Omid Sarbishei

Analytical Optimization of Bit-Widths in Fixed-Point LTI Systems

Analyses of range and precision are important for high-level synthesis and verification of fixed-point circuits. Conventional range and precision analysis methods mostly focus on combinational arithmetic circuits and suffer from major inefficiencies when dealing with sequential linear-time-invariant circuits. Such problems mainly include inability to analyze precision when quantization of constant coefficients is taken into account, and lacking efficient word-length optimization algorithms to handle both variables and constants, while satisfying the error metrics. The algorithms presented in this paper solve these problems. Experiments illustrate the efficiency and robustness of our algorithms.

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

Fixed-point accuracy analysis and optimization of polynomial data-flow graphs with respect to a reference model is a challenging task in many digital signal processing applications. Range and precision analysis are two important steps of this process to assign suitable integer and fractional bit-widths to the fixed-point variables and constant coefficients in a design such that no overflow occurs and a given error bound on maximum mismatch (MM) or mean-square-error (MSE) and signal-to-quantization-noise ratio (SQNR) is satisfied. This paper explores efficient optimization algorithms based on robust analyses of MM and MSE/SQNR for fixed-point polynomial data-flow graphs. Experimental results illustrate the robustness of our analyses and the efficiency of the optimization algorithms compared to previous work.

Analysis of range and precision for fixed-point linear arithmetic circuits with feedbacks

Analysis of range and precision is always an important task for high level synthesis and verification. Although several researches have been dedicated to these two problems, in the case of linear fixed-point arithmetic circuits with feedbacks such as an Infinite Impulse Response (IIR) filter, conventional approaches are either constituting major overestimations or cannot handle arbitrary order feedback circuits. In this paper we focus on this problem and propose two efficient heuristics for range and precision analysis of such circuits, when the input and error bounds are given. The methods can be used for efficient integer and fractional bit-width allocation in the optimization flow. Moreover, for the purpose of module reusability and matching, verification algorithms have been proposed. Experimental results prove robust computations of range and precision.

Analysis of precision for scaling the intermediate variables in fixed-point arithmetic circuits

This paper presents a new technique for scaling the intermediate variables in implementing fixed-point polynomial-based arithmetic circuits. Analysis of precision has been used first to set the input and coefficient bit-widths of the polynomial so that a given error bound is satisfied. Then, we present an efficient approach to scale and truncate different intermediate variables with no need of re-computing precision at each stage. After applying it to all the intermediate variables, a final precision computation and sensitivity analysis is performed to set the final values of truncation bits so that the given error bound remains satisfied. Experimental results on a set of polynomial benchmarks show the robustness and efficiency of the proposed technique.

MSE minimization and fault-tolerant data fusion for multi-sensor systems

Multi-sensor data fusion is an efficient method to provide both accurate and fault-tolerant sensor readouts. Furthermore, detection of faults in a reasonably short amount of time is crucial for applications dealing with high risks. In order to deliver high accuracies for the sensor measurements, it is required to perform a calibration for each sensor. This paper focuses on designing a fault-tolerant calibrated multisensor system. First, the least squares method is applied to calibrate each sensor using a linear curve fitting function. Next, an analytical technique is proposed to carry out a fault-tolerant multi-sensor data fusion, while minimizing the Mean-Square-Error (MSE) for the final sensor readout. While our data fusion approach is applicable to different multi-sensor systems, the experimental results are shown for 16 temperature sensors, where an environmental thermal chamber was used as the reference model to calibrate the sensors and perform the measurements.

On the Fixed-Point Accuracy Analysis and Optimization of FFT Units with CORDIC Multipliers

Fixed-point Fast Fourier Transform (FFT) units are widely used in digital communication systems. The twiddle multipliers required for realizing large FFTs are typically implemented with the Coordinate Rotation Digital Computer (CORDIC) algorithm to restrict memory requirements. Recent approaches aiming to optimize the bit-widths of FFT units while satisfying a given maximum bound on Mean-Square-Error (MSE) mostly focus on the architectures with integer multipliers. They ignore the quantization error of coefficients, disabling them to analyze the exact error defined as the difference between the fixed-point circuit and the reference floating-point model. This paper presents an efficient analysis of MSE as well as an optimization algorithm for CORDIC-based FFT units, which is applicable to other Linear-Time-Invariant (LTI) circuits as well.

Analysis of Mean-Square-Error (MSE) for fixed-point FFT units

Range and precision analysis are important steps in assigning suitable integer and fractional bit-widths to the fixed-point variables in a design such that no overflow occurs and a given error bound on maximum mismatch and (or) Mean-Square-Error (MSE) is satisfied. Although, range and maximum mismatch analysis of linear arithmetic circuits has been studied before [8], regarding analysis of MSE, the previous works [9,10,12] cannot analyze the error, when it is defined as the difference between the fixed-point circuit and the reference model, e.g., floating-point format. This paper presents an efficient analysis of MSE for linear arithmetic circuits narrowing on Fast Fourier Transform (FFT) units. Furthermore, an optimization algorithm is introduced to set the bit-widths in an FFT unit while satisfying a given maximum bound on MSE. Experimental results prove the robustness of our MSE analysis and the efficiency of the optimization algorithm compared to [12] for an 8K FFT unit.

Challenges in verifying and optimizing fixed-point arithmetic-intensive designs

Arithmetic circuit plays a key role in digital signal processing (DSP). A datapath is used to implement the specification usually represented as a polynomial. The two most important problems are verification and optimization of the arithmetic circuits. Circuit verification confirms whether the implementation can realize the specification with correct behavior or two implementations match well, and optimization generates suitable bit-widths according to constraints. This paper depicts specification of arithmetic circuits, explains the techniques of verification and optimization, and describes current challenges in arithmetic circuit designs.

Verification of fixed-point datapaths with comparator units using Constrained Arithmetic Transform (CAT)

Arithmetic Transform (AT) [1, 16, 17] is an efficient spectral technique, to analyze range and precision of fixed-point polynomial datapaths, among other methods including AA [4, 15] and SMT [5]. However, the major inefficiency of AT is that it cannot handle the datapaths with comparator units, which imply the non-arithmetic if-statements. This paper presents the approach, Constrained Arithmetic Transform (CAT), to perform range and precision analysis of fixed-point datapaths with comparator units. A custom branch-and-bound search is also introduced to provide more cutting branches and perform faster analyses of range and precision, by making use of safe and negligible overestimations. Experimental results prove the efficiency of our approach.

Fixed-point accuracy analysis of datapaths with mixed CORDIC and polynomial computations

Fixed-point accuracy analysis of imprecise datapaths in terms of Maximum-Mismatch (MM) [1], or Mean-Square-Error (MSE) [14], w.r.t. a reference model is a challenging task. Typically, arithmetic circuits are represented with polynomials; however, for a variety of functions, including trigonometric, hyperbolic, logarithm, exponential, square root and division, Coordinate Rotation Digital Computer (CORDIC) units can result in more efficient implementations with better accuracy. This paper presents a novel approach to robustly analyze the fixed-point accuracy of an imprecise datapath, which may consist of a combination of polynomials and CORDIC units. The approach builds a global polynomial for the error of the whole datapath by converting the CORDIC units and their errors into the lowest possible order Taylor series. The previous work for almost accurate analysis of MM [1] and MSE [14, 15] in large datapaths can only handle polynomial computations.