Effects of Bandwidth on a Transmission Channel
All transmission channels of any practical interest are of limited frequency bandwidth. The limitations arise from the physical properties of the channel or from deliberate limitations on the bandwidth to prevent interference from other sources. For primarily economic reasons, most data communications systems seek to maximize the amount of data that can be sent on a channel.
Handling Capacity of a Channel
Claude Shannon masterminded a formula to prove the maximum capacity of an ideal channel whose only impairments are finite bandwidth and noise randomly distributed over that finite bandwidth. That formula is shown here:
C = W x Log2[1 + (P/N)] bits per second
In this formula, P is the power in watts of the signal through the channel, N is the power in watts of the noise out of the channel, and W is the bandwidth of the channel in hertz.
Neglecting all other impairments, some typical values for a voice-grade analog circuit used for data are W = 3000 hertz, P = 0.0001 watts (10 dBm), and N = 0.0000004 watts (34 dBm). According to Shannon's Law, the value of C is as shown here:
3000 x Log2(1 + 250) = about 24,000 bits per second
Shannon's value of C is normally not achievable because there are numerous impairments in every real channel besides those taken into account in Shannon's Law. Also, there are no ideal modems. However, Shannon's Law provides an upper theoretical limit to a binary channel. It is important to note that, due to the nature of the function Log2, the value of C in the formula can be increased more readily by increasing W than by increasing (P/N).
Readers familiar with the latest generation of modems might question how they achieve an operating rate of 33.6Kbps in the upstream direction when, according to Shannon's Law, the operating rate should be limited to approximately 24000bps. The answer to this question involves the actual bandwidth used by such modems. Figure 3.12 illustrates the amplitude-frequency response curve for a voice-grade telephone channel. At a 0db level, the bandwidth is very close to 3000Hz; however, at lower levels, the bandwidth slightly increases, enabling a higher operating rate to be achieved.
One of the factors that tends to reduce the achievable capacity of a channel below the value of C in the formula is a problem called intersymbol (or interbit) interference. If a rectangular pulse like that shown in Figure 3.13 is input to a band-limited channel, the bandwidth limitation of the channel rounds the "corners" of the pulse, as shown in the output waveform, and causes an undesired signal to appear. The "tail" or overshoot part of the new signal interferes with previous and subsequent pulses, adding uncertainty to the signal; that is, the signal might be incorrectly interpreted at the destination.
Figure 3.12 The voice-grade channel amplitude-frequency response curve.
Figure 3.13 A pulse response through a band-limited channel.
Harry Nyquist analyzed the problem of intersymbol interference and developed an ideal rounded pulse shape for which that impairment is minimized. Nyquist also did much theoretical research dealing with sampling of analog signals for representation in binary form. Nyquist's Sampling Theorem (also known as Shannon's Sampling Theorem) says that if an analog signal is sampled 2f times per second, the samples can be used to perfectly reconstruct the original signal over a spectrum of hertz. For example, if a signal is sampled at the rate of 8,000 times per second, those samples can be used to reconstruct the original signal with perfect accuracy over the range of 04000 hertz.