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

Abstract

Abstract-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 timevarying 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 fixedpoint 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.

Accelerated Performance Evaluation of Fixed-Point Systems With Un-Smooth Operations

“…This translates to calling the function emulating fixed-point behavior for each and every fixed-point operation in the system. If the system under consideration consists of only smooth operations, it is possible to avoid performing fixed-point simulation by using the analytical techniques discussed in [6]. But, in the presence of even one un-smooth operation, the lack of applicability of analytical methods makes it inevitable to use fixed-point simulation for the entire system.…”

“…The analytical noise power evaluation [6] technique is applied to all clusters consisting of smooth operations. In this technique, an expression corresponding to the total output noise as a function of input noise variance and mean from each fixed-point arithmetic source is obtained analytically.…”

“…In [14], [16], and [17], techniques for expressing the PDF by primarily qualifying it with the fourth moment of total quantization error are described. Similar to the technique described in [6], these techniques essentially require computation of the path function (transfer function in the case of LTI systems) of the path from all operations to the system output. The actual values, however, depend on the fixed-point formats assigned to each of the arithmetic operations.…”

“…The technique for identification of all paths involving the preprocessing step is discussed in [7]. The generalization of the same is described in [6]. The algorithm adopted to find all paths in the system graph is described in [18].…”

“…In the case of large systems, both the size of input data set and the fixed-point search can be very large, and hence, performance evaluation by simulation proves difficult to be useful practically. Analytical techniques [6]- [8] for error estimation make use of well established stochastic models for errors due to finite precision [9]. Such techniques incur an initial overhead of evaluating the closed-form expression λ(w) specific for a given system or signal processing algorithm.…”