By Bruce A. Fette

The following is excerpted from Chapter 5 of RF & Wireless Technologies by Bruce Fette. If you order a copy of this book before December 31, 2007 you can receive additional 20% off. Visit www.newnespress.com or call 1-800-545-2522 and use code 91137.

Part 1 introduces radio propagation.

We have seen that reflection of a signal from the ground has a significant effect on the strength of the received signal. The nature of short-range radio links, which are very often installed indoors and use omnidirectional antennas, makes them accessible to a multitude of reflected rays, from floors, ceilings, walls, and the various furnishings and people that are invariably present near the transmitter and receiver. Thus, the total signal strength at the receiver is the vector sum of not just two signals, but of many signals traveling over multiple paths.

In most cases indoors, there is no direct line-of-sight path, and all signals are the result of reflection, diffraction, and scattering. From the point of view of the receiver, there are several consequences of the multipath phenomena:

1. Variation of signal strength. Phase cancellation and strengthening of the resultant received signal cause an uncertainty in signal strength as the range changes, and even at a fixed range when there are changes in furnishings or movement of people. The receiver must be able to handle the considerable variations in signal strength.
2. Frequency distortion. If the bandwidth of the signal is wide enough so that its various frequency components have different phase shifts on the various signal paths, then the resultant signal amplitude and phase will be a function of sideband frequencies. This is called frequency selective fading.
3. Time delay spread. The differences in the path lengths of the various reflected signals cause a time delay spread between the shortest path and the longest path. The resulting distortion can be significant if the delay spread time is of the order of magnitude of the minimum pulse width contained in the transmitted digital signal. There is a close connection between frequency selective fading and time-delay distortion, since the shorter the pulses, the wider the signal bandwidth. Measurements in factories and other buildings have shown multipath delays ranging from 40 to 800 ns.
4. Fading. When the transmitter or receiver is in motion, or when the physical environment is changing (tree leaves fluttering in the wind, people moving around), there will be slow or rapid fading, which can contain amplitude and frequency distortion, and time delay fluctuations. The receiver AGC and demodulation circuits must deal properly with these effects.

Flat Fading
In many of the short-range radio applications covered in this chapter, the signal bandwidth is narrow and frequency distortion is negligible. The multipath effect in this case is classified as flat fading. In describing the variation of the resultant signal amplitude in a multipath environment, we distinguish two cases: (1) there is no line-of-sight path and the signal is the resultant of a large number of randomly distributed reflections; (2) the random reflections are superimposed on a signal over a dominant constant path, usually the line of sight.

Short-range radio systems that are installed indoors or outdoors in built-up areas are subject to multipath fading essentially of the first case. Our aim in this section is to determine the signal strength margin that is needed to ensure that reliable communication can take place at a given probability. While in many situations there will be a dominant signal path in addition to the multipath fading, restricting ourselves to an analysis of the case where all paths are the result of random reflections gives us an upper bound on the required margin.

Rayleigh Fading
The first case can be described by a received signal R (t), expressed as

where r and θ are random variables for the peak signal, or envelope, and phase. Their values may vary with time, when various reflecting objects are moving (people in a room, for example), or with changes in position of the transmitter or receiver that are small in respect to the distance between them. We are not dealing here with the large-scale path gain that is expressed in Eqs. (5.5) and (5.6). For simplicity, Eq. (5.7) shows a continuous wave (CW) signal as the modulation terms are not needed to describe the fading statistics. The envelope of the received signal, r, can be statistically described by the Rayleigh distribution whose probability density function is:

where σ² represents the variance of R(t) in Eq. (5.7), which is the average received signal power. This function is plotted in Figure 5.6. We normalized the curve with σ equal to 1. In this plot, the average value of the signal envelope, shown by a dotted vertical line, is 1.253.

Note that it is not the most probable value, which is 1 (σ). The area of the curve between any two values of signal strength r represents the probability that the signal strength will be in that range. The average for the Rayleigh distribution, which is not symmetric, does not divide the curve area in half. The parameter that does this is the median, which in this case equals 1.1774. There is a 50% probability that a signal will be below the median and 50% that it will be above.

5.6. Rayleigh probability density function.

As stated previously, the Rayleigh distribution is used to determine the signal margin required to give a desired communication reliability over a fading channel with no line of sight. The curve labeled "1 Channel" in Figure 5.7 is a cumulative distribution function with logarithmic axes. For any point on the curve, the probability of fading below the margin indicated on the abscissa is given as the ordinate. The curve is scaled such that "0 dB" signal margin represents the point where the received signal equals the mean power of the fading signal, 2, making the assumption that the received signal power with no fading equals the average power with fading. Some similar curves in the literature use the median power, or the power corresponding to the average envelope signal level, r_a, as the reference, "0 dB" value.

5.7.Fading Margins.

