Signal-fading phenomena can drastically affect the performance of a terrestrial communications system. Often caused by multipath conditions, fading can degrade the bit-error-rate (BER) performance of a digital communications system, resulting in lost data or dropped calls in a cellular system. A key to preventing loss of radio performance is to understand the nature of multipath fading phenomena in terrestrial communications systems and how to anticipate when such phenomena may be a concern.

Fading can occur in many forms, including a phenomenon called flat fading. In flat fading, the same degree of fading takes place for all of the frequency components transmitted through a radio channel and within the channel bandwidth. That is, all the frequency components of the transmitted signal rise and fall in unison.

In contrast, frequency-selective fading causes different frequencies of an input signal to be attenuated and phase shifted differently in a channel. Frequently, channels experiencing frequency-selective fading may require an equalizer to achieve the desired performance. Frequency-selective fading gives rise to notches in the frequency response of the channel. Equalization techniques attempt to restore the memory less (flat fading) nature of the channels. With the proper equalization, it is possible to transmit at higher data rates before the onset of intersymbol interference (ISI) is apparent in the time domain.

Frequency-selective fading can be viewed in the frequency domain, although in the time domain, it is called multipath delay spread. The simplest measure of multipath is the overall time span of path delays from the first pulse to arrive at the receiver (the bona fide direct signal) to the last pulse to arrive at the receiver (the multipath echo). This applies to a particular threshold (not thermal) above which the last echo is significant (Fig. 1). This spread has also been referred to as the excess delay spread. The two parameters most often used as statistical designators of the multipath channels are the average time delay and the delay spread. The first moment of the time-delay profile is the mean delay, and the square root of its central moment (about the mean) is defined as the delay spread. The delay spread of different systems may be comparable, but the activity of signal components (the number of echoes and their amplitudes) may be much different. The first moment of a delay spread is:

and the second central moment (variance) is given as:

Therefore, the root-mean-square (RMS) delay spread is:

Figure 1a offers a simple example of delay spread. It shows the pulse for the direct path and for the indirect path (multipath). The first pulse arrives at τ_{o} after the transmitted pulse and the echo at τ_{2}. The difference in time is the excess delay spread. In an actual system, there are multiple paths and in the interval of the delay spread there is a raft of echo pulses, and not just two discrete paths as shown in the figure. This may also be a continuum and may not be uniform, but more concentrated near τ_{o}, and falling off in amplitude as the signals travel further in time. Of course, it is possible that some farther-out pulses may be bigger than the original pulse caused, for example, by specular reflections (forward scatter).

Frequency fading due to time dispersion is also known as ISI. Delay spread in time causes ISI, in which there is time dispersion of the signal. The time dispersion sets a limit on the speed at which modulated symbols can be transmitted in the channel. Because of the dispersion, symbols can collide and result in distorted output data. In this type of fading, the differences in delay between the various reflections arriving at the receiver can be a significant fraction of the data symbol interval, establishing conditions for overlapping symbols.

If the time-delay spread equals zero, there is no selective fading. As a rule of thumb, a channel can be considered flat when τ_{rms}/T is less than 0.1 where T is the symbol period. Statistical analysis can determine the range of frequencies over which a channel can be considered flat—that is, all received signal levels are approximately comparable in magnitude and the phases are approximately in unison (do not negate each other). Using the statistical approach, the coherence bandwidth is defined as the bandwidth over which the fading statistics are correlated to better than 90 percent (Fig. 2). Clearly, if two frequencies do not fall within the coherence bandwidth, they will fade independently.

Consider the multipath model depicted in Fig. 1b. It shows a single reflector of the transmitted signal. A pulse is propagated toward the receiver via a direct path and a single bounce path. The figure portrays simple conditions; in fact, the echoes are numerous. The delay is indicated by τ_{2} relative to the direct path where τ_{o} is assumed to be the epoch point. Clearly, the direct path delay is shorter than the echo delay. At the receiver, the two signals are combined, with the differential delay being τ_{Δ} seconds in length. From signal theory, this is analogous to a delay-line canceller (Fig. 2).

