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6.18 Digital Communication System Properties24 Results from Section 6.17 reveal several properties about digital communication systems. • As the received signal becomes increasingly noisy, whether due to increased distance from the transmitter (smaller α) or to increased noise in the channel (larger N0 ), the probability the receiver makes an error approaches 1/2. In such situations, the receiver performs only slightly better than the “receiver” that ignores what was transmitted and merely guesses what bit was transmitted. Consequently, it becomes almost impossible to communicate information when digital channels become noisy. • As the signal-to-noise ratio increases, performance gains–smaller probability of error pe – can be easily obtained. At a signal-to-noise ratio of 12 dB, the probability the receiver makes an error equals 10−8 . In words, one out of one hundred million bits will, on the average, be in error. • Once the signal-to-noise ratio exceeds about 5 dB, the error probability decreases dramatically. Adding 1 dB improvement in signal-to-noise ratio can result in a factor of ten smaller pe . • Signal set choice can make a significant difference in performance. All BPSK signal sets, baseband or modulated, yield the same performance for the same bit energy. The BPSK signal set does perform much better than the FSK signal set once the signal-to-noise ratio exceeds about 5 dB. Exercise 6.20 (Solution on p. 256.) Derive the expression for the probability of error that would result if the FSK signal set were used. The matched-filter receiver provides impressive performance once adequate signal-to-noise ratios occur. You might wonder whether another receiver might be better. The answer is that the matched-filter receiver is optimal: No other receiver can provide a smaller probability of error than the matched filter regardless of the SNR. Furthermore, no signal set can provide better performance than the BPSK signal set, where the signal representing a bit is the negative of the signal representing the other bit. The reason for this result rests in the dependence of probability of error pe on the difference between the noise-free integrator outputs: For a given Eb , no other signal set provides a greater difference. How small should the error probability be? Out of N transmitted bits, on the average N pe bits will be received in error. Do note the phrase “on the average” here: Errors occur randomly because of the noise introduced by the channel, and we can only predict the probability of occurrence. Since bits are transmitted at a rate R, errors occur at an average frequency of Rpe . Suppose the error probability is an impressively small number like 10−6 . Data on a computer network like Ethernet is transmitted at a rate R = 100Mbps, which means that errors would occur at a rate of roughly 100 per second. This error rate is very high, requiring a much smaller pe to achieve a more acceptable average occurrence rate for errors occurring. Because Ethernet is a wireline channel, which means the channel noise is small and the attenuation low, obtaining very small error probabilities is not difficult. We do have some tricks up our sleeves, however, that can essentially reduce the error rate to zero without resorting to expending a large amount of energy at the transmitter. We need to understand digital channels and what implications Shannon’s Noisy Channel Coding Theorem has for them.
6.19 Digital Channels25 Let’s review how digital communication systems work within the Fundamental Model of Communication (Figure 1.3). As depicted in Figure 6.17, the message is a single bit. The analog components of a digital transmission/reception system can be lumped into a single system known as the digital channel. Digital channels are described by transition diagrams, which indicate the output alphabet symbols that result for each possible transmitted symbol and the probabilities of the various reception possibilities. The probabilities on transitions coming from the same symbol must sum to one. For the matched-filter receiver and the signal sets we have seen, the depicted transition diagram, known as a binary symmetric channel, captures how transmitted bits are received. The probability of error pe is the sole parameter of the digital channel, and it encapsulates signal set choice, channel properties, and the matched-filter receiver. With this simple but entirely accurate model, we can concentrate on how bits are received. 24 This 25 This
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