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' Time Discretization
Discretization Quantization Amplitude Figure 1.4.8 Sampling and quantization of a sinusoidal signal. 1.4.4 Quantization of Sinusoidal Signals Figure 1.4.8 illustrates the sampling and quantization of an analog sinusoidal signal
xa (t) = A cos Slot using a rectangular grid. Horizontal lines within the range of
the quantizer indicate the allowed levels of quantization. Vertical lines indicate the
sampling times. Thus, from the original analog signal xa (t) we obtain a discrete—time
signal x(n) _= xa(nT) by sampling and a discretetime, discreteamplitude signal
xq (n T) after quantization. In practice, the staircase signal xq (t) can be obtained by
using a zero—order hold. This analysis is useful because sinusoids are used as test
signals in A/D converters. “ If the sampling rate F5 satisﬁes the sampling theorem, quantization is the only
error in the A/D conversion process. Thus we can evaluate the quantizatiOn error by quantizing the analog signal xa (t)
instead of the discrete—time signal x(n) = xa (nT). Inspection of Fig. 1.4.8 indicates that the signal xa (t) is almost linear between quantization levels (see Fig. 14.9). The (a) (b) Figure 1.4.9 The quantization error eq (1‘) = xa (t) — xq(t). 6.3 Analogto—Digital and DigitaltoAnalog Converters 405 Output 5% = Q[x]
Quantization levels \A 2A Two's—complement
code words
01 1
010
001
000
1 1 1
Input 1 10
101
100 Decision levels l‘ Range R = FSR (Peaktopeak range) Figure 6.3.3 Example of a midtread quantizer. There are various binary coding schemes, each with its advantages and disad
vantages. Table 6.1 illustrates some existing schemes for 3—bit binary coding. These
number representation schemes are described in more detail in Section 9.4. The two’s—complement representation is used in most digital signal processors.
Thus it is convenient to use the same system to represent digital signals because we
can operate on them directly without any extra format conversion. In general, a
(b + 1)bit binary fraction of the form [80,81,132    ,6], has the value ‘ﬁO‘ZO—i‘ﬂl'2_1+,52'2—2+'”+,3b'Z—b if we use the two’scomplement representation. Note that ,80 is the most signiﬁcant
bit (MSB) and ,6b is the least signiﬁcant bit (LSB). Although the binary code used
to represent the quantization levels is important for the design of the A/D converter
and the subsequent numerical computations, it does not have any effect in the per—
formance of the quantization process. Thus in our subsequent discussions we ignore
the process of coding when we analyze the performance of A/D converters. The only degradation introduced by an ideal converter is the quantization error,
which can be reduced by increasing the number of bits. This error, which dominates
the performance of practical A/D converters, is analyzed in the next section. Practical A/D converters differ from ideal converters in several ways. Various
degradations are usually encountered in practice. Speciﬁcally, practical A/D convert—
ers may have an offset errbr (the ﬁrst transition may not occur at exactly +% LSB), TABLE 6.1 Commonly Used Bipolar Codes
Decimal Fraction 7 Positive Negative Sign + Two’s Offset One’s
Number Reference Reference Magnitude Complement Binary Complement +7 +§ —§ 0111 0111 1111 0111
+6 +§ +3 0110 0110 1110 0110
+5 +3 —3 0101 0101 1101 0101
+4 +§ +3 0100 0100 1100 0100
+3 +§ —§ 0011 0011 1011 0011
+2 +§ —§ 0010 0010 1010 0010
+1 +§ —% 0001 0001 1001 0001 0 0+ 0+ 0000 0000 1000 0000 0 0— 0+ 1000 (0000) (1000) 1111
—1 —§ +§ 1001 1111 0111 1110
+2 ~§ +§ 1010 1110 0110 1101
—3 —§ +3 1011 1101 0101 1100
—4 —3 +3 1100 1100 0100 1011
—5 —§ +§ 1101 1011 0011 1010
—6 —g +3 1110 1010 0010 1001
—7 —g +§ 1111 1001 0001 1000
—8 —§ +3 (100m @000) scalefactor (or gain) error (the difference between the values at which the ﬁrst tr
sition and the last transition occur is not equal to FS — ZLSB), and a linearity er
(the differences between transition values are not all equal or uniformly changii
If the differential linearity error is large enough, it is possible for one or more cc
words to be missed. Performance data on commercially available A/D conver
are speciﬁed in manufacturers’ data sheets. 6.33 v Analysis of Quantization Errors To determine the effects of quantization on the performance of an A/D conver
we adopt a statistical approach. The dependence of the quantization error on
characteristics of the input signal and the nonlinear nature of the quantizer mal
deterministic analysis intractable, except in very simple cases. In the statistical approach, we assume that the quantization error is randon
nature. We model this error as noise that is added to the original (unquantized) sig
If the input analog signal is within the range of the quantizer, the quantization e1 ...
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 Spring '06
 V."Ragu"Balakrishnan
 Digital Signal Processing, quantization, quantization error, Analysis of Quantization Errors

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