458 Chapter 15
The composite square wave has ripples in its shape, due to band limiting, since this
example uses only the fi rst four terms, up to the seventh harmonic. For a signal that has
been limited to an audio bandwidth of approximately 21 kHz, this square wave must be
considered as giving an ideal response even though the fundamental is only 3 kHz. The
9% overshoot followed by a 5% undershoot, the Gibbs phenomenon, will occur whenever
a Fourier series is truncated or a bandwidth is limited.
Instead of sending a stream of numbers that describe the wave shape at each regularly spaced
point in time, we fi rst analyze the wave shape into its constituent frequency components and
then send (or store) a description of the frequency components. At the receiving end these
numbers are unraveled and, after some calculation, the wave shape is reconstituted. Of course
this requires that both the sender and the receiver of the information know how to process it.
Thus the receiver will attempt to apply the inverse, or opposite, process to that applied during
coding at the sending end. In the extreme it is possible to encode a complete Beethoven
symphony in a single 8-bit byte. First, we must equip both ends of our communication link
with the same set of raw data, in this case a collection of CDs containing recordings of
Beethoven’s work. We then send the number of the disc that contains the recording which
we wish to “ send. ” At the receiving end, the decoding process uses the received byte of
information, selects the disc, and plays it. A perfect reproduction using only one byte to
encode 64 minutes of stereo recorded music is created ... and to CD quality!
A very useful signal is the impulse. Figure 15.12 shows an isolated pulse and its attendant
spectrum. Of equal value is the waveform of the signal that provides a uniform spectrum.
Note how similar these wave shapes are. Indeed, if we had chosen to show in Figure
15.12(a) an isolated square-edged pulse then the pictures would be identical, save that
references to the time and frequency domains would need to be swapped. You will
encounter these wave shapes in diverse fi elds such as video and in the spectral shaping
of digital data waveforms. One important advantage of shaping signals in this way is
that since the spectral bandwidth is better controlled, the effect of the phase response of
a band-limited transmission path on the waveform is also limited. This will result in a
waveform that is much easier to restore to clean “ square ” waves at the receiving end.
15.7 Arithmetic
We have seen how the process of counting in binary is carried out. Operations using the
number base of 2 are characterized by a number of useful tricks that are often used. Simple