U. Glaeser

where
From Eqs. (27.32) and (27.33), the average quantization noise power p q can be calculated as
The signal-to-noise ratio in decibels is then
© 2002 by CRC Press LLC
(27.33)
(27.34)
(27.35)
where P s is the time-averaged signal power. Assume that the ADC has a full scale of m bits. Then the
maximum input signal amplitude is
and thus
p( x)
⎧1/Q
for x∈[ – Q/2,Q/2 ]
= ⎨
⎩ 0 otherwise
From Eqs. (27.34), (27.35), and (27.37) it follows that
P q
∫
∞
x 2 = p( x)dx
= --- dx =
Q
– ∞
(27.36)
(27.37)
(27.38)
Thus, each additional bit in the quantized signal resolution means ca. 6 dB improvement in the SNR
(or equivalently in the dynamic range). The “const” in expression (27.38) is of secondary importance.
Its value depends on the signal probability density distribution and the ADC range. For instance, for an
ADC range equal to ( – 4 Ps,4 Ps) , the respective value is const ≈ −7.3 dB.
Representing a signal just as a stream of uniformly quantized samples is referred to as the pulse code
modulation (PCM). Typical resolutions in bits per sample (bps) are 16 bps, 24 bps, and even 30 bps.
For instance, for a CD standard with two stereo channels, 44.1 ksamples/s sampling rate and 16 bit
resolution, the resulting audio bit rate is 2 × 44, 100 × 16 = 1.41 Mb/s. In reality, the CD standard has
a large overhead bit rate due to 49-bit representation of every 16-bit sample. The resulting total bit rate
is thus equal to (49/16) × 1.41 = 4.32 Mb/s.
PCM representation is not an efficient method for high quality audio. In order to reduce the required
bit rate, various data compression and coding techniques can be used. Simple but not very efficient
approaches preserve the signal waveform and are therefore referred to as lossless coding techniques
(section 27.8). Data compression facility of lossless audio coders is rather moderate. Average achievable
bit per sample values are only slightly greater than 4.5 bps [35]. Sophisticated techniques, which are still
subject of an intensive research, allow for a drastic reduction of this value—at least by one order of
magnitude. These coding techniques are lossy in the sense that they corrupt the signal; however, this
corruption, can be controlled in such a way that it is inaudible. Such audio coders are called transparent
(section 27.9). In order to efficiently and transparently compress audio and/or speech, the knowledge
about the speech and audio production (the parametric audio coding discussed in section 27.3) as well
as the knowledge concerning the human auditory perception (discussed in section 27.4, resulting in the
perceptual audio coding) should be exploited (Fig. 27.17).
∫
SNR = 10 log
Q/2
– Q/2
x 2
Ps ---- 10 Pq ⎛ ⎞
⎝ ⎠
A 2 m
= ( – 1)Q
P s
2 m
[ ( – 1)Q]
2
∝
Q 2
-----
12
SNR( m)
= 20m log 2 + const ≈ 6.02m + const
10