Handbook for Sound Engineers

DSP Technology 1171

 9 ' e  and 7' /2. Values larger than 7' / 2 are set to 3'
and values smaller than  9 ' /2 are set to  4 '—i.e., the
numbers saturate at the maximum and minimum values,
respectively.

The step size, ', has an impact on the resulting
quality of the quantization. If ' is large, fewer bits will
be required for each sample to represent the range, 2XM,
but there will be more quantization errors. If ' is small,
more bits will be required for each sample, although
there will be less quantization error. Normally, the
system design process determines the value of XM and
the number of bits required in the converter, %. If XM is
chosen too large, then the step size, ' will be large and
the resulting quantization error will be large. If XM is
chosen too small, then the step size,' will be small, but
the signal may clip the A/D converter if the actual range
of the signal is larger than XM.

This loss of information during quantization can be
modeled as noise signal that is added to the signal as
shown in Fig. 31-15. The amount of quantization noise
determines the overall quality of the signal. In the audio
realm, it is common to sample with 24 bits of resolution
on the A/D converter. Assuming a ±15 V swing of an
analog signal, the granularity of the digitized signal is
30V/2^24 , which comes to 1.78μV.

With certain assumptions about the signal, such as

the peak value being about four times the rms signal

value, it can be shown that the signal to noise ratio

(SNR) of the A/D converter is approximately 6 dB per

bit.^1 Each additional bit in the A/D converter will

contribute 6 dB to the SNR. A large SNR is usually

desirable, but that must be balanced with overall system

requirements, system cost, and possibly other noise

issues inherent in a design that would reduce the value

of having a high-quality A/D converter in the system.

The dynamic range of a signal can be defined as the

range of the signal levels over which the SNR exceeds a

minimum acceptable SNR.

There are cost-effective A/D converters that can

shape the quantization noise and produce a high-quality

signal. Sigma-Delta converters, or noise-shaping

converters, use an oversampling technique to reduce the

amount of quantization noise in the signal by spreading

the fixed quantization noise over a bandwidth much

larger than the signal band.^5 The technique of oversam-

pling and noise shaping allows the use of relatively

imprecise analog circuits to perform high-resolution

conversion. Most digital audio products on the market

use these types of converters.

31.6.5 Sample Rate Selection

The sampling rate, 1/T, plays an important role in deter-

mining the bandwidth of the digitized signal. If the ana-

log signal is not sampled often enough, then high-

frequency information will be lost. At the other

extreme, if the signal is sampled too often, there may be

more information than is needed for the application,

causing unnecessary computation and adding unneces-

sary expense to the system.

In audio applications it is common to have a

sampling frequency of 48 kHz = 48,000 Hz, which

yields a sampling period of 1/48,000 = 20.83μs. Using

a sample rate of 48 kHz is why, in many product data

sheets, the amount of delay that can be added to a signal

is an integer multiple of 20.83μs.

The choice of which sample rate to use depends on

the application and the desired system cost. High-quality

audio processing would require a high sample rate while

low bandwidth telephony applications require a much

lower sample rate. A table of common applications and

their sample rate and bandwidths are shown in Table

31-3. As shown in the sampling process, the maximum

bandwidth will always be less than ½ the sampling

frequency. In practice, the antialiasing filter will have

some roll-off and will band limit the signal to less than ½

the sample rate. This bandlimiting will further reduce the

Figure 31-14. The quantization of an input signal, x, into

Q(x).

Figure 31-15. The sampling process of Fig. 31-9 with the

addition of an antialiasing filter and modeling the quantiza-

tion process as an additive noise signal.

7 $ 2

x

x=Q(x)

$

2 $

3 $

3 $

2 $

$

5 $ 23 $ 2 $ 23 $ 25 $ 27 $ 2

$ 2

4 $

9 $ 2

Two’s

complement

code

011

010

001

000

111

110

101

100

9 $ 2

2 XM

Anti-

aliasing

Filter

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Bandlimit Sample Quantize

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