Audio Engineering

Representation of Audio Signals 459

counting demonstrates the process of addition and, at fi rst sight, the process of subtraction
would need to be simply the inverse operation. However, since we need negative numbers
in order to describe the amplitude of the negative polarity of a waveform, it seems sensible
to use a coding scheme in which the negative number can be used directly to perform
subtraction. The appropriate coding scheme is the two’s complement coding scheme.

(a)

Amplitude dB

Frequency

Δf (^1) T (linear scale)
f 2 π
τ
T
2 τ
Figure 15.12 : (a) A pulse with a period of 2  seconds is repeated every T seconds, producing the
spectrum as shown. The spectrum appears as having negative amplitudes, as alternate “ lobes ”
have the phase of their frequency components inverted, although it is usual to show the modulus
of the amplitude as positive and to refl ect the inversion by an accompanying plot of phase against
frequency. The shape of the lobes is described by the simple relationship: Ak xx (sin )/.
(b) A further example of the duality between time and frequency showing that a widely spread
spectrum will be the result of a narrow pulse. The sum of the energy must be the same for each
so that we would expect a narrow pulse to be of large amplitude if it is to carry much energy. If
we were to use such a pulse as a test signal we would discover that the individual amplitude of
any individual frequency component would be quite small. Thus when we do use this signal for
just this purpose we will usually arrange to average the results of a number of tests.
(b)
2 fc Frequency
Amplitude
1
Time
2 fc
π
2 fc
π