# Coverage prediction for digital mobile systems (Part 2)

Propagation as a channel filter An instructive and useful way to look at the propagation mechanism is to consider it as a filter, and the propagation path itself as a channel containing this filter, as illustrated in Figure 1 below left. The notations h(t) and H(f) shown in Figure 1 are standard engineering terms for the transfer function of the filter. The transfer function is simply a way of describing what happens to the signal as it passes through the filter.

The propagation channel filter transfer function certainly has attenuation (the familiar path loss), but it also has other characteristics that can have important effects on the signal that is detected at the receiver.

How do we find the other characteristics of this filter? One approach is to find all the ways the signal can travel from the transmitter to the receiver, rather than just assuming it gets there via a single path as discussed in Part 1 of this article. If we take into account all the ways the signal arrives at the receiver, we have completely described the filter transfer function. An approach to the problem is the multiray or "ray-tracing" method.1 The multiray concept is illustrated in Figure 2 below right. Signals leaving the transmitter encounter a wide variety of objects in the propagation environment including buildings, mountains, the ground and vehicles. The signals can "bounce" (reflect and diffract) off these objects and get to the receiver via many paths. This is multipath propagation, and it causes the familiar signal fading at the receiver which every mobile radio engineer has observed.

Using some mathematics, we can develop a way to use these rays to find the transfer function, h(t), of the channel filter. First, let's look at the case for the single path to the receiver. In this situation we have only path loss so we can write a simple equation for the signal strength at the receiver. Under these circumstances, the signal at the receive antenna terminals, Er, would be the same as the signal at the transmitter, except weaker. This can be written as shown in Equation 1 [See sidebar on page 42.], where Er is the (complex) electric field voltage or magnetic field current at the receive antenna, Et is the magnitude of the transmitted signal (voltage or current), v is the carrier frequency in radians, and t is time. The multiplicative factor, A, is the propagation loss, while u is some phase delay or phase shift introduced by the channel. The expressionexp(2jvt1u) is just a convenient way to describe the transmitted carrier wave in this case. For simplicity only the electric field will be represented in the following equations with the understanding that there is an associated magnetic field.

If the channel is now considered as a filter with some lowpass impulse response, that impulse response would be given by Equation 2, where the "d(t2t)" means there is impulse in the channel response at time t5t. A sinewave signal at frequency v leaving the transmit antenna would arrive at the receiver reduced in amplitude by factor A, shifted in phase by u and delayed by t seconds where the delay is a direct function of the path length from the transmitter to the receiver. Such a model of the transmission channel is applicable for free-space propagation conditions where the signal energy arrives at the receiver directly (via one path) from the transmitter.

If the channel consisted of two transmission paths for the transmit energy to arrive at the receiver (for example, with the addition of a single ground reflection), the channel impulse response would be the sum of the effect of the two paths as shown in Equation 3.

This is the impulse response of the so-called "two-ray" channel model. If we have N possible transmission paths, h(t) transforms as shown in Equation 4. This is the channel impulse response to the receiver at a particular coordinate point in space, p2(x2,y2,z2), from the transmitter located at some other coordinate point, p1(x1,y1,z1). The more general impulse response as a function of this geometry can thus be written as shown in Equation 5, where the number, amplitude, phase and time delay of the components of the summation are a function of the location of the transmit and receive antenna points in the propagation space.

The channel impulse response given by Equation 5 is for a single static point in space. For mobile communication, the receiver is often moving, and that motion can affect the phase relationship of the components of Equation 5 in a way that may be important to digital data being transmitted over the channel. This motion will result in a frequency or Doppler shift of the received signal that will be a function of speed and direction of motion, and the angle of arrival of the signal energy. Equation 5 can be modified to include Doppler shift and thus account for this motion as shown in Equation 6, where Dun is a phase displacement due to the motion.

It should be kept in mind that this is motion of either the transmitter or receiver relative to every other element in the propagation environment. The transmitter or receiver may be fixed, but a signal from a moving reflection source (such as a moving bus) may result in a nonzero Dun for a particular component of Equation 6. For a mobile receiver, Dun = ( 2pvt ) cos ( wn – wv ) l where wn is the arrival angle of the nth ray component, v is the speed of motion, wv is the direction of motion, and l is the wavelength.2

In general, the amplitudes, An, and phase shifts, un, will be functions of the carrier frequency, v, because they are controlled by the interaction of the transmitted energy with the features of the propagation environment. Inserting the frequency dependence into Equation 5 produces Equation 7.

We can now apply this approach to a practical situation. Figure 3 on page 43 shows an overhead drawing of a downtown area with several buildings. The multiple rays from the transmitter to the receiver are shown reflecting and diffracting off the buildings. (The author's article describing the methods involved in such "ray-tracing" is listed in the references.) If we were to plot the channel filter impulse response for this case, as given by Equation 7, we would have the graph shown in Figure 4 at the right. This comprehensive description of the propagation channel filter (often called a power delay profile) includes not only path loss information but several other characteristics of the propagation channel. Now that we have the filter response, how do we use it to predict coverage or performance in a digital system?

