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On the other hand, for a two's complement representation, the truncation error is always negative and falls in the range -A(e(O
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IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
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With rounding, the quantization error is independent of the type of fixed-point representation and falls in the range
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For floating-point numbers. the mantissa is either rounded or truncated, and the size of the error depends on the value of the exponent.
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8.6.2 Quantization of Filter Coefficients
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In order to implement a filter on a digital processor. the filter coefficients must be converted into binary form. This conversion leads to movements in the pole and zero locations and a change in the frequency response of the filter. The accuracy with which the filter coefficients can be specified depends upon the word length of the processor, and the sensitivity of the filter to coefficient quantization depends on the structure of the filter, as well as on the locations of the poles and zeros. For a second-order section with poles at z = r e f J e ,
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the filter coefficients in a direct form realization are
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If a(1) and a(2) are quantized to B 1 bits, the real part of the pole location is restricted to 2B+' possible values, and the radius squared is restricted to 2 B values. The set of allowable pole locations for a bbit processor is shown in Fig. 8- 16.
Fig. 8-16. The set of allowable pole locations in the first quadrant 1 of the z-plane for a second-order 1 R filter implemented in direct form using a 4-bit processor.
A general sensitivity analysis of a pth-order polynomial
CHAP. 81
IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
shows that the root locations are more sensitive to coefficient quantization errors when the roots are tightly clustered. For example, if the coefficients a(k) are quantized,
then the sensitivity of the location of the ith pole to changes Aa(k) in the coefficients a(k) is approximately
Aa(k)
With
where
then
Thus, if the poles are tightly clustered, la; - njI is small, and small changes in a ( k )will result in large changes in the pole locations. The movement of the poles may be minimized by maximizing the distance between the poles, lai - a / 1 . This may be accomplished by implementing a high-order filter as a combination of first- or second-order &stems. For example, with a cascade of second-order sections. each pair of complex conjugate poles and zeros may be realized separately, thereby localizing the coefficient quantization errors to each section. For an FIR filter,
H (z) =
h(n)zCn
when the coefficients are quantized, the system function becomes
Thus, the quantization errors may be modeled as H ( z ) in parallel with A H ( z ) as shown in Fig. 8-17. If we assume that the coefficients h(n)are less than I in magnitude, and that the coefficients are rounded to B 1 bits,
Therefore, a loose bound on the error in the frequency response is
As with IIR filters, if the zeros are tightly clustered, the zero locations will be sensitive to coefficient quantization errors. However, FIR filters are commonly implemented in direct fonn for two reasons:
1. The zeros of FIR filters are not generally tightly clustered. 2. In direct form, linear phase is easily preserved.
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[CHAP. 8
Fig. 8-17.
filters.
Model for the coefticient quunliza~ion error in FIR
8.6.3 Round-Off Noise
Round-off noise is introduced into a digital filter when products or sums of products are quantized. For example, if two ( B + I)-bit numbers are multiplied, the producl is a ( 2 8 I)-bit number. I f the product is to be saved in a ( B + I)-bit register or used in a ( B + I)-bit adder, it must be quantized to ( B I)-bits, which results in the addition of round-qfnoiw. This round-off noise propagates lhrough the filter and appears at the output of the of filter as round-off noise. I n this section, we illustrate the analys~s round-off noise effects by example. Consider the second-order IIR filter ~mplementedin direct form I shown in Fig. 8-lS(u). The difference equation For this network is
If we assume that all numbers are represented by B I fixed-point numbers and that the network uses ( B I)-bit adders, each ( 2 R I)-bit product must be quantized to B I bits by either truncation or rounding. Fig. 8-18(h) shows the quantizers explicitly. The difference equation corresponding to this system is the nonlinear equation
If the quantizers are replaced with noise sources that are equal to the quantization error, we have an alternative representation shown in Fig. 8-18(c). This representation is particularly useful when it is assumed that the quantization noise has the following properties: Each quantization noise source is a w i d e - , ~ e nstationary white n o i s ~ s~ process. The probability distribution Function of each noise source is uniformly distributed over the quantization interval. 3. Each noise source is uncorrelated with the input to the quantizer. all other noise sources, and the input to the system.
With B I bits, and a fractional representation for all numbers, the second property implies that the quantization noise for rounding has a zero mean and o variance equal to
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