We consider an n-element array formingK \leq nbeams in order to receive signals from K sources whose positions and power levels are known. The array employs K sets of n weights to segregate the K signals. These weights are determined by direct computation based on the available position information. Currently known techniques require (for n = K) about (4/3)K4computer "operations" (multiply and add). We present here an improved technique that accomplishes that in 8K3operations, yielding a reduction by a factor of K/6.

This paper proposes a single-frequency communication system consisting ofKmobile users simultaneously communicating with a central station equipped with a computationally adapted array consisting ofn≥Kantennas. The array employsKsets ofnweights to segregate the signals received from theKusers. The weights are determined by direct computation based on position information determined by direct computation based on position information transmitted by the users. Currently known techniques require (forn=K) about (4/3)K4computer `operations' (multiply and add). An improved technique is developed here which accomplishes this in 8/K3operations, yielding a reduction by a factor ofK/6.

Winograd's DFT algorithm is applied to the design of a fast DFT machine. The main feature of the proposed machine in comparison with alternative designs is a shift in hardware investment from computing elements to storage elements. There are indications that this could lead to significant cost reductions. A more detailed version of this paper [1] also describes a spectrum analyzer variant.

Examination of rounding and truncation errors in radix (−2) systems establishes the following facts: 1) truncation error is 33 percent lower than in the positive radix case. 2) There is no 1-bit rounding algorithm. 3) The error bound of the standard (1-bit) positive radix rounding algorithm can be approached asymptotically by using more than one of the discarded bits. For N bits, this error bound is multiplied by 1 + (3. 2N−l)−1.

The digital filtering algorithm of W. D. Little, which employs a large RAM to obtain high speed, is implemented in a simple hardware configuration.

The solution of a set of m linear equations with a non-Hermitian Toeplitz associated matrix is considered. Presently available fast algorithms solve this set with 4 m 2 “operations” (an “operation” is defined here as a set of one addition and one multiplication). An improved algorithm requiring only 3 m 2 “operations” is presented.

The Appendix of the above paper1 presents a serial sign flipper for radix (-2). Recently, a simpler realization has been brought to the attention of the author. The improved circuit has been developed independently of the published design by H. C. Wilck of the Jet Propulsion Laboratory, Pasadena, Calif. His approach to the problem is quite different from the author's.

Analysis of the bit-level operations involved in the convolution realizing a nonrecursive digital filter leads to hardware designs of digital filters based on the operation of counting.

Analysis of the bit-level operations involved in the convolutions realizing recursive digital filters leads to hardware designs of such filters based on the operation of counting. Various designs realizing both the canonic and "direct" forms are presented with particular emphasis on low-cost low-speed high-flexibility machines.

Hardware Fourier transformations are considered with the goal of increasing speed through parallel operation. Two designs in which the basic element is a fast counter are developed. One is applicable when the number of transformed points N is prime while the other, requiring somewhat more hardware, is applicable to all N.

The direct conversion of an analog signal to its digital representation in base (-2) is considered. A successive approximation version strategy is developed and applied to a practical converter that is described in detail.

My paper "Negative Radix Conversion," contained the only reference known to me at the time [1].

Adoption of a negative radix for number representation results in a system that is totally indifferent to the sign of the number. Compared to the sign-magnitude representation, the approach considered here requires one extra digit.

The algorithm of W. F. Trench for the inversion of Toeplitz matrices is presented with a detailed proof for the case of non-Hermitian matrices. The only condition necessary to insure the validity of the algorithm is that all principal minors be nonzero.