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

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