16
D. Rocchesso: Sound Processing
root-mean-square value
RMS value
signal-to-quantization noise
ratio
SNR
dither
many actual behaviors. The uniformly-distributed white noise has a zero mean
but it has a nonzero quadratic mean (i.e., a power) with value
2
=
1
q/2
q/2
0
2
d =
q
2
12
.
(43)
In the frequency domain, the quantization noise is interpreted by means of a
spectrum such as that depicted in fig. 5, which represents the square of the
magnitude of the Fourier transform. The area of the dashed rectangle is equal
to the power
2
. Usually the root-mean-square value (or RMS value) of the
0
f
2
|E|
-F /2
F /2
s
s
2
/F
s
Figure 5: Squared magnitude spectrum of an ideal quantization noise
quantization noise is given, and this is defined as
rms
=
2
=
q
12
,
(44)
which can be directly compared with the maximal representable value in order
to get the signal-to-quantization noise ratio (or SNR)
SN R = 20 log
10
q2
b-1
q/
12
= 20 log
10
(2
b
3) 4.7 + 6 b dB .
(45)
As a general rule, each further quantization bit increases the SNR by 6dB.
Therefore, with 16 bits we have a signal-to-quantization noise ratio of about
101.1dB. When we are given a SNR of 96.3dB with 16 bits, it means that the
ratio has been computed using the maximum value q/2 of the quantization
noise and not its RMS value, which is more significant for the human ear. The
definition (45) is that proposed by Steiglitz [102].
The assumptions on the statistical properties of the quantization noise are
better verified if the signal is large in amplitude and wide in its frequency
extension. For quasi-sinusoidal signals the quantization noise is heavily colored
and correlated with the unquantized signal, in such an extent that some additive
noise called dither is sometimes introduced in order to whiten and decorrelate
the quantization noise. In this way, the perceptual effects of quantization turn
out to be less severe.
By considering the quantization noise as an additive signal we can easily
study its effects within linear systems. The operations performed by a discrete-
time linear system, especially when done in fixed-point arithmetics, can indeed
modify the spectral content of noise signals, and different realizations of the
same transfer functions can behave very differently as far as their immunity
to quantization noise is concerned. Several quantizations can occur within the
realization of a linear system. For instance, the multiplication of two fixed-point