Digital errors due to multipath Errors in a digital system occur when the receiver mistakenly interprets a 0 for a 1 or vice versa. The receiver needs to make a decision about which value it has received. If the only signal the receiver has to work with is a perfect replica of the transmitted signal, it could make this decision flawlessly every time, and there would be no errors. But it doesn't have a perfect replica to work with. First, and most common, there is noise introduced by the receiver and, perhaps, external sources. As we've determined from Equation 7 and Figure 3, there are also a lot of other signals arriving at the receiver that can confuse the detection process and cause an error. The effect of noise on the error rate performance is well-known and exhaustively treated in communication engineering textbooks. We will thus focus on the errors due to the multipath signals.

Let's say we want to transmit a series of digital pulses or symbols over a propagation channel like that shown in Figure 4. We will assume our digital pulse has been smoothed off with a filter at the transmitter so it uses less bandwidth. When it gets to the receiver it looks like the first pulse waveform in Figure 5 above. There is a strong peak in the signal that comes from the strongest ray arriving at the receiver, but there are also other peaks in the signal due to reflected signals arriving at some later time.

Now, what happens when we send a second pulse, and a third and a fourth? As Figure 5 shows, when the receiver tries to detect the fourth pulse, the decision is corrupted by reflected energy from pulses transmitted earlier. This is known as inter-symbol interference (ISI). For a given data rate and propagation channel response, it can result in error rates that make the signal totally unusable even if the average signal power is more than adequate to overcome errors due to receiver noise. You could raise your transmitter power by a factor of 100 and your received data would still be full of errors!

Because increasing transmitter power doesn't reduce these errors, the unfortunate misnomer describing them as "irreducible errors" is sometimes used. If fact, using various techniques in the receiver, such as channel equalizers, many multipath-related effects can be reduced before the signal reaches the decision-making process. A common example is the European digital cellular system called GSM, which has a data rate of about 270kbps and uses an equalizer to improve performance in multipath conditions. Even more sophisticated are CDMA cellular systems, that use so-called RAKE receivers where the multipath energy is actually constructively combined to improve overall fading performance. On the other end of the scale is a European short-range telephone technology called DECT that has a data rate of 1Mbps but uses no equalizer. DECT systems are known for having performance problems in multipath urban areas even when the signal strength at the receiver is well above the level needed for acceptable error rates if only noise were present.

An approach to explicitly calculating error rates for this type of inter-symbol interference requires detailed knowledge of the multipath components. For this reason, some simplified ways of describing the degree of multipath in the channel have been devised. One of the most common is RMS delay spread. The RMS delay spread, st, is a statistical measure of the amount of time dispersion, or spreading, found in the multipath signal. Formally, it is calculated as the second central moment of a power delay profile such as that illustrated in Figure 3.

The RMS delay spread, st, is calculated as shown in Equations 8, 9 and 10. An is the amplitude of ray n, tn is the time delay to ray n, and N is the total number of rays.

Using a single number like RMS delay spread to find error rates is often misleading, because the statistical averaging process glosses over important details about when the multipath energy arrives and what its magnitude and phase are. These factors cannot be addressed looking at RMS delay spread alone. As an example, Figures 6 and 7 above show two simple power delay profiles with two rays each. One has a strong echo delayed a short period of time after the main signal; the other has a much weaker multipath echo delayed a much longer period of time. Both channel responses could have exactly the same RMS delay spread value, but by using comprehensive analysis it can be shown that the channel in Figure 6 will produce a high error rate dominated by multipath, and the channel in Figure 7 will have a much lower error rate controlled primarily by noise. For this reason, convenient measures of channel time dispersion like RMS delay spread should be used with caution and recognized for the significant approximations they represent.

As mentioned above, the data rate is important in determining whether multipath will cause errors. Reviewing Figure 5, if we envisioned a lower data rate (much greater pulse width), then all of the multipath signals would have come and gone before the decision time for the next pulse, and no errors would occur due to multipath. Depending on the data rate and the time delays for the multipath, there may or may not be errors due to this effect. For reflections along city streets, multipath delays on the order of several hundred nanoseconds may occur. With longer range systems, reflection paths from mountains may result in path delays of several microseconds. Indoor wireless systems where multipath comes from reflections off relatively closely spaced interior walls, delay times on the order to 10 to 100 nanoseconds are found. Using RMS delay spread (a quantity in time), a common rule-of-thumb says that if the RMS delay spread is greater than 1/5 the time between the digital symbols or pulses, then errors due to multipath may be significant if no equalizer or other correction device is used in the receiver.