An example of using the curve is as follows. Say you require a communication reliability of 99%. Then the minimum usable signal level is that for which there is a 1% probability of fading below that level. On the curve, the margin corresponding to 1% is 20 dB. Thus, you need a signal strength 20 dB larger than the required signal if there was no fading.

Assume you calculated path loss and found that you need to transmit 4 mW to allow reception at the receiver’s sensitivity level. Then, to ensure that the signal will be received 99% of the time during fading, you’ll need 20 dB more power or 6 dBm (4 mW) plus 20 dB equals 26 dBm or 400 mW. If you don’t increase the power, you can expect loss of communication 63% of the time, corresponding to the "0 dB" margin point on the "1 Channel" curve of Figure 5.7.

The following table shows signal margins for different reliabilities.

Part 3 will cover Diversity Techniques

References
Gibson, J. D. (ed.), The Mobile Communications Handbook, CRC Press, Inc., 1996.
Rappaport, T. S., Wireless Communications, Principles and Practice, Prentice Hall, Upper Saddle River, NJ, 1996.
Spix, G. J., "Maxwell’s Electromagnetic Field Equations," unpublished tutorial, copyright 1995 http://www.connectos.com/spix/rd/gj/nme/maxwell.htm

Related Articles

• Test strategies for 2-by-2 MIMO in 802.11n systems
• NIST researchers tackle wireless interference problem
• Fundamentals of the 802.11 and Bluetooth protocols

Copyright: Printed with permission from Newnes, a division of Elsevier. Copyright 2008. "RF & Wireless Technologies" by Bruce A. Fette. For more information about this title and other similar books, please visit www.newnespress.com.

By Bruce A. Fette

It is fitting in a book about wireless communication to look at the phenomena that lets us transfer information from one point to another without any physical medium—the propagation of radio waves. If you want to design an efficient radio communication system, even for operation over relatively short distances, you should understand the behavior of the wireless channel in the various surroundings where this communication is to take place. While the use of "brute force" —increasing transmission power—could overcome inordinate path losses, limitations imposed on design by required battery life, or by regulatory authorities, make it imperative to develop and deploy short-range radio systems using solutions that a knowledge of radio propagation can give.

The overall behavior of radio waves is described by Maxwell’s equations. In 1873, the British physicist James Clerk Maxwell published his Treatise on Electricity and Magnetism in which he presented a set of equations that describe the nature of electromagnetic fields in terms of space and time. Heinrich Rudolph Hertz performed experiments to confirm Maxwell’s theory, which led to the development of wireless telegraph and radio. Maxwell’s equations form the basis for describing the propagation of radio waves in space, as well as the nature of varying electric and magnetic fields in conducting and insulating materials, and the flow of waves in waveguides.

From them, you can derive the skin effect equation and the electric and magnetic field relationships very close to antennas of all kinds. A number of computer programs on the market, based on the solution of Maxwell’s equations, help in the design of antennas, anticipate electromagnetic radiation problems from circuit board layouts, calculate the effectiveness of shielding, and perform accurate simulation of ultra-high-frequency and microwave circuits. While you don’t have to be an expert in Maxwell’s equations to use these programs (you do in order to write them!), having some familiarity with the equations may take the mystery out of the operation of the software and give an appreciation for its range of application and limitations.

Mechanisms of Radio Wave Propagation
Radio waves can propagate from transmitter to receiver in four ways: through ground waves, sky waves, free space waves, and open field waves. Ground waves exist only for vertical polarization, produced by vertical antennas, when the transmitting and receiving antennas are close to the surface of the earth. The transmitted radiation induces currents in the earth, and the waves travel over the earth’s surface, being attenuated according to the energy absorbed by the conducting earth. The reason that horizontal antennas are not effective for ground wave propagation is that the horizontal electric field that they create is short circuited by the earth.

Ground wave propagation is dominant only at relatively low frequencies, up to a few MHz, so it needn’t concern us here. Sky wave propagation is dependent on reflection from the ionosphere, a region of rarified air high above the earth’s surface that is ionized by sunlight (primarily ultraviolet radiation).

The ionosphere is responsible for long-distance communication in the high-frequency bands between 3 and 30 MHz. It is very dependent on time of day, season, longitude on the earth, and the multiyear cyclic production of sunspots on the sun. It makes possible long-range communication using very low power transmitters. Most short-range communication applications that we deal with in this chapter use VHF, UHF, and microwave bands, generally above 40 MHz. There are times when ionospheric reflection occurs at the low end of this range, and then sky wave propagation can be responsible for interference from signals originating hundreds of kilometers away. However, in general, sky wave propagation does not affect the short-range radio applications that we are interested in.

The most important propagation mechanism for short-range communication on the VHF and UHF bands is that which occurs in an open field, where the received signal is a vector sum of a direct line-of-sight signal and a signal from the same source that is reflected off the earth. Later we discuss the relationship between signal strength and range in line-of-sight and open field topographies.

