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However, if (x(n)lexceeds X , , then x(n) will be clipped, and the quantization error could be very large. ,,
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Fig. 3-4. A 3-bit uniform quantizer.
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A useful model for the quantization process is given in Fig. 3-5. Here, the quantization error is assumed to be an additive noise source. Because the quantization error is typically not known, the quantization error is described statistically. It is generally assumed that e(n) is a sequence of random variables where I. 2. 3. 4. The statistics of e(n) do not change with time (the quantization noise is a stationary random process). The quantization noise e(n) is a sequence of uncorrelated random variables. The quantization noise e(n) is uncorrelated with the quantizer input x(n). The probability density function of e(n) is uniformly distributed over the range of values of the quantization error.
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Although it is easy to find cases in which these assumptions do not hold (e.g., if x(n) is a constant), they are generally valid for rapidly varying signals with fine quantization ( A small).
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Fig. 3-5. A quantization noise model.
With rounding, the quantization noise is uniformly distributed over the interval [-A/2, A/2], and the quantization noise power (the variance) is
,, .: 12 =
With a step size
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[CHAP. 3
: , and a signal power the signal-to-quantization noise ratio, in decibels (dB), is
Thus, the signal-to-quantization noise ratio increases approximately 6 dB for each bit. The output of the quantizer is sent to an encoder-,which assigns a unique binary number (codeword) to each quantization level. Any assignment of codewords to levels may be used, and many coding schemes exist. Most digital signal processing systems use the two's-complement representation. In this system, with a (B 1) bit codeword, c = [bo, l ,. . . , b B ] h
the leftmost or most significant bit, bo, is the sign bit, and the remaining bits are used to represent either binary integers or fractions. Assuming binary fractions, the codeword bobl . . . bs has the value b2
An example is given below for a 3-bit codeword.
Binary Symbol
Numeric Value
3.3 DIGITAL-TO-ANALOG CONVERSION
As stated in the sampling theorem, if x,(t) is strictly bandlimited so that Xa(jSZ) = 0 for I 2 > no, if s1 and T, < T /QO,then x a ( t )may be uniquely reconstructed from its samples x(n) = x,(nT,). The reconstruction process involves two steps, as illustrated in Fig. 3-6. First, the samples x(n)are converted into a sequence of impulses,
and then x,(t) is filtered with a reconstructionfilter, which is an ideal low-pass filter that has a frequency response given by
This system is called an ideal discrete-to-continuous (DIC) converter. Because the impulse response of the reconstruction filter is
CHAP. 31
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Fig. 3-6. (a) A discrete-to-continuous converter with an ideal low-pass reconstruction filter. (h) The frequency response of the ideal reconstruction filter.
the output of the filter is
xa(t) =
n=-00
t nTs)/TT C x(n)hAr - nTs) = C x(n)sin n (- nT,$)/Ts
n=-m
This interpolation formula shows how x,(t) is reconstructed from its samples x(n) = x,(nTs). In the frequency domain, the interpolation formula becomes
which is equivalent to
XAjW
T ~ X ( ~ J * ~ S )
I1 < n n
otherwise TS
Thus, x ( e i w )is frequency scaled (o= QTS), and then the low-pass filter removes all frequencies in the periodic above the cutoff frequency Q. = TIT,. , spectrum x(eiQTr) Because it is not possible to implement an ideal low-pass filter, many D/A converters use a zero-order hold for the reconstruction filter. The impulse response of a zero-order hold is
OitlT,
otherwise
ho(0 =
and the frequency response is
After a sequence of samples xa(nT,)has been converted to impulses, the zero-order hold produces the staircase approximation to xu(!)shown in Fig.3-7. With a zero-order hold, it is common to postprocess the output with a reconstruction compensation filter that approximates the frequency response
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[CHAP. 3
-2T. -T.
2. T
3. T
4. T
-2T. -T.
2. T
3. T
4. T
Fig. 3-7. The use of a zero-order hold to interpolate between the samples in x , ( t ) .
so that the cascade of Ho(ejo) with H C ( e j w )approximates a low-pass filter with a gain of T, over the passband. Figure 3-8 shows the magnitude of the frequency response of the zero-order hold and the magnitude of the frequency response of the ideal reconstruction compensation filter. Note that the cascade of H , ( j n ) with the zero-order hold is an ideal low-pass filter.
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