The Theoretical Basis for Data Communication

2.1 The Theoretical Basis for Data Communication Information can be transmitted on wires by varying some physical property such as voltage or current. By representing the value of this voltage or current as a single-valued function of time, f(t), we can model the behavior of the signal and analyze it mathematically. This analysis is the subject of the following sections. 2.1.1 Fourier Analysis In the early 19th century, the French mathematician Jean-Baptiste Fourier proved that any reasonably behaved periodic function, g(t) with period T can be constructed as the sum of a (possibly infinite) number of sines and cosines: Equation 2 where f = 1/T is the fundamental frequency, an and bn are the sine and cosine amplitudes of the nth harmonics (terms), and c is a constant. Such a decomposition is called a Fourier series. From the Fourier series, the function can be reconstructed; that is, if the period, T, is known and the amplitudes are given, the original function of time can be found by performing the sums of Eq. (2-1). A data signal that has a finite duration (which all of them do) can be handled by just imagining that it repeats the entire pattern over and over forever (i.e., the interval from T to 2T is the same as from 0 to T, etc.). The an amplitudes can be computed for any given g(t) by multiplying both sides of Eq. (2-1) by sin(2pkft) and then integrating from 0 to T. Since only one term of the summation survives: an. The bn summation vanishes completely. Similarly, by multiplying Eq. (2-1) by cos(2pkft) and integrating between 0 and T, we can derive bn. By just integrating both sides of the equation as it stands, we can find c. The results of performing these operations are as follows: 2.1.2 Bandwidth-Limited Signals To see what all this has to do with data communication, let us consider a specific example: the transmission of the ASCII character ''b'' encoded in an 8-bit byte. The bit pattern that is to be transmitted is 01100010. The left-hand part of Fig. 2-1(a) shows the voltage output by the transmitting computer. The Fourier analysis of this signal yields the coefficients: Figure 2-1. (a) A binary signal and its root-mean-square Fourier amplitudes. (b)-(e) Successive approximations to the original signal. The root-mean-square amplitudes, , for the first few terms are shown on the right-hand side of Fig. 2-1(a). These values are of interest because their squares are proportional to the energy transmitted at the corresponding frequency. No transmission facility can transmit signals without losing some power in the process. If all the Fourier components were equally diminished, the resulting signal would be reduced in amplitude but not distorted [i.e., it would have the same nice squared-off shape as Fig. 2-1(a)]. Unfortunately, all transmission facilities diminish different Fourier components by different amounts, thus introducing distortion. Usually, the amplitudes are transmitted undiminished from 0 up to some frequency fc [measured in cycles/sec or Hertz (Hz)] with all frequencies above this cutoff frequency attenuated. The range of frequencies transmitted without being strongly attenuated is called the bandwidth. In practice, the cutoff is not really sharp, so often the quoted bandwidth is from 0 to the frequency at which half the power gets through. The bandwidth is a physical property of the transmission medium and usually depends on the construction, thickness, and length of the medium. In some cases a filter is introduced into the circuit to limit the amount of bandwidth available to each customer. For example, a telephone wire may have a bandwidth of 1 MHz for short distances, but telephone companies add a filter restricting each customer to about 3100 Hz. This bandwidth is adequate for intelligible speech and improves system-wide efficiency by limiting resource usage by customers. Now let us consider how the signal of Fig. 2-1(a) would look if the bandwidth were so low that only the lowest frequencies were transmitted [i.e., if the function were being approximated by the first few terms of Eq. (2-1)]. Figure 2-1(b) shows the signal that results from a channel that allows only the first harmonic (the fundamental, f) to pass through. Similarly, Fig. 2-1(c)-(e) show the spectra and reconstructed functions for higher-bandwidth channels. Given a bit rate of b bits/sec, the time required to send 8 bits (for example) 1 bit at a time is 8/b sec, so the frequency of the first harmonic is b/8 Hz. An ordinary telephone line, often called a voice-grade line, has an artificially-introduced cutoff frequency just above 3000 Hz. This restriction means that the number of the highest harmonic passed through is roughly 3000/(b/8) or 24,000/b, (the cutoff is not sharp). For some data rates, the numbers work out as shown in Fig. 2-2. From these numbers, it is clear that trying to send at 9600 bps over a voice-grade telephone line will transform Fig. 2-1(a) into something looking like Fig. 2-1(c), making accurate reception of the original binary bit stream tricky. It should be obvious that at data rates much higher than 38.4 kbps, there is no hope at all for binary signals, even if the transmission facility is completely noiseless. In other words, limiting the bandwidth limits the data rate, even for perfect channels. However, sophisticated coding schemes that make use of several voltage levels do exist and can achieve higher data rates. We will discuss these later in this chapter. Figure 2-2. Relation between data rate and harmonics. 2.1.3 The Maximum Data Rate of a Channel As early as 1924, an AT&T engineer, Henry Nyquist, realized that even a perfect channel has a finite transmission capacity. He derived an equation expressing the maximum data rate for a finite bandwidth noiseless channel. In 1948, Claude Shannon carried Nyquist's work further and extended it to the case of a channel subject to random (that is, thermodynamic) noise (Shannon, 1948). We will just briefly summarize their now classical results here. Nyquist proved that if an arbitrary signal has been run through a low-pass filter of bandwidth H, the filtered signal can be completely reconstructed by making only 2H (exact) samples per second. Sampling the line faster than 2H times per second is pointless because the higher frequency components that such sampling could recover have already been filtered out. If the signal consists of V discrete levels, Nyquist's theorem states: For example, a noiseless 3-kHz channel cannot transmit binary (i.e., two-level) signals at a rate exceeding 6000 bps. So far we have considered only noiseless channels. If random noise is present, the situation deteriorates rapidly. And there is always random (thermal) noise present due to the motion of the molecules in the system. The amount of thermal noise present is measured by the ratio of the signal power to the noise power, called the signal-to-noise ratio. If we denote the signal power by S and the noise power by N, the signal-to-noise ratio is S/N. Usually, the ratio itself is not quoted; instead, the quantity 10 log10 S/N is given. These units are called decibels (dB). An S/N ratio of 10 is 10 dB, a ratio of 100 is 20 dB, a ratio of 1000 is 30 dB, and so on. The manufacturers of stereo amplifiers often characterize the bandwidth (frequency range) over which their product is linear by giving the 3-dB frequency on each end. These are the points at which the amplification factor has been approximately halved (because log103 0.5). Shannon's major result is that the maximum data rate of a noisy channel whose bandwidth is H Hz, and whose signal-to-noise ratio is S/N, is given by For example, a channel of 3000-Hz bandwidth with a signal to thermal noise ratio of 30 dB (typical parameters of the analog part of the telephone system) can never transmit much more than 30,000 bps, no matter how many or how few signal levels are used and no matter how often or how infrequently samples are taken. Shannon's result was derived from information-theory arguments and applies to any channel subject to thermal noise. Counterexamples should be treated in the same category as perpetual motion machines. It should be noted that this is only an upper bound and real systems rarely achieve it.