, so we might as well also assume that the differential area is a linear strip, rather than a wedge, as shown in the second part of the sketch. That makes an easy power density:

for

where the leading factors make its area s_g². Unfortunately, a closed form for the inverse transform is not available. At this point, we could substitute (8.3.15) into (8.3.10) and perform the integration numerically - easy! Of course, if we are willing to perform a numerical integration, we might as well substitute (8.3.15) into the more accurate (8.3.7) and be done with it - but we'll continue with (8.3.10), in order to compare the three models on an equal footing.

Instead of continuing with (8.3.15), we'll employ an alternative calculation of the spatial autocorrelation R_dos. Consider the disc of scatterers as a set of concentric rings of scatterers. Then we can simply average R_ros frome (8.3.14) over the radius, with a weighting that reflects the increasing area at larger values of radius.

The reason for using this form is that it opens the way to other radially dependent models in which the power of scatterers in a given ring is determined by some path loss law (see Section 1 ). You can take this idea the rest of the way yourself if you need it.

We can compare the three models graphically. First, the angular power densities. Assume that the mobile is at Q=0 (the broadside direction) and that the angular dispersion is relatively large, W=0.1. Then plot the angular power densities (8.3.11), (8.3.13) and (8.3.15) for a specific parameter choice.

From basic Fourier theory, we know that the disc of scatterers should give a wider main lobe (greater coherence distance) than the uniform distribution, but its decay will be somewhat faster because the step discontinuity is softened. The ring of scatterers should have the slowest decay, because of its singularity.

Next, we'll look at the corresponding autocorrelation functions in the broadside direction (endfire just changes the phase). The plot below shows that the all three angular power densities allow complete decorrelation in 8 to 12 wavelengths. In fact, much of the diversity gain can be obtained with correlation coefficients as high as 0.5 (as we will see in the text Detection and Diversity), so it appears that a separation of 5l to 7l is sufficient. However, this plot is for angle spread or a fairly large value. From Fourier theory, and consistent with our conclusions at the end of Section 8.1 , correlation distances scale inversely with the angular dispersion W. Therefore W=0.01 requires antenna separations of 50l or more just for decorrelation to the 0.5 level. The DOS model drops even more slowly. A rule of thumb in base station design is a 20l separation for diversity. At 900 MHz, this is about 7 metres - a large structure for a roof or tower! Fortunately, the shorter wavelength of PCS bands at 1.9 GHz will reduce it by a factor of 2.

A generalization of the simple angular scattering models we have just considered places the main contribution from the cluster in an environment of isotropic low-level clutter. [ Kalk97 ] follows just this approach, modeling the angular power density as a strong uniform distribution of width W, just as above, but on a low-level pedestal of width 2p. As you might expect, the pedestal produces somewhat faster decorrelation.

Finally, note that all of these models - uniform, ring of scatterers and disc of scatterers - are very idealized. They represent notions of how an average over many mobile locations might look. In any specific location, however, they might look quite different. In particular, the question of whether the received power is dominated by contributions from a few point scatterers is critical, since the answer tends to direct our thinking toward diversity, as above, or to eigenmethods that attempt to resolve the individual contributions and combine them coherently, thereby avoiding fading. These questions will be examined in the book Smart Antennas in this series; unfortunately, though, there are few published measurements yet on this topic, and much depends on such data.

Quadratic Expansion

Here is a simple and useful result. The plots of spatial autocorrelation invite a quadratic approximation for small displacements, just like the quadratic expansions for autocorrelation function in time due to Doppler spread, and autocorrelation function in frequency due to delay spread. Like those expansions, we will find that it depends on the rms measure of the corresponding spread - in this case the angular dispersion. From the approximation (8.3.10), we have

We are interested in the broadside direction, the one from which we obtain diversity. Therefore we set Q=0. The second derivative, evaluated at x=0, is

where f_rms is the rms angle spread of the incoming signal, and s_g² normalizes the angular power density. This gives us an easy Taylor series expansion

and resulting approximation

Therefore, when you estimate the diversity antenna correlations for small spacings ( x/l<1/(2pf_rms) ), all you need to know is the rms angle spread - a simple and useful result. As a quick application of it, we note from (8.3.20) that the correlation distance must scale inversely with the rms angle spread.