U. Glaeser

8 ksamples/s for telephony (the signal spectrum extends up to 4 kHz, and thus covers most of the
frequencies contained in speech), 32 ksamples/s for medium quality digital audio (audible frequency band
up to 16 kHz is covered), 44.1 ksamples/s for a CD standard (audio frequency band up to 22.05 kHz is
represented), 48 and 96 ksamples/s for high quality digital audio (the represented frequencies range up to
24 and 48 kHz, respectively).
An important generalization of the classic sampling theory applies to band signals [8]. A continuoustime
signal, whose spectrum is limited to some frequency band
© 2002 by CRC Press LLC
(27.30)
can be sampled with a sampling rate of at least F s = 2∆f only (i.e., critically sampled), if both spectrum
border frequencies f 1 and f 2 in Eq. (27.30) are consecutive multiples of the Nyquist frequency F s/2, i.e., if
(27.31)
where k is an integer. Such a signal is referred to as the integer-band signal. Audio signals are not by themselves
integer-band signals but they can be split with an analysis filter bank to some subband signals, which all
are integer-band signals, and thus, can be critically sampled. This is indeed the case in many digital audio
coders, e.g., in the MUSICAM standard the input audio signal, initially sampled with 48 ksamples/s, is split
into 32 subbands with bandwidths of 24,000/32 = 750 Hz each. Signals in each subband are sampled with
48/32 = 1.5 ksamples/s sampling rate.
Another signal discretization process is quantization, i.e., the procedure of converting a signal with
continuously distributed values into a signal with discrete values. Unlike sampling, which, under some
conditions, can be considered lossless, i.e., the original signal can—at least theoretically—be perfectly
recovered after sampling, quantization is an inherently lossy operation [8,37].
The error due to the quantization has a nature of noise and is referred to as the quantization noise.
Although this noise is unavoidable and cannot be removed from the signal, it can be made inaudible by
controlling its level and forcing it to lie under the threshold of audibility. Masking effects, discussed in
section 27.4, can be very effectively exploited with this end in view.
The quantization noise is usually analyzed under the following simplifying assumptions:
• The quantization steps are uniform.
• The number of quantization levels is high.
∆f = f1– f2, f2 > f1 f1 = kFs/2 and f2 = ( k + 1)Fs/2
The first assumption is not fulfilled in many quantization techniques for audio signals. This is because
the perception of noise does not depend on its absolute power but on the signal-to-noise ratio (SNR).
Thus, it is reasonable to quantize audio signals nonuniformly, with quantization steps proportional to the
signal values. If the steps are not uniform, then the quantization error will be a function of the input
signal, and consequently, it will not be an additive noise any more. Fortunately, in most procedures for
the quantization of audio signals, quantization steps are at least range by range uniform and the first
assumption can be considered as approximately valid. The second assumption is usually satisfactorily
fulfilled. Due to this assumption the quantization noise has a uniform probability density distribution
and is not correlated with the signal.
Denote by Q the quantization step and by p(x) the probability distribution function of the quantization
error. Then
∫
Q/2
p( x)dx
=
1
– Q/2
(27.32)