The range of line-of-sight signals, when there are no reflections from the earth or ionosphere, is a function of the dispersion of the waves from the transmitter antenna. In this free-space case the signal strength decreases in inverse proportion to the distance away from the transmitter antenna. When the radiated power is known, the field strength is given by equation (5.1):

where P_t is the transmitted power, G_t is the antenna gain, and d is the distance. When P_t is in watts and d is in meters, E is volts/meter. To find the power at the receiver (P_r) when the power into the transmitter antenna is known, use (5.2):

G_t and G_r are the transmitter and receiver antenna gains, and λ is the wavelength.

Range can be calculated on this basis at high UHF and microwave frequencies when high-gain antennas are used, located many wavelengths above the ground. Signal strength between the earth and a satellite, and between satellites, also follows the inverse distance law, but this case isn’t in the category of short-range communication! At microwave frequencies, signal strength is also reduced by atmospheric absorption caused by water vapor and other gases that constitute the air.

Open Field Propagation
Although the formulas in the previous section are useful in some circumstances, the actual range of a VHF or UHF signal is affected by reflections from the ground and surrounding objects. The path lengths of the reflected signals differ from that of the line-of-sight signal, so the receiver sees a combined signal with components having different amplitudes and phases.

The reflection causes a phase reversal. A reflected signal having a path length exceeding the line-of-sight distance by exactly the signal wavelength or a multiple of it will almost cancel completely the desired signal ("almost" because its amplitude will be slightly less than the direct signal amplitude). On the other hand, if the path length of the reflected signal differs exactly by an odd multiple of half the wavelength, the total signal will be strengthened by "almost" two times the free space direct signal.

In an open field with fl at terrain there will be no reflections except the unavoidable one from the ground. It is instructive and useful to examine in depth the field strength versus distance in this case. The mathematical details are given in the Mathcad worksheet "Open Field Range."

In Figure 5.1 we see transmitter and receiver antennas separated by distance d and situated at heights h₁ and h₂. Using trigonometry, we can find the line of sight and reflected signal path lengths d₁ and d₂. Just as in optics, the angle of incidence equals the angle of reflection θ. We get the relative strength of the direct signal and reflected signal using the inverse path length relationship. If the ground were a perfect mirror, the relative reflected signal strength would exactly equal the inverse of d₂. In this case, the reflected signal phase would shift 180 degrees at the point of reflection. However, the ground is not a perfect reflector. Its characteristics as a reflector depend on its conductivity, permittivity, the polarization of the signal, and its angle of incidence.

5.1. Open field signal paths

In the Mathcad worksheet we have accounted for polarization, angle of incidence, and permittivity to find the reflection coefficient, which approaches -1 as the distance from the transmitter increases. The signals reaching the receiver are represented as complex numbers, since they have both phase and amplitude. The phase is found by subtracting the largest interval of whole wavelength multiples from the total path length and multiplying the remaining fraction of a wavelength by 2π radians, or 360 degrees.

Figure 5.2 gives a plot of relative open field signal strength versus distance using the following parameters :
Polarity: horizontal
Frequency: 300 MHz
Antenna heights: both 3 meters
Relative ground permittivity: 15

5.2. Field strength versus range at 300 MHz

Also shown is a plot of free space field strength versus distance (dotted line). In both plots, signal strength is referenced to the free space field strength at a range of 3 meters. Notice in Figure 5.2 that, up to a range of around 50 meters, there are several sharp depressions of field strength, but the signal strength is mostly higher than it would be in free space. Beyond 100 meters, signal strength decreases more rapidly than for the free space model. Whereas there is an inverse distance law for free space, in the open field beyond 100 meters (for these parameters) the signal strength follows an inverse square law. Increasing the antenna heights extends the distance at which the inverse square law starts to take effect. This distance, dm, can be approximated by

where h₁ and h₂ are the transmitting and receiving antenna heights above ground and λ is the wavelength, all in the same units as the distance d_m. In plotting Figure 5.2, we assumed horizontal polarization. Both antenna heights, h₁ and h₂, are 3 meters. When vertical polarization is used, the extreme local variations of signal strengths up to around 50 meters are reduced because the ground reflection coefficient is less at larger reflection angles. However, for both polarizations, the inverse square law comes into effect at approximately the same distance. This distance in Figure 5.2 where λ is 1 meter is, from equation (5.3): d_m _ (12 _ 3 _ 3)/ λ _ 108 meters. In Figure 5.2 we see that this is approximately the distance where the open-field field strength falls below the free-space field strength.

Part 2 will cover Multipath Phenomena
Part 3 will cover Diversity Techniques

References
Gibson, J. D. (ed.), The Mobile Communications Handbook, CRC Press, Inc., 1996.
Rappaport, T. S., Wireless Communications, Principles and Practice, Prentice Hall, Upper Saddle River, NJ, 1996.
Spix, G. J., "Maxwell’s Electromagnetic Field Equations," unpublished tutorial, copyright 1995 (http://www.connectos.com/spix/rd/gj/nme/maxwell.htm).