On the Fixed-Point Accuracy Analysis and Optimization of Polynomial Specifications

Sarbishei, O.; Radecka, Katarzyna

doi:10.1109/tcad.2013.2238290

Cited by 19 publications

(17 citation statements)

References 49 publications

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“…where Q is the orthogonal matrix, each column of which corresponds to Lanczos vectors, A is the dense symmetric matrix with size n × n and T is the symmetric tridiagonal matrix. The most commonly used criterion for evaluating accuracy of fixed-point representation is the SQNR stated in Equation (22). Variations in SQNR of the fixed-point algorithm for different matrix dimensions are shown in Figure 3.…”

Section: Numerical Results Of Lanczos Algorithm In Constant Variablementioning

confidence: 99%

“…On the other hand, analytical approaches provide IWLs that can represent all possible input matrices, but generates conservative IWLs. There are again a number of issues associated with analytical methods that are used to find the variable fixed‐point format: IA ignores correlation among signals resulting in overestimation, the ranges determined through AA explodes during non‐affine forms like division and SMT and polynomial chaos expansion (PCE) come with higher costs . Moreover, these tools limit their usefulness to problems with lower dimension .…”

Section: Lanczos Algorithm In Various Fixed‐point Formatsmentioning

confidence: 99%

“…The MSE can be computed using Equation and can be used to obtain SQNR, as stated in Equation . MM, MSE and SQNR can also be determined analytically . Penalty is given by

italicPenalt y_{p} = \frac{M S E_{p} - M S E_{64}}{M S E_{64}} \times 100 %,

where Penalty p is the percentage of penalty paid by a fixed‐point system for using p order precision.…”

Section: Numerical Accuracy Evaluationmentioning

confidence: 99%

“…The MSE can be computed using Equation (23) and can be used to obtain SQNR, as stated in Equation (22). MM, MSE and SQNR can also be determined analytically [2,22,43]. Penalty is given by…”

Section: Fixed-point Simulation-based Approachmentioning

confidence: 99%

See 3 more Smart Citations

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

Pradhan

Kabi

Mohanty

et al. 2015

Circuit Theory & Apps

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SUMMARYProper range and precision analysis play an important role in the development of fixed-point algorithms for embedded system applications. Numerical linear algebra algorithms used to find singular value decomposition of symmetric matrices are suitable for signal and image-processing applications. These algorithms have not been attempted much in fixed-point arithmetic. The reason is wide dynamic range of data and vulnerability of the algorithms to round-off errors. For any real-time application, the range of the input matrix may change frequently. This poses difficulty for constant and variable fixed-point formats to decide on integer wordlengths during float-to-fixed conversion process because these formats involve determination of integer wordlengths before the compilation of the program. Thus, these formats may not guarantee to avoid overflow for all ranges of input matrices. To circumvent this problem, a novel dynamic fixed-point format has been proposed to compute integer wordlengths adaptively during runtime. Lanczos algorithm with partial orthogonalization, which is a tridiagonalization step in computation of singular value decomposition of symmetric matrices, has been taken up as a case study. The fixed-point Lanczos algorithm is tested for matrices with different dimensions and condition numbers along with image covariance matrix. The accuracy of fixed-point Lanczos algorithm in three different formats has been compared on the basis of signal-toquantization-noise-ratio, number of accurate fractional bits, orthogonality and factorization errors. Results show that dynamic fixed-point format either outperforms or performs on par with constant and variable formats. Determination of fractional wordlengths requires minimization of hardware cost subject to accuracy constraint. In this context, we propose an analytical framework for deriving mean-square-error or quantization noise power among Lanczos vectors, which can serve as an accuracy constraint for wordlength optimization. Error is found to propagate through different arithmetic operations and finally accumulate in the last Lanczos vector. It is observed that variable and dynamic fixed-point formats produce vectors with lesser round-off error than constant format. All the three fixed-point formats of Lanczos algorithm have been synthesized on Virtex 7 field-programmable gate array using Vivado high-level synthesis design tool. A comparative study of resource usage and power consumption is carried out.

