DSP ASSIGNMENT.docx
Transcript of DSP ASSIGNMENT.docx
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time-variant quantizers are frequently used.
If zero is assigned a decision level, the quantizer is called a
midtread type.If zero is assigned a decision level, the quantizeris called a midrise type.
Figure 9.7 (b) illustrates a midtread quantizer with L = 8 levels.
In theory, the extreme decision levels are taken as x1 = -and
xl+1= , to cover the total dynamic range of the input signal.
However, practical A/D converters can handle only a finite
range. Hence we define the range R of the quantizer by
assuming that I1 = IL = . For example, the range of the
quantizer shown in Fig. 9.7(b) is equal to 8 , In practice, thetermfull-scale range (FSR) is used to describe the range of an
A/D converter for bipolar signals (i.e., signals with both
positive and negative amplitudes). The term full scale (FS) is
used for unipolar signals.
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It can be easily seen that the quantization error eq (n) is always
in the range - / 2 to /2 :
-/2< eq(n) /2 (9.2.4)
In other words, the instantaneous quantization error cannotexceed half of the quantization step. If the dynamic range of the
signal, defined as xmaxxmin ,is larger than the range of the
quantizer, the samples that exceed the quantizer range are
clipped, resulting in a large (greater than /2) quantization error.
The operation of the quantizer is better
described by the quantization characteristic function, illustrated
in Fig. 9.8 for a midtread quantizer with eight quantization
levels. This characteristic is preferred in practice over themidriser because it provides an output that is insensitive to
infinitesimal changes of the input signal about zero.
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Note that the input amplitudes o f a midtread quantizer
are rounded to the nearest quantization levels.
The coding process in an A/D converter assigns aunique binary number to each quantization level. If we have L
levels, we need at leastL different binary numbers. With a word
length o f b + 1 bits w e can represent 2b+l distinct binary
numbers. Hence we should have 2b+1 L or, equivalently,
b+1 log2L. Then the step size or the resolution of the A/D
converter is given by
Where R is the range of quantizer
There are various binary coding schemes, each with its
advantages and disadvantages.Table 9.1 illustrates some existing
schemes for 3-bit binary coding.
The twos-com plement 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 012.b
has the value
-0> -2 + 12-1 +2 2-2 +--------b.2-b
If we use the twos-complement representation. Note that 0 is
the most significant bit (MSB) and bis the least significant bit
(LSB). Although the binary code used to represent the
quantization levels is important for the design of the A/D
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converter and the subsequent numerical computations, it does
not have any effect in the performance of the quantization
process. Thus in our subsequent discussions we ignore the
process of coding when we analyze the performance of A/Dconverters.
Figure 9.9(a) show s the characteristic of an ideal 3-bit A/Dconverter. The only degradation introduced by an ideal
converter is the quantization error, which can be reduced by
increasing the number of bits.
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To determine the effects of quantization on the performance of
an A/D converter ,we adopt a statistical approach.The
dependence of the quantization error on the characteristics of the
input signal and the nonlinear nature of the quantizer make
a deterministic analysis intractable, except in very simple cases.
In the statistical approach, we assume that
the quantization error is random in nature. We model this erroras noise that is added to the original (unquantized) signal. If the
input analog signal is within the range o f the quantizer, the
quantization error eq(n) is bounded in magnitude [i.e.,Ieq {n)I