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

205K - views

Published bytatyana-admore

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

Download Pdf

Download Pdf - The PPT/PDF document "DSP Eect of Product Roundo in FinitePrec..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Page 1

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

Page 2

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

Page 3

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

Page 4

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

Page 5

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

Page 6

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

Page 7

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

Page 8

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

Page 9

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

Page 10

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

Page 11

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

Â© 2020 docslides.com Inc.

All rights reserved.