233 what this code might be; it has never been found. Codes such as the Hamming code work quite well in practice to keep error rates low, but they remain greater than zero. Until the “magic” code is found, more important in communication system design is the converse. It states that if your data rate exceeds capacity, errors will overwhelm you no matter what channel coding you use. For this reason, capacity calculations are made to understand the fundamental limits on transmission rates. Exercise 6.34 (Solution on p. 259.) The first definition of capacity applies only for binary symmetric channels, and represents the number of bits/transmission. The second result states capacity more generally, having units of bits/second. How would you convert the first definition’s result into units of bits/second? Example 6.5 The telephone channel has a bandwidth of 3 kHz and a signal-to-noise ratio exceeding 30 dB (at least they promise this much). The maximum data rate a modem can produce for this wireline channel and hope that errors will not become rampant is the capacity. C = 3 × 103 log2 1 + 103 (6.62) = 29.901 kbps Thus, the so-called 33 kbps modems operate right at the capacity limit. Note that the data rate allowed by the capacity can exceed the bandwidth when the signal-to-noise ratio exceeds 0 dB. Our results for BPSK and FSK indicated the bandwidth they require exceeds T1 . What kind of signal sets might be used to achieve capacity? Modem signal sets send more than one bit/transmission using a number, one of the most popular of which is multi-level signaling. Here, we can transmit several bits during one transmission interval by representing bit by some signal’s amplitude. For example, two bits A can be sent with a signal set comprised of a sinusoid with amplitudes of ±A and ± . 2
6.32 Comparison of Analog and Digital Communication41 Analog communication systems, amplitude modulation (AM) radio being a typifying example, can inexpensively communicate a bandlimited analog signal from one location to another (point-to-point communication) or from one point to many (broadcast). Although it is not shown here, the coherent receiver (Figure 6.6) provides the largest possible signal-to-noise ratio for the demodulated message. An analysis of this receiver thus indicates that some residual error will always be present in an analog system’s output. Although analog systems are less expensive in many cases than digital ones for the same application, digital systems offer much more efficiency, better performance, and much greater flexibility. • Efficiency: The Source Coding Theorem allows quantification of just how complex a given message source is and allows us to exploit that complexity by source coding (compression). In analog communication, the only parameters of interest are message bandwidth and amplitude. We cannot exploit signal structure to achieve a more efficient communication system. • Performance: Because of the Noisy Channel Coding Theorem, we have a specific criterion by which to formulate error-correcting codes that can bring us as close to error-free transmission as we might want. Even though we may send information by way of a noisy channel, digital schemes are capable of error-free transmission while analog ones cannot overcome channel disturbances; see this problem (Problem 6.15) for a comparison. • Flexibility: Digital communication systems can transmit real-valued discrete-time signals, which could be analog ones obtained by analog-to-digital conversion, and symbolic-valued ones (computer data, for example). Any signal that can be transmitted by analog means can be sent by digital means, with the only issue being the number of bits used in A/D conversion (how accurately do we need to represent signal amplitude). Images can be sent by analog means (commercial television), but better communication performance occurs when we use digital systems (HDTV). In addition to digital 41 This
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