DSP Eect of Product Roundo in FinitePrecision Filters Digital Signal Processing Eect of Product Roundo in FinitePrecision Filters D

DSP Eect of Product Roundo in FinitePrecision Filters Digital Signal Processing Eect of Product Roundo in FinitePrecision Filters D - Description

Richard Brown III D Richard Brown III 1 11 brPage 2br DSP E64256ect of Product Roundo64256 in FinitePrecision Filters Product RoundO64256 Linear Model When you take the product of two 1 bit 64257xedpoint numbers the result requires 2 bits to store ID: 27106 Download Pdf

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DSP Eect of Product Roundo in FinitePrecision Filters Digital Signal Processing Eect of Product Roundo in FinitePrecision Filters D

Richard Brown III D Richard Brown III 1 11 brPage 2br DSP E64256ect of Product Roundo64256 in FinitePrecision Filters Product RoundO64256 Linear Model When you take the product of two 1 bit 64257xedpoint numbers the result requires 2 bits to store

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DSP Eect of Product Roundo in FinitePrecision Filters Digital Signal Processing Eect of Product Roundo in FinitePrecision Filters D




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Presentation on theme: "DSP Eect of Product Roundo in FinitePrecision Filters Digital Signal Processing Eect of Product Roundo in FinitePrecision Filters D"— Presentation transcript:


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DSP: Effect of Product Roundoff in Finite-Precision Filters Digital Signal Processing Effect of Product Roundoff in Finite-Precision Filters D. Richard Brown III D. Richard Brown III 1 / 11
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DSP: Effect of Product Roundoff in Finite-Precision Filters Product Round-Off: Linear Model When you take the product of two +1 -bit fixed-point numbers, the result requires +2 bits to store. If we don’t have enough bits, we typically round off the least significant bits of the product, which leads to another

source of quantization error in finite precision filters. We will use a linear model for round-off error analysis. For example, consider a second-order all-pole filter: Product Round-O Model for Product Round-O Error Analysis () () The main idea is that we insert a quantization noise source after each non-unity product. If an extended precision accumulator is used, then a quantization noise source is placed only after the final sum. D. Richard Brown III 2 / 11
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DSP: Effect of Product Roundoff in Finite-Precision Filters Product

Round-Off: Linear Model Assumptions To facilitate analysis, the product round-off quantization errors are modeled as random sequences just like input quantization errors: 1. Each product round-off error is uniformly distributed on 2. The product round-off error is independent of for all 3. The product round-off error is independent of for all and 4. The product round-off error is independent of for all Procedure : Denote the round-off noise variance (assumed the same for all products). For each product round-off error source =1 ,...,L 1.

Determine the “noise transfer function )= 2. Compute the output noise variance caused by roundoff error as j d The total round-off noise variance at output is then tot =1 D. Richard Brown III 3 / 11
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DSP: Effect of Product Roundoff in Finite-Precision Filters Effect of Product Round-Off: Simple Example Product Round-O Model for Product Round-O Error Analysis () () We denote the round-off noise variance for both products as . We have )= )= 1+ Hence, assuming and are such that is stable, tot =2 j d =2 1+  1+2 D. Richard Brown

III 4 / 11
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DSP: Effect of Product Roundoff in Finite-Precision Filters Product Roundoff Noise: DF-II Second Order Section Given the realization structure with product roundoff errors: The roundoff errors appear at the output as )= )+ )+ )+ )( )+ )) where )= 1+ . Hence the variance of the product roundoff noise at the output is =3 +2 j dω. D. Richard Brown III 5 / 11
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DSP: Effect of Product Roundoff in Finite-Precision Filters Product Roundoff Noise: General DF-I and DF-II Assume )= with non-unity

denominator coefficients and +1 non-unity numerator coefficients. For a general direct-form II filter, similar analysis techniques can be used to show N j d {z noises that propagate through filter +( +1) {z noises directly connected to output Note that j d if an extended-precision accumulator is used. For a direct-form I filter, inspection of the signal flow graph shows that all of the roundoff noises propagate through )= . Hence, +1+ j d standard accumulation j d extended-precision accumulation. D. Richard Brown III 6 / 11
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DSP:

Effect of Product Roundoff in Finite-Precision Filters 2nd Order DF-I vs. DF-II Example Suppose =1 and )= 6+0 54 +0 108 +0 with ROC (causal and stable). With these numbers, we can compute the relevant integrals (using, for example, the algebraic technique) to be j d 12 7719 j d 6420 which means that 3+2 12 7719=28 5438 DF-II 6420=43 2099 DF-I D. Richard Brown III 7 / 11
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DSP: Effect of Product Roundoff in Finite-Precision Filters Cascaded DF-II Realization Structure Now, what if we split this realization structure up into a cascade of two first

order sections? There are (at least) four possibilities. Here are two: 36 11 21 31 12 22 32 11 21 31 36 12 22 32 Note both realizations have the same )= )= Is there any difference? D. Richard Brown III 8 / 11
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DSP: Effect of Product Roundoff in Finite-Precision Filters Analysis of Cascaded DF-II Realization Structure In each realization structure, some of the noises propagate through , some propagate through , some propagate through )= , and some are directly connected to the output. Given )= 6+0 36 )= 1+0 we can use algebraic methods (or the usual series

convergence results) to compute j d 32 j d 8533 Also recall our earlier result j d 12 7719 D. Richard Brown III 9 / 11
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DSP: Effect of Product Roundoff in Finite-Precision Filters Analysis of First Cascaded DF-II Realization Structure For the first realization: 36 11 21 31 12 22 32 11 propagates through to get to the output. 21 31 , and 12 propagate through to get to the output. 22 ]=0 23 is directly connected to the output. Hence, assuming =1 , we have =12 7719+3 8533+0+1 19 33 which is better than the regular DF-II realization with 28 54 D. Richard Brown

III 10 / 11
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DSP: Effect of Product Roundoff in Finite-Precision Filters Analysis of Second Cascaded DF-II Realization Structure In the second realization: 11 21 31 36 12 22 32 11 propagates through to get to the output. 21 ]=0 31 , and 12 propagate through to get to the output. 22 and 23 are directly connected to the output. Hence, assuming =1 , we have =12 7719+0+2 32+0+2 19 41 which is slightly worse than the first ordering. D. Richard Brown III 11 / 11