The Rectangular Transform (RT) method for computing convolutions belongs to a family of Reduced Computational Complexity (RCC) algorithms. Convolution calculations by the RT method were programmed for the Research Signal Processor (RSP) and run on the RSP simulator, giving tabulations of numbers of RSP machine cycles. One of the original objectives was to see how well the original RSP architecture was suited to the RCC algorithms and to be able to make suggestions for possible changes. The results are also intended to demonstrate the efficiency of the RT convolution algorithms on a microprocessor with a limited instruction set and to show how to construct efficient RT programs for digital convolution. All results are given for the original RSP, as it was before the modification which resulted in the Real-time Signal Processor described in another paper in this issue.

Rectangular transforms for digital convolution on the research signal processor

Polynomial transforms (PT's) are known to be efficient for computing convolutions. In this correspondence, the use of polynomial transforms for computing autocorrelation has been considered. A method has been described for using polynomial transforms instead of the FFT in the Rader's algorithm for autocorrelation, and the number of arithmetic operations compared; it is found that the use of PT's does not result in the expected computational advantage over the FFT method.

http://ieeexplore.ieee.org/iel6/29/26183/01164299.pdf

On the computation of autocorrelation using polynomial transforms

It can be shown that the mean frequency of a real-valued stochastic signal can be expressed as an integral of the normalized autocorrelation function r(τ) weighted by a function equal to 1/τ2. The fast decline of the weighting function implies that the behavior of the autocorrelation function for small values of τ is the most important portion for estimation of the mean frequency of a signal. It is demonstrated in a simulation study that estimates of the mean frequency with mean squared error equal to the error in estimates obtained via a FFT derived mean frequency estimate can be obtained by using just a few lags of the normalized autocorrelation function with a computational effort substantially less than that required for estimation via FFT. Upper bounds, that can be used as guidelines when implementing the estimator, are given for the bias error introduced by using just a few lag values of the autocorrelation function.

Computationally efficient estimation of the mean frequency for real-valued signals

A technique based on the recently developed rectangular transforms is proposed for computing the autocorrelation function of a data sequence. The refinements suggested with the technique make it computationally more efficient than Rader's algorithm using the fast Fourier transform and the Winograd-Fourier transform algorithm, when the number of desired lag values is smaller than 30. However, the technique as such can be used for higher values.

http://ieeexplore.ieee.org/iel6/29/26147/01163436.pdf

V. Reddy

Papers

High-speed computation of autocorrelation using rectangular transforms