Handbook for Sound Engineers

1170 Chapter 31

The sampling analysis can be extended to the
frequency response of the discrete time sequence, x[n],
by using the relationships x[n]=xc(nT ) and

The result is that

(31-20)

X(e^ jZ is a frequency-scaled version of the contin-
uous-time frequency response, Xs( j:), with the
frequency scale specified by Z=:T. This scaling can
also be thought of as normalizing the frequency axis by
the sample rate so that frequency components that
occurred at the sample rate now occur at 2S. Because
the time axis has been normalized by the sampling
period T, the frequency axis can be thought of as being
normalized by the sampling rate 1e 7.

31.6.4 Quantization

The discussion up to this point has been on how to

quantify the effects of periodically sampling a continu-

ous-time signal to create a discrete-time version of the

signal. As shown in Fig. 31-9, there is a second

step—namely, mapping the infinite-resolution dis-

crete-time signal into a finite precision representation

(i.e., some number of bits per sample) that can be

manipulated in a computer. This second step is known

as quantization. The quantization process takes the sam-

ple from the continuous-to-discrete conversion and

finds the closest corresponding finite precision value

and represents this level with a bit pattern. This bit pat-

tern code for the sample value is usually a binary

two’s-complement code so that the sample can be used

directly in arithmetic operations without the need to

convert to another numerical format (which takes some

number of instructions on a DSP processor to perform).

In essence, the continuous-time signal must be both

quantized in time (i.e., sampled), and then quantized in

amplitude.

The quantization process is denoted mathematically as

where,

Q(•) is the nonlinear quantization operation,

x[n] is the infinite precision sample value.

Quantization is nonlinear because it does not satisfy

Eq. 31-7—i.e., the quantization of the sum of two

values is not the same as the sum of the quantized

values due to how the nearest finite precision value is

generated for the infinite-precision value.

To properly quantize a signal, it is required to know

the expected range of the signal— i.e., the maximum

and minimum signal values. Assuming the signal ampli-

tude is symmetric, the most positive value can be

denoted as XM. The signal then ranges from +XM to XM

for a total range of 2XM. Quantizing the signal to % bits

will decompose the signal into 2B different values. Each

value represents 2XMe 2 B in amplitude and is repre-

sented as the step size '=2XM 2 B=XM 2 (B^1 ). As a

simplified example of the quantization process, assume

that a signal will be quantized into eight different values

which can be conveniently represented as a 3-bit value.

Fig. 31-14 shows one method of how an input signal,

x[n], can be converted into a 3-bit quantized value,

Q(x[n]). In this figure, values of the input signal

between ' /2 and '/2 are given the value 0. Input

signal values between '/2 and 3/2 are represented by

their average value ', and so forth. The eight output

values range from  4 ' to 3' for input signals between

Figure 31-12. The spectrum replicas and the ideal low-pass
filter that will remove the copies except for the desired
baseband spectrum.

Figure 31-13. The final result of reconstructing the analog
signal from the sampled signal.

(^7) S
T
1
0
s(jX^7 )
7S (^7) S (^7) N (^7) N (^7) S 7S 7
( (^7) N)
7
T
0
HLP ( j 7

(^7) c (^7) c
(^7) C (^7) C
(^7) S (^7) S
(^7) N (^7) N
7
1
Xe
jZ
1
T
--- xn>@e–jZn
k –= f
f

= ¦

Xe

jZ

1

T

--- XcjZ

T

--- -^2 Sk

T

©¹§·©¹§·–---------

k –= f

f

= ¦

xn>@=Qxn >@