MODULATION PERFORMANCE IN FADING AND MULTIPATH CHANNELS
(i) Indication of Performance
As discussed earlier, the mobile radio channel is characterized by various impairments such as fading, multipath, and Doppler spread. In order to study the effectiveness of any modulation scheme in a mobile radio environment, it is required to evaluate the performance of the modulation scheme over such channel conditions. Although bit error rate (BER) evaluation gives a good indication of the performance of a particular modulation scheme, it does not provide information about the type of errors. For example, it does not give incidents of bursty errors. In a fading mobile radio channel, it is likely that a transmitted signal will suffer deep fades which can lead to outage or a complete loss of the signal.
(ii) Probability of Outage
Evaluating the probability of outage is another means to judge the effectiveness of the signaling scheme in a mobile radio channel. An outage event is specified by a specific number of bit errors occurring in a given transmission. Bit error rates and probability of outage for various modulation schemes under various types of channel impairments can be evaluated either through analytical techniques or through simulations. While simple analytical techniques for computing bit error rates in slow flat-fading channels exist, performance evaluation in frequency selective channels and computation of outage probabilities are often made through computer simulations. Computer simulations are based on convolving the input bit stream with a suitable channel impulse response model and counting the bit errors at the output of the receiver decision circuit.
Before a study of the performance of various modulation schemes in multipath and fading channels is made, it is imperative that a thorough understanding of the channel characteristics be obtained.
11.8 PERFORMANCE OF DIGITAL MODULATION IN SLOW FLAT-FADING CHANNELS
(i) Received Signal
As discussed earlier, flat-fading channels cause a multiplicative (gain) variation in the transmitted signal s (t). Since slow flat-fading channels change much slower that the applied modulation, it can be assumed that the attenuation and phase shift of the signal constant over at least one symbol interval. Therefore, the received signal r (t) may be expressed as
r(t) = α(t) exp (- jq(t) s(t) + n(t) 0 ≤ t ≤ T …(11.31)
where α (t) is the gain of the channel, q (t) is the phase shift of the channel, and n(t) is additive Gaussian noise.
Depending on whether it is possible to make an accurate estimate of the phase q(t), coherent or noncoherent matched filter detection may be employed at the receiver.
(ii) Probability of Error
To evaluate the probability of error of any digital modulation scheme in a slow flat-fading channel, one must average the probability of error of the particular modulation in AWGN channels over the possible ranges of signal strength due to fading. In other words, the probability of error in AWGN channels is viewed as a conditional error probability, where the condition is that α is fixed. Hence, the probability of error in slow flat-fading channels can be obtained by averaging the error in AWGN channels over the fading probability density function. In doing so, the probability of error in a slow flat-fading channel can be evaluated as under:
(X) p (X) dX …(11.32)
where Pc (X) is the probability of error for an arbitrary modulation at a specific value of signal-to-noise ratio X, X = and p (X) is the probability density function of X due to the No Boling channel. Eb, and N0 are constants that represent the average energy per bit and noise power density in a non-fading AWGN channel, and the random variable a2 is used to represent instantaneous power values of the channel, with respect to the non-fading . It is convenient to assume is one, for a unity gain fading channel. Then, p (X) can simply be viewed as the distribution of the instantaneous value of in a fading channel, and Pe (X) can be seen to be the conditional probability of bit errors for a given value of the random due to fading.
(iii) Rayleigh Fading Channel Case
For Rayleigh fading channels, the fading amplitude α has a Rayleigh distribution, so the fading power α2 and consequently X have a chi-square distribution with two degrees of freedom. Therefore, we have
p(X) = exp X > 0 …(11.33)
where Γ = is the average value of the signal-to-noise ratio. For = 1, note that Γ corresponds to the average for the fading channel.
By using Equation (11.22) and the probability of error of a particular modulation scheme in AWGN, the probability of error in a slow flat-fading channel can be evaluated. It can be shown that for coherent binary PSK and coherent binary FSK, Equation (11.21) evaluates to
Pe , PSK = (coherent binary PSK) …(11.34)
Pe , FSK = (coherent binary FSK) …(11.35)
(iv) Average Error Probability
It can also be shown that the average error probability of DPSK and orthogonal noncoherent FSK in a slow, flat, Rayleigh fading channel are given by
Pe, DPSK = (differential binary PSK) …(11.36)
Pe, DPSK = (noncoherent orthogonal binary FSK) …(11.37)
Figure 11.6 illustrates how the BER for various modulations changes as a function of in a Rayleigh flat-fading environment. The figure was produced using simulation instead of analysis, but agrees closely with equations (11.34) to (11.37).
For large values of (i.e., large values of X), the error probability equations may be simplified as
Pe, PSK = (coherent binary PSK) …(11.38)
Pe, FSK = (coherent FSK) …(11.39)
Pe, DPSK = (differential PSK) …(11.40)
Pe, NCFSK = (noncoherent orthogonal binary FSK) …(11.41)
FIGURE 11.10 Bit error rate performance of binary modulation schemes in a Rayleigh flat-fading channel as compared to a typical performance clime in AWGN
(v) BER for GMSK
For GMSK, the expression for BER in Rayleigh fading channel as under:
Pe, GSMK = (coherent GMSK) …(11.42)
As seen from Equations (11.39) to (11.43), for lower error rates, all five modulation techniques exhibit an inverse algebraic relation between error rate and mean SNR. This is in contrast with the exponential between error rate and SNR in an AWGN channel. According to these results, it is seen that operating at BERs of 10-3 to 10-6 requires roughly a 30 dB to 60 dB mean SNR. This is significantly larger than that required when operating over a nonfading Gaussian noise channel (20 dB to 50 dB more link is required). However, it can easily be shown that the poor error performance is due to the non-zero probability of very deep fades, when the instantaneous BER can become as low as 0.5. Significant improvement in BER can be achieved by using efficient techniques such as diversity or error control coding to totally avoid the probability of deep fades.
MODULATION PERFORMANCE IN FADING AND MULTIPATH CHANNELS