Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
12.3 Error Probability in Delay- and Frequency-Dispersive
12.3.1 Physical Cause of Error Floors
In wireless propagation channels, transmission errors are caused not only by noise but also by signal distortions. These distortions are created on one hand by delay dispersion (i.e., echoes of the transmit signal arriving with different delays), and on the other hand by frequency dispersion (Doppler effect – i.e., signal components arriving with different Doppler shifts). For high data rates, delay dispersion is dominant; at low data rates, frequency dispersion is the main reason for signal distortion errors. In either of these cases, an increase in transmit power does not lead to a reduction of the BER; for that reason, these errors are often called error floor or irreducible errors. Of course, these errors can also be reduced or eliminated, but this has to be done by methods other than increasing power (e.g., equalization, diversity, etc.). In this section, we only treat the case when the receiver does not use any of these countermeasures, so that dispersion leads to increased error rates. Later chapters discuss in detail the fact that dispersion can actually be a benign effect if specific receiver structures are used.
Frequency Dispersion
We first consider errors due to frequency dispersion. For FSK, it is immediately obvious how frequency dispersion leads to errors: random Frequency Modulation (FM) (see Section 5.7.3) leads to a frequency shift of the received signal, and can push a bit over the decision boundary. Assume that a+1 was sent (i.e., the frequencyfc+fmod). Due to the random FM effect, the frequencyfc+ fmodfinstis received. If this is smaller thanfc, the receiver opts for a−1. Note thatinstantaneous frequency shifts can be significantly larger than the maximum Doppler frequency even though the
statistics of the random FM are determined by the Doppler spectrum of the channel. Consider the following equation for the instantaneous frequency:
finst(t )= Im
r∗(t )dr(t )dt
|r(t )|2 (12.88)
Obviously, this can become very large when the amplitude becomes very small. In other words, deep fading dips lead to large shifts in the instantaneous frequency, and thus higher error probability.
A somewhat different interpretation can be given for differential detection. As mentioned above, differential detection assumes that the channel does not change between two adjacent symbols.
However, if there is a finite Doppler, then the channeldoes change – remember that the Doppler spectrum gives a statistical description of channel changes. Thus, a nonzero Doppler effect implies a wrong reference phase for differential detection. If this effect is strong, it can lead to erroneous decisions. Also in this case it is true that channel changes are strongest near fading dips.6
For MSK with differential detection, Hirade et al. [1979] determined the BER due to the Doppler effect:
BERDoppler= 1
2(1−ξs(TB)) (12.89)
whereξs(t )is the normalized autocorrelation function of the channel (so thatξs(0)=1)– i.e., the Fourier transform of the normalized Doppler spectrum. For smallvmaxTB we then get a BER that is proportional to the squared magnitude of the product of Doppler shift and bit duration:
BERDoppler=1
2π2(vmaxTB)2 (12.90)
This basic functional relationship also holds for other Doppler spectra and modulation formats; it is only the proportionality constant that changes.
From this relationship, we find that errors due to frequency dispersion are mainly important for systems with a low data rate. For example, paging systems and sensor networks exhibit data rates on the order of 1 kbit/s, while Doppler frequencies can be up to a few hundred Hz. Error floors of 10−2 are thus easily possible. This has to be taken into account when designing the coding for such systems. For high-data-rate systems (which include almost all current cellular, cordless, and Wireless Local Area Network (WLAN) systems), errors due to frequency dispersion do not play a noticeable role.7Even for the Japanese JDC cellular system, which has a symbol duration of 50μs the BER due to frequency dispersion is only on the order of 10−4, which is negligible compared with errors due to noise.
Delay Dispersion
In contrast to frequency dispersion, delay dispersion has great importance for high-data-rate systems.
This becomes obvious when we remember that the errors in unequalized systems are determined by the ratio of symbol duration that is disturbed by InterSymbol Interference (ISI) to that of the undisturbed part of the symbol. The maximum excess delay of a channel impulse response is determined by the environment, and independent of the system; let us assume in the following a maximum excess delay of 1μs. In a system with a symbol duration of 20μs, the ISI can disturb 5% of each symbol, while it can disturb 20% if the symbol duration is 5μs.
