An Introduction to Ultra Wideband Communication Systems

3.5. Spatial Behavior and Modeling of UWB Signals

3.5.1. Introduction

The previous sections described the large-scale and small-scale temporal variations of the UWB channel. Specifically, the discussion centered on the statistics of the received signal and models that result in similar statistics. Another important aspect of wireless channels is the spatial variation. It is important to know how the received signal will vary in local area both in time and space. These characteristics are important for understanding applications that attempt to exploit multiple antennas as well as for understanding the channel behavior in a mobile scenario. Models that attempt to capture spatial behavior must model the spatial correlation, spatial fading statistics, and angle-of-arrival statistics in addition to the standard temporal characteristics.

One of the attractive features of UWB is its immunity to multipath fading. Due to the narrow temporal pulses, the interaction between multipath components is severely limited as compared to narrowband signals. This substantially reduces multipath fading. The spatial fading characteristics (that is, the fading variation in a local area) of UWB signals were examined in [11], [12], [13], [25], [72], [79] and [80]. A common procedure to measure fading in a local area is to take measurements at a number of points in a local area (typically a 1 m² grid). Using such measurements three related aspects of the UWB channel are examined. First, the robustness of the UWB signal to local fading is studied by examining the cumulative distribution function of the received signal power (or energy) in the measurement area. Second, the spatial correlation is studied by examining the correlation between received signals at two spatially separated locations (assuming that the channel is stationary). Finally, the angle-of-arrival (AOA) statistics are also studied. These aspects are usually examined in combination with the temporal aspects of the channel. The AOA statistics are particularly useful in creating two-dimensional channel models as we will discuss.

3.5.2. Spatial Fading

Spatial fading (sometimes called local fading) refers to the variation in received power over a local area. To define spatial fading we first must define the metric to be examined. For spatial fading in UWB we will examine the entire received signal energy. Note that this is equivalent to comparing received power for a fixed observation interval. The total received energy , at a position (i, j) of location l is calculated as

Equation 3.68

where is the received signal at the grid location, (i, j) at the measurement location l, and T is the observation duration. A local fade at a location l and position (i, j) can be defined as [50]

Equation 3.69

where e_ref is defined as the energy in the LOS path (that is, excluding multipath) measured by the receiver at a reference distance of 1 m from the transmitter. An example measurement local area (that is, location) is shown in Figure 3.28. The example uses 49 points equi-spaced over an area of approximately 1 m² (15 cm separation between consecutive grid points).

First- and second-order statistics of the total received energy can be calculated for each location as follows. The sample mean is calculated as

Equation 3.70

while the sample standard deviation is calculated as

Example cumulative distribution functions of received signal energies are plotted for six locations in Figure 3.29, taken over an approximately 1 m² measurement grid consisting of 49 points [72]. From the plots, we can clearly see the immunity of UWB signals toward local fading. This lack of variation was first shown in [50] and is in dramatic contrast to the spatial fading observed in narrowband signals [26]. Thus, we note that signal fading is significantly reduced as compared to traditional narrowband systems. This means that diversity mechanisms are not as necessary in UWB systems, and fading margins can be substantially reduced.

3.5.3. Spatial Fading of Signal Components

As mentioned previously, a classic receiver structure for channels with resolvable multipath is the RAKE receiver [43]. Because a RAKE receiver has a limited number of correlators and thus only captures a fraction of the energy, the fading seen by a RAKE receiver may be very different from the fading observed over the entire signal. This is an important consideration. While the total received signal energy may exhibit little variation, this may be irrelevant if a receiver cannot capture the entire received signal energy. Thus, it is important to understand the variation in portions of the received signal. Specifically, one should characterize the fading of the dominant signal components. This would correspond to the fading observed by a RAKE receiver with a limited number of correlators. The immunity to fading seen in the total received signal energy may be mitigated when only a portion of the received signal is captured.

In particular, examining the statistics of a single time delay bin provides insight into the performance of a single finger RAKE receiver collecting energy at a specific delay as it moves through the environment. As an example analysis, consider the time delay bin of the received signal with the highest average energy (over the spatial grid) and the statistics for that bin calculated as discussed in [11], [12], [42], and [72]. An example measured CDF [72] for a Gaussian pulse of approximate duration 200 ps is shown in Figure 3.30. It is clearly observed that the variance of the received signal energy for a single dominant component (that is, a 1 finger RAKE receiver) is much higher than the variance of the total energy. An example set of measurements for a larger number of dominant signal components is presented in Table 3.13 and Figure 3.31 (taken from [72]). We can see that while the variance of the total received signal can be extremely small (~0.5 dB), the variance of a single dominant component is more pronounced (~3 dB). Adding additional components quickly reduces the variation. Again, the spatial model must account for this variation, especially when RAKE receiver structures are to be examined.

3.5.4. Spatial Correlation

Spatial correlation can be defined in various ways, but a standard definition is the amount of correlation between received waveforms separated by some distance, d. Spatial correlation is an important parameter of the channel when using multiple antennas at the transmitter or receiver. The use of multiple antenna arrays is a classic technique in communications for combating mulitpath fading, obtaining SNR improvements, reducing the impact of interference, and more recently, for increasing the achievable channel capacity. Antenna technologies such as diversity techniques and multiple input multiple output (MIMO) applications require very low spatial correlation, whereas beamforming applications require high spatial correlation. Several researchers have investigated the spatial correlation of UWB channels [11,12,25,42,72]. On the surface, one would expect that low spatial fading would correspond to high spatial correlation. However, this is not the case. UWB channels generally exhibit low spatial correlation.