Random FM and doppler shift errors Again referring to Figure 5, if we envision even lower data rates where all multipath reflections have died out, you might think that no errors would be introduced in the system due to the propagation channel filter. If the receiver is stationary (and the propagation environment is unchanging), that's true. But if the receiver is moving, another kind of error can occur that actually gets worse as the data rate is lowered.

Errors due to random FM arise in narrowband transmissions due to the phase shift of the carrier from one symbol to the next. If the data transmission rate is high, the amount of phase change that is possible from one symbol to the next, even with high mobile speeds, is still very small, so that errors due to random FM are not important compared to errors from amplitude fading in noise and errors due to intersymbol interference. For coherent detection, depending on the receiver carrier reference recovery techniques, the random phase changes can be tracked so that errors due to random FM are minimized or reduced to zero. Random FM errors are therefore of primary concern for mobile systems with relatively low data rates that employ differential modulation and detection techniques.

The usual analysis of random FM errors assumes that the signal is arriving from a single direction and that the mobile is moving in a direction w relative to the arrival angle of the signal. Under these conditions, the Doppler frequency fd (frequency shift) for a given mobile speed is

fd = v cosw l

where v is the speed of the mobile in meters per second. In determining error rates, the traditional assumption is that the worst case Doppler frequency,

fd = v l

occurs and leads to an error rate that will depend on the modulation type. In a complex environment with energy arriving from many different directions, as illustrated in Figure 3, the rate of phase change, and hence frequency deviation, can vary considerably. In fact, a phase change of 180° can occur in deep fades over an arbitrarily short distance increment resulting in a possibility of infinite frequency deviation.

Rather than make assumptions about random FM deviation, the site-specific physical ray-tracing channel model provides detailed information about the arrival angles of signals at the mobile unit. The specific nature of the phase shift may be estimated and used to find fd. To simplify the analysis, an average value of fd can be found and used in a way similar to RMS delay spread to estimate when errors due to random FM become important. For this purpose the term fdTs is used, where Ts is the duration of the transmitted symbol. The term fdTs is an angle error and has the units of cycles. For example, an fdTs of 0.1 represents a phase error introduced by the channel of 368.

When considered together with the errors due to intersymbol interference, the overall error rate picture looks something the drawing in Figure 8 on page 48. For a given signal-to-noise ratio, modulation type, propagation channel and mobile speed, as the data rate increases, the errors increase due to ISI. As the data rate decreases, the errors increase due to random FM. In between, the error rate is largely a function of noise.

Conclusions The contrast between empirical measurement-based models, as discussed in Part 1 of this article (January 1997), and physical propagation models for predicting the coverage of digital mobile systems has been presented. While empirical models are simple, they do not explicitly take into account many important elements of the propagation environment, and they do not currently include information about channel delay spread or random FM, which are important in predicting error rates in many kinds of digital systems.

Physical propagation models in the form of ray-tracing offer a means of acquiring the necessary propagation information for predicting the performance of digital systems in any given environment. However, to make predictions, physical models rely on detailed descriptions of the environment and require relatively intensive calculations. Ever-increasing computer processor power and storage space for data make the calculations required by physical models less burdensome. Physical modeling such as ray-tracing, therefore, offers the best way for predicting coverage and error-rate performance in current and future digital systems. A software prediction tool, called EDX SignalPro, developed by our company, offers physical modeling calculations for such digital system design.

The digital system performance results presented here have generally assumed that no techniques are used in the receiver or system to combat the linear distortions caused by the propagation channel. With modern receiver design and the increasing economic feasibility of using sophisticated digital signal processing (DSP) chips in handsets, long-known techniques for countering channel impairments can now be widely employed; nevertheless, it will remain important to model the channel accurately to gain insight into the magnitude of channel impairments, which must be addressed by hardware and system designs, and to assess the overall efficacy of those designs.

References Anderson, H.R. "A Ray-tracing Propagation Model for Digital Broadcast Systems in Urban Areas," IEEE Transactions on Broadcasting, Sept. 1993. Anderson, H.R. "Site-specific BER Analysis in Frequency-selective Channels Using a Ray-tracing Propagation Model," Proceedings of the 1994 Globecom Conference, San Francisco, Dec. 1994. Balanis, C.A. Advanced Engineering Electromagnetics. John Wiley, New York, NY, 1989. Code of Federal Regulations Title 47, FCC Rules, Part 73.313, U.S. Government Printing Office. Hata, M. "Empirical Formula for Propagation Loss in Land Mobile Radio Services", IEEE Transactions on Vehicular Technology, Sept. 1981. Jakes, W.C. Microwave Mobile Communications. IEEE Press, Piscataway, NJ, 1994 (republished). Okumura, Y. et al. "Field Strength and its Variability in VHF and UHF Land-mobile Radio-service," Rev. Elec. Commun. Lab., Sept.-Oct. 1968. VHF and UHF propagation curves for the frequency range 30 MHz and 1000 MHz. ITU-R, Recommendation 370-6, 1994 PN Series Volume, Propagation in Non-Ionizing Media, 1994.