An ideal delay line will produce an output signal, which is an exact replica of the input signal, but delayed in time by τ_{2}. A network which had the property that e(t)_{out} = e(t)_{in} + ein(t−τ_{1}) would have to possess exp(−jτω) as a transcendental transfer function and the delayed signal input at t = τ_{2} in response to a input at τ_{o} (epoch). The similarity to the propagation model illustrated in Fig. 1b should be apparent.

To understand the nature of signal delays for a given network, it is necessary to find the frequency-domain response (the frequency selectivity) of that network. Since the delayed pulse is assumed to fall with the direct pulse upon their arrival at the receiver, the end result will be frequency-selective fading. Mathematically, taking the Fourier transform of both sides of the system function, H(jω), from Eq. 1:

If the multipath signal, a, is equal to a(t), the multipath function is time dependent, as would be prevalent for a mobile communications receiver. The system function H(jω) is complex, but it is the amplitude (real) part that is of interest, and the frequency-domain characteristic of the network follow from analysis. Applying Euler's identity for exp(−−jωt) results in:

For a =1 (Eq. 3),

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Equation 3 is the amplitude frequency response of the channel, which can be plotted as shown in Fig. 3. Equation 3 goes to zero ("fade notches") when the value of τ is:

For example at a frequency of τ = 0.5f, there will be a null in the transmission, and the receiver will exhibit zero output (frequency-selective fading). Other applicable frequencies will have various degrees of fading.

Coherence bandwidth has been defined as the bandwidth over which the fading statistics are correlated to better than 90 percent (Fig. 3). Going back to Eq. 3 for values of a < 1, there is no complete cancellation at the notches. The transfer output is undulatory (dotted curve in Fig. 3). As the multipath content becomes highly attenuated, the multipath approaches zero and receiver input reverts to a "flat fade" condition (|H(ω)|=1). Clearly, the output is equal to the transmitted signal, but attenuated by the free-space loss.

The multipath null locations are also a function of frequency and dynamic differential delays between the direct and multipath signal components. The time τ required for a radio wave to travel a given distance d in free space is τ = d/c, where c is the speed of the electromagnetic (EM) wave (in a vacuum). The delay time of the multipath signal after the arrival of the direct signal is found from the difference between the direct and multipath signal distances, τ = (d_{m} − d_{d})/c, where τ is the differential delay.

The multipath model described here is rather simplistic since only a single discrete signal path has been considered. Under real-world conditions, a network would exhibit a multitude of discrete multipath signals and possibly even a continuum of multipath signals (Fig. 1). It should be noted that the echo signals need not be smaller than the direct signal wave. If the difference in path length is large, the fading characteristics will vary greatly even with small frequency separations.

Frequency-selectivity fading in the time domain is manifested as ISI or smearing in the time domain. Multipath is not always a bad thing, since there would be no cellular industry without it. A multiplicity of randomly reflected and diffracted signals, reaching the cellular handset, with random amplitude and uniform phase distribution, assumes Rayleigh statistics. For the rare occasion when there is a line-of-sight signal path to the base station, the statistics are Ricean in nature.

Combating time-dependent fading in a dynamic channel could be daunting. One solution is the use of adaptive equalization.^{6} In this approach, an adaptive transversal equalizer filter is used to track the fading multipath signals. Another approach is the use of multicarrier modulation, where the spectrum of the frequency selective channel is divided into a large number of parallel, independent, and approximately flat subchannels. One example of this is orthogonal frequency division multiplexing (OFDM). OFDM is a multitone system using a multiplicity of juxtaposed tones, transmitted at some rate R, where M independent tones in parallel will result in information transmitted at a rate of MR. These contiguous tones can combat ISI since the fading is flat for each tone over the entire ensemble of (independent) tones. In this case, the symbol transmission rate (R) is much smaller than the coherence bandwidth of the channel (the tones). Since a condition results in which the narrowband tones are subject to flat fading, there is therefore no ISI.

Spread-spectrum signals can also effectively minimize the effects of multipath distortion. For example, Fig. 4 shows two received signals: one is a direct signal while the other is a multipath signal. If the multipath signal is delayed by one chip, it will be rejected by the receiver since it is no longer in sync with the timing of the reference source in the receiver. It is thus seen as "hash" by the receiver.

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