Relationship between Data Rate and Bandwidth

Relationship between Data Rate and Bandwidth

There is a direct relationship between the information-carrying capacity of a signal
and its bandwidth: The greater the bandwidth, the higher the information-carrying
capacity. As a very simple example, consider the square wave of Figure 2.2b. Suppose
that we let a positive pulse represent binary 0 and a negative pulse represent binary 1.
Then the waveform represents the binary stream 0101. ... The duration of each pulse
is 1I(2f); thus the data rate is 2fbits per second (bps). What are the frequency components
of this signal? To answer this question, consider again Figure 2.4. By adding
together sine waves at frequenciesfand 3f, we get a waveform that begins to resemble
the square wave. Let us continue this process by adding a sine wave of frequency 5f,
as shown in Figure 2.5a, and then adding a sine wave of frequency 7f, as shown in
Figure 2.5b. As we add additional odd multiples of f, suitably scaled, the resulting
waveform approaches that of a square wave more and more closely.
Indeed, it can be shown that the frequency components of the square wave
with amplitudes A and -A can be expressed as follows:
4 .; sin(21rkft)
set) = A X - L.J
1r k odd k=l k
This waveform has an infinite number of frequency components and hence an infinite
bandwidth. However, the peak amplitude of the kth frequency component, kf, is
only 11k, so most of the energy in this waveform is in the first few frequency components.
What happens if we limit the bandwidth to just the first three frequency components?
We have already seen the answer, in Figure 2.5a.As we can see, the shape of
the resulting waveform is reasonably close to that of the original square wave.
We can use Figures 2.4 and 2.5 to illustrate the relationship between data rate
and bandwidth. Suppose that we are using a digital transmission system that is capable
of transmitting signals with a bandwidth of 4 MHz. Let us attempt to transmit a
sequence of alternating Os and Is as the square wave of Figure 2.5c. What data rate
can be achieved? We look at three cases.
Case I. Let us approximate our square wave with the waveform of Figure
2.5a. Although this waveform is a "distorted" square wave, it is sufficiently
close to the square wave that a receiver should be able to discriminate
between a binary 0 and a binary 1. If we let f = 106 cycles/second = 1 MHz,
then the bandwidth of the signal