6For general QAM, not only the reference phase but also the reference amplitude is relevant. However, in the following we will restrict our considerations to PSK and FSK, and thus ignore amplitude distortions.
7However, this does not mean that time variations in the channel are unimportant in such systems. Channel variations can also have an impact on coding, on the validity of channel estimation, etc.
Demodulation 241
Many theoretical and experimental investigations have shown that the error floor due to delay dispersion is given by the following equation:
BER=K Sτ
TB 2
(12.91) whereSτ is the rms delay spread of the channel (see Chapter 6). Just as for frequency dispersion, errors mainly occur near fading dips. Section 12.3.2 gives an interpretation of this fact in terms of group delay, which reaches its largest values near fading dips (see also Chapter 5).
Equation (12.91) is only valid if the maximum excess delay of the channel is much smaller than the symbol duration, and the channel is Rayleigh fading. The proportionality constantK depends on the modulation method, filtering at transmitter and receiver, the form of the average impulse response, and choice of the sampling instant, as we will discuss in the sections below.
Choice of the Sampling Instant In a flat-fading channel, the choice of sampling instant is obvious – sampling should always occur at those times where the SNR at the decision device is largest; this usually occurs either at the bit transitions or exactly in the middle between bit transitions; in either case we call this time henceforthts=0.
For channels with delay dispersion, the choice of sampling time is no longer that obvious.
Most theoretical derivations assume that eitherts=0 (i.e., sampling occurs at the minimum excess delay),8 or at theaverage mean delay. The latter actually is the optimum sampling time for some Power Delay Profiles (PDPs, see Chapter 6), as demonstrated in Figure 12.7.
0.0
0.0 0.25 0.5
Normalized sampling time ts/t2
BER
0.75 1.0
0.005 0.01 0.015 0.02
Figure 12.7 Dependence of delay-dispersion-induced error probability BER on choice of sampling instant in a two-spike channel.
Reproduced with permission from Molisch [2000]©Prentice Hall.
When the sampling instant is chosen adaptively, according to the instantaneous state of the chan- nel, the error floor can be decreased considerably, and – in unfiltered systems – even completely eliminated.
Impact of the Shape of the Power Delay Profile To a first approximation, only the rms delay spread determines the BER due to delay dispersion. A closer look reveals, however, that the actual
8Without restriction of generality, we assume thatτ0=0.
0.01
BER
10–5 10–4 10–3 10–2
0.1
two-spike channel rectangular PDP exponential PDP
Normalized delay spread St/TB
Figure 12.8 Impact of the shape of the PDP on the error floor due to delay dispersion. The modulation method is MSK with differential detection.
Reproduced with permission from Molisch [2000]©Prentice Hall.
shape of the PDP also has an impact. The variation of the BER for different PDPs (for equalSτ) is usually less than a factor of 2, and is thus often neglected. Figure 12.8 shows that a rectangular PDP leads to a slightly larger BER than a two-spike PDP; the difference is 75%.
An exponential PDP leads to an even larger error floor; however, for this shape, sampling at the average mean delay is not optimum.
Filtering Filtering at the transmitter and/or receiver also leads to signal distortion, and thus makes the signal more susceptible to errors by the additional distortions caused by the channel. The narrower the filtering, the larger the error floor. Figure 12.9 shows the effect of filtering on QAM with Raised Cosine (RC) filters.
The question naturally arises as to the optimum filter bandwidth. Very narrow filters (bandwidth on the order of the inverse symbol duration or less) lead to strong ISIs by themselves. Even if all decisions can be made correctly in the absence of further disturbances, such a filter makes the system more susceptible to channel-induced ISI and noise. For wide filters, a lot of noise can pass through the filter, which also leads to high error rates. The optimum filter bandwidth depends on the ratio of delay dispersion to noise. Figure 12.10 shows an example of such a tradeoff.
Modulation Method The modulation method also has an impact on the error floor: obviously, a modulation format is more sensitive to distortions by the channel the closer the signals are in the signal constellation diagram. QPSK shows a higher error floor than BPSK when the rms delay spread is normalized to the same symbol duration (see Figure 12.11). Higher order modulation formats fare better when equalbit durations are assumed.