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Section: Numerical Results Of Lanczos Algorithm In Constant Variablementioning

confidence: 99%

Section: Lanczos Algorithm In Various Fixed‐point Formatsmentioning

confidence: 99%

“…The MSE can be computed using Equation and can be used to obtain SQNR, as stated in Equation . MM, MSE and SQNR can also be determined analytically . Penalty is given by

italicPenalt y_{p} = \frac{M S E_{p} - M S E_{64}}{M S E_{64}} \times 100 %,

where Penalty p is the percentage of penalty paid by a fixed‐point system for using p order precision.…”

Section: Numerical Accuracy Evaluationmentioning

confidence: 99%

“…The MSE can be computed using Equation (23) and can be used to obtain SQNR, as stated in Equation (22). MM, MSE and SQNR can also be determined analytically [2,22,43]. Penalty is given by…”

Section: Fixed-point Simulation-based Approachmentioning

confidence: 99%

See 2 more Smart Citations

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

Pradhan

Kabi

Mohanty

et al. 2015

Circuit Theory & Apps

View full text Add to dashboard Cite

show abstract

“…Signal-to-quantization-noise-ratio (SQNR) is chosen as an error measure to evaluate the accuracy of the proposed method [25], [44]. It is given by…”

Section: Proposed Solutionmentioning

confidence: 99%

An overflow free fixed-point eigenvalue decomposition algorithm: Case study of dimensionality reduction in hyperspectral images

Kabi

Sahadevan

Pradhan

2017

2017 Conference on Design and Architectures for Signal and Image Processing (DASIP)

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Abstract-We consider the problem of enabling robust range estimation of eigenvalue decomposition (EVD) algorithm for a reliable fixed-point design. The simplicity of fixed-point circuitry has always been so tempting to implement EVD algorithms in fixed-point arithmetic. Working towards an effective fixed-point design, integer bit-width allocation is a significant step which has a crucial impact on accuracy and hardware efficiency. This paper investigates the shortcomings of the existing range estimation methods while deriving bounds for the variables of the EVD algorithm. In light of the circumstances, we introduce a range estimation approach based on vector and matrix norm properties together with a scaling procedure that maintains all the assets of an analytical method. The method could derive robust and tight bounds for the variables of EVD algorithm. The bounds derived using the proposed approach remain same for any input matrix and are also independent of the number of iterations or size of the problem. Some benchmark hyperspectral data sets have been used to evaluate the efficiency of the proposed technique. It was found that by the proposed range estimation approach, all the variables generated during the computation of Jacobi EVD is bounded within ±1.

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Analysis of Finite Word-Length Effects in Fixed-Point Systems

Ménard

Caffarena

López

et al. 2018

Handbook of Signal Processing Systems

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Systems based on fixed-point arithmetic, when carefully designed, seem to behave as their infinite precision analogues. Most often, however, this is only a macroscopic impression: finite word-lengths inevitably approximate the reference behavior introducing quantization errors, and confine the macroscopic correspondence to a restricted range of input values. Understanding these differences is crucial to design optimized fixed-point implementations that will behave "as expected" upon deployment. Thus, in this chapter, we survey the main approaches proposed in literature to model the impact of finite precision in fixed-point systems. In particular, we focus on the rounding errors introduced after reducing the number of least-significant bits in signals and coefficients during the so-called quantization process.

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On the Fixed-Point Accuracy Analysis and Optimization of Polynomial Specifications

Cited by 19 publications

References 49 publications

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

Development of numerical linear algebra algorithms in dynamic fixed‐point format: a case study of Lanczos tridiagonalization

An overflow free fixed-point eigenvalue decomposition algorithm: Case study of dimensionality reduction in hyperspectral images

Analysis of Finite Word-Length Effects in Fixed-Point Systems

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