A number of impulse response profiles collected in the same local area may be similar because the main features of the channel are essentially similar within the local area. However, due to the fine resolution of UWB pulses, the relative delays between paths observed by different antennas can be quite different. This can be true even for small spatial movements. If the multipath components arrive from a single direction, we would expect that the received signal at one antenna would simply be a time shifted version of the signal seen at a second antenna. However, if the received signal is coming from several directions, the signal seen at a neighboring antenna will be very different. This can be understood by examining Figure 3.32, which demonstrates an example receive geometry for three antennas and a single multipath component. Figure 3.33 presents an example of a measured received signal at the three locations shown in Figure 3.32. It can be seen from the shift in the profiles that (1,4) is closest to the transmitter and (7,4) is farthest away. As the signal propagates across the three sensors, the main signal components are simply shifted in time, as they come from a single direction. However, the weaker components clearly come from various directions and thus do not show a simple time shift.

Spatial fading is dependent on the entire received signal, which is the sum of many resolvable multipath components. Because the interaction between the components is limited due to the short duration of the pulses used, the fading is limited. However, the relative positions of the paths can change substantially over a short distance, leading to low spatial correlation despite the low spatial variation.

It was noted in [72] and [79] that the spatial correlation in UWB signals is very low when considering the entire received signal. However, it was found that the spatial correlation is significantly higher in the early part of the received signal than in the entire received signal. This implies that the early part of the signal is dominated by components arriving from the same direction, whereas the components in the later part of the received signal come from several directions. This leads to low correlation in the later-arriving components of the received signal. Because this correlated portion was relatively short (on the order of 20 ns), the overall signal correlation was heavily influenced by the second part of the received signal.

Figure 3.34 plots an example set of correlation measurements taken over a 1 m² grid with a 200 ps pulse [72]. The example demonstrates that there is a marked difference in spatial correlation between the first arriving components and the later arriving components. From the plots we can see that the initial portion of the received pulse can be highly spatially correlated (relatively). This means that the direct path and the first few multipath components exhibit high correlation. This, however, does not extend to the entire profile. The longer the time duration of the received profile used for correlation, the smaller the coefficient values. Thus, we can conclude that while UWB signals as a whole exhibit low spatial correlation, portions of the signal can exhibit strong spatial correlation. This has some impact on the use of diversity arrays. If a RAKE receiver is used that collects only a portion of the signal energy, it is possible that diversity antennas will be ineffective. Again, any channel model used to simulate UWB signals should take this into account.

3.5.5. A Two-Dimensional Channel Model for UWB Indoor Propagation

Most proposed statistical models for the UWB indoor multipath channel include characteristics related only to the time-of-arrival (TOA). Notable exceptions are [11], [12], [42] [46], [47], and [72], which present statistics for the Angle-of-Arrival (AOA) in addition to the TOA for a collection of UWB indoor channel measurements. In order to use statistical models in simulating or analyzing the performance of systems employing spatial diversity combining, MIMO, or other multiantenna techniques, information about AOA statistics is required in addition to TOA information. Ideally, it would be desirable to characterize the full space time nature of the channel.

In this section, we discuss possible spatial models in terms of TOAs and AOAs of multipath components. Two-dimensional models have been previously proposed for wideband channels, but few models have been presented for UWB channels [11,72]. The standard approach is to use the temporal models discussed previously and add AOA information. In [9], a model was developed that modeled cluster AOA and TOA independently. Each cluster from the Saleh-Valenzuela model is assumed to come from a separate direction and be uniformly distributed over [0,2p]. In [72] and [76] it is proposed that the Split-Poisson model be extended by adding AOA information to the two clusters. To illustrate the basic ideas behind spatial modeling, we will discuss the models of [72] and [76] further.

From the measured indoor bicone NLOS data it was observed in [72] that the following trends emerged in the distribution of the AOA of multipath components:

1. The AOAs corresponding to the (stronger) earliest arriving paths (first 20 ns) appear to arrive principally from the same direction, as shown in Figures 3.35 and 3.36. Therefore, the AOAs of the first cluster were modeled using a Laplacian distribution with a mean value equal to the LOS direction.

   Figure 3.35. Example AOA Distribution for the First 20 ns and After 20 ns (Polar Plot).

   Figure 3.36. Example AOA Distribution for the First 20 ns and After 20 ns (Linear Plot).

2. The AOAs corresponding to the subsequent diffuse multipath do not appear to be concentrated in any specific direction. Thus, the second cluster AOAs were modeled using a uniform distribution on [0,2p].

The mean and variance of the Laplacian AOA distribution for the first cluster were determined empirically using measurement data. The two-dimensional CLEAN algorithm [11, 72] was used to determine the TOA and AOA of each mulitpath component.

These observations motivated an extension of the Split-Poisson channel model into a two-dimensional (space time) model that also takes the AOA information into account. In order to account for the spatial dimension in this model, each path in the first cluster is assigned an AOA taken from a Laplacian distribution (the mean corresponds to the LOS direction, and the standard deviation was found to be approximately five degrees). Each path in the second cluster is assigned an AOA taken from a uniform distribution over [0°, 360°] as shown in Figure 3.37.

The model was compared with measurements in [72] and [76] in order to examine the performance of the two-dimensional model. The spatial correlation of both the measured and model received signals over distance were examined. The correlation was calculated separately for the early and late arriving paths. The pulse used to generate the received signals was a 500 ps Gaussian pulse. Figure 3.38 shows the results of this comparison. A good match between the model and the measured data was observed. The correlation is stronger in the first cluster because the variance of the AOAs is small. The second cluster, however, shows little spatial correlation since the paths come from 360°. Note that a similar model was proposed for 802.15.4a [77]. Specifically, the SV model was proposed for time-of-arrival modeling. Individual clusters are modeled as uniformly distributed, while the angular power spectrum is modeled using a Laplacian distribution.