Romuald Rocher

Analytical Approach for Numerical Accuracy Estimation of Fixed-Point Systems Based on Smooth Operations

In embedded systems using fixed-point arithmetic, converting applications into fixed-point representations requires a fast and efficient accuracy evaluation. This paper presents a new analytical approach to determine an estimation of the numerical accuracy of a fixed-point system, which is accurate and valid for all systems formulated with smooth operations (e.g., additions, subtractions, multiplications and divisions). The mathematical expression of the system output noise power is determined using matrices to obtain more compact expressions. The proposed approach is based on the determination of the time-varying impulse-response of the system. To speedup computation of the expressions, the impulse response is modelled using a linear prediction approach. The approach is illustrated in the general case of time-varying recursive systems by the Least Mean Square (LMS) algorithm example. Experiments on various and representative applications show the fixed-point accuracy estimation quality of the proposed approach. Moreover, the approach using the linear-prediction approximation is very fast even for recursive systems. A significant speed-up compared to the best known accuracy evaluation approaches is measured even for the most complex benchmarks.

Accuracy constraint determination in fixed-point system design

Most of digital signal processing applications are specified and designed with floatingpoint arithmetic but are finally implemented using fixed-point architectures. Thus, the design flow requires a floating-point to fixed-point conversion stage which optimizes the implementation cost under execution time and accuracy constraints. This accuracy constraint is linked to the application performances and the determination of this constraint is one of the key issues of the conversion process. In this paper, a method is proposed to determine the accuracy constraint from the application performance. The fixed-point system is modeled with an infinite precision version of the system and a single noise source located at the system output. Then, an iterative approach for optimizing the fixed-point specification under the application performance constraint is defined and detailed. Finally the efficiency of our approach is demonstrated by experiments on an MP3 encoder.

Fixed-point configurable hardware components

To reduce the gap between the VLSI technology capability and the designer productivity, design reuse based on IP (intellectual properties) is commonly used. In terms of arithmetic accuracy, the generated architecture can generally only be configured through the input and output word lengths. In this paper, a new kind of method to optimize fixed-point arithmetic IP has been proposed. The architecture cost is minimized under accuracy constraints defined by the user. Our approach allows exploring the fixed-point search space and the algorithm-level search space to select the optimized structure and fixed-point specification. To significantly reduce the optimization and design times, analytical models are used for the fixed-point optimization process.

A Hierarchical Methodology for Word-Length Optimization of Signal Processing Systems

The problem of converting floating point algorithms to implementation friendly fixed point formats is often solved as an optimization problem where the precision is traded to gain in the implementation cost. The complexity of the problem is known to grow exponentially with more optimizable variables. This paper proposes a divide and conquer technique to solve the growing size of the problem. The approach in this technique is original in the sense that it is formulated from a designers perspective rather than merely attempting to divide and conquer at the algorithmic level. This paper introduces the single noise source model based on which the proposed technique is built. A mixed approach for error propagation is also explained keeping in view of the elements in the circuit that cannot be handled analytically.

Accuracy evaluation of fixed-point LMS algorithm

The implementation of adaptive filters with fixed-point arithmetic requires the computation quality to be evaluated. The accuracy may be determined by calculating the global quantization noise power in the system output. A new model for evaluating analytically the global noise power in the LMS algorithm and in the NLMS algorithm is developed. Two existing models are presented, then the model is detailed and compared with the ones before. The accuracy of our model is analyzed by simulation.

Fast performance evaluation of fixed-point systems with un-smooth operators

Fixed-point refinement of signal processing systems is an essential step performed before implementation of any signal processing system. Existing analytical techniques to evaluate performance of fixed-point systems are not applicable to the errors due to quantization in the presence of un-smooth operators. Thus, it is inevitable to use simulation to evaluate performance of fixed-point systems in the presence un-smooth operators. This paper proposes a hybrid technique which can be used in place of pure simulation to accelerate the performance evaluation. The principle idea in the proposed hybrid approach is to selectively simulate parts of the system only when un-smooth errors occur but use analytical results otherwise. The acceleration thus obtained reduces the performance evaluation time which can be used to explore a wider word-length design space or speedup the optimization process. This method has been tried on a complex MIMO sphere decoding algorithm and the results obtained show several orders of magnitude improvement in terms of evaluation time.

Analytical approach for analyzing quantization noise effects on decision operators

The presence of decision operators has proved to be a serious impediment for a fully analytical noise power estimation technique. This paper proposes a generalized decision operator which can potentially capture the behavior of all possible types of decision operators and provides a fully analytical technique to handle them while performing quantization noise power estimation. The proposed method is applied to BPSK and 16-QAM decision operators. The total error rate and the PDF of the error signal are found to follow the simulation to a great degree of accuracy.

Integer word-length optimization for fixed-point systems

Time-to-market and implementation cost are high-priority considerations in the automation of digital hardware design. Nowadays, digital signal processing applications are implemented into fixed-point architectures due to its advantage of manipulating data with lower word-length (WL). Thus, floating-point to fixed point conversion is mandatory. However, this conversion is translated into optimizing the integer word length (IWL) and fractional word length (FWL). Optimizing the IWL can significantly reduce the cost when the application is tolerant to a low probability of overflows. In this paper, we propose a new IWL optimization algorithm that exploits selective simulation technique to reduce both the implementation cost and optimization time. The efficiency of the algorithm is illustrated through experiments, where 17 to 22 % of cost reduction with respect to interval arithmetic and acceleration factor up to 617 with respect to classical max-1 algorithm are reported.

Accuracy evaluation of fixed-point based LMS algorithm

The implementation of adaptive filters with fixed-point arithmetic requires computation quality evaluation. The accuracy may be determined by computing the global quantization noise power at the system output. In this paper, a new model for evaluating analytically the global noise power in LMS-based algorithms is presented. Thus, the model is developed for LMS and NLMS algorithms. The accuracy of our model is analyzed by simulations.

Shaping probability density function of quantization noise in fixed point systems

Word-length optimization provides opportunities for minimization of implementation cost metrics such as power, area and delay. The constraints on cost in case of implementation on miniature embedded systems platforms continues to grow more stringent. Implementation of complex systems such as wireless communication transceivers is known to greatly benefit from optimal word-lengths. However, WL optimization of such complex systems poses a NP hard combinatorial problem. In line with a divide and conquer technique already explored, this paper proposes to study the probability density function of quantization noise at the output of an arithmetic operator based system. The proposed technique is based on clustering of noise sources according to their power using Kurtosis of the total noise as the clustering criteria. It is noted in this paper that the noise PDF becomes important only at the un-smooth boundaries in the system. The proposed technique is tested on a synthetic 32 tap FIR filter and on a MIMO algorithm. Results obtained show a marked improvement in the behavior of analytical model with the use of the proposed density distribution shaping algorithm.