Charles M. Rader

The chirp z-transform algorithm

A computational algorithm for numerically evaluating the z -transform of a sequence of N samples is discussed. This algorithm has been named the chirp z -transform (CZT) algorithm. Using the CZT algorithm one can efficiently evaluate the z -transform at M points in the z -plane which lie on circular or spiral contours beginning at any arbitrary point in the z -plane. The angular spacing of the points is an arbitrary constant, and M and N are arbitrary integers. The algorithm is based on the fact that the values of the z -transform on a circular or spiral contour can be expressed as a discrete convolution. Thus one can use well-known high-speed convolution techniques to evaluate the transform efficiently. For M and N moderately large, the computation time is roughly proportional to (N+M) \log_{2}(N+M) as opposed to being proportional to N . M for direct evaluation of the z -transform at M points.

Discrete Fourier transforms when the number of data samples is prime

The discrete Fourier transform of a sequence of N points, where N is a prime number, is shown to be essentially a circular correlation. This can be recognized by rearranging the members of the sequence and the transform according to a rule involving a primitive root of N. This observation permits the discrete Fourier transform to be computed by means of a fast Fourier transform algorithm, with the associated increase in speed, even though N is prime.

Digital filter design techniques in the frequency domain

Digital filtering is the process of spectrum shaping using digital components as the basic elements. Increasing speed and decreasing size and cost of digital components make it likely that digital filtering, already used extensively in the computer simulation of analog filters, will perform, in real-time devices, the functions which are now performed almost exclusively by analog components. In this paper, using the z-transform calculus, several digital filter design techniques are reviewed, and new ones are presented. One technique can be used to design a digital filter whose impulse response is like that of a given analog filter; other techniques are suitable for the design of a digital filter meeting frequency response criteria. Another technique yields digital filters with linear phase, specified frequency response, and controlled impulse response duration. The effect of digital arithmetic on the behavior of digital filters is also considered.

Discrete Convolutions via Mersenne Transrorms

A transform analogous to the discrete Fourier transform is defined in the ring of integers with a multiplication and addition modulo a Mersenne number. The arithmetic necessary to perform the transform requires only additions and circular shifts of the bits in a word. The inverse transform is similar. It is shown that the product of the transforms of two sequences is congruent to the transform of their circular convolution. Therefore, a method of computing circular convolutions without quantization error and with only very few multiplications is revealed.

Hyperbolic householder transformations

A class of transformation matrices, analogous to the Householder matrices, is developed, with a nonorthogonal property designed to permit the efficient deletion of data from least-squares problems. These matrices, which we term hyperbolic Householder, are shown to effect deletion, or simultaneous addition and deletion, of data with much less sensitivity to rounding errors than for techniques based on normal equations. When the addition/deletion sets are large, this numerical robustness is obtained at the expense of only a modest increase in computations, and when only a relatively small fraction of the data set is modified, there is a decrease in required computations. Two applications to signal processing problems are considered. First, these transformations are used to obtain a square root algorithm for windowed recursive least-squares filtering. Second, the transformations are employed to implement the rejection of spurious data from the weight vector estimator in an adaptive array.

A new principle for fast Fourier transformation

An alternative form of the fast Fourier transform (FFT) is developed. The new algorithm has the peculiarity that none of the multiplying constants required are complex-most are pure imaginary. The advantages of the new form would, therefore, seem to be most pronounced in systems for which multiplication are most costly.

Adaptive beamformer orthogonal rejection test

Research in the area of signal detection in the presence of unknown interference has resulted in a number of adaptive detection algorithms. Examples of such algorithms include the adaptive matched filter (AMF), the generalized likelihood ratio test (GLRT), and the adaptive coherence estimator (ACE). Each of these algorithms results in a tradeoff between detection performance for matched signals and rejection performance for mismatch signals. This paper introduces a new detection algorithm we call the adaptive beamformer orthogonal rejection test (ABORT). Our test decides if an observation contains a multidimensional signal belonging to one subspace or if it contains a multidimensional signal belonging to an orthogonal subspace when unknown complex Gaussian noise is present. In our analysis, we use a statistical hypothesis testing framework to develop a generalized likelihood ratio decision rule. We evaluate the performance of this decision rule in both the matched and mismatched signal cases. Our results show that for constant power complex Gaussian noise, if the signal is matched to the steering vector, ABORT, GLRT, and AMF give approximately equivalent probability of detection, higher than that of ACE, which trades detection probability for an extra invariance to scale mismatch between training and test data. Of these four tests, ACE is most selective and, therefore, least tolerant of mismatch, whereas AMF is most tolerant of mismatch and, therefore, least selective, ABORT and GLRT offer compromises between these extremes, with ABORT more like ACE and with GLRT more like AMF.

A Simple Method for Sampling In-Phase and Quadrature Components

We present a method of developing in-phase and quadrature samples of a band-limited RF waveform. The problem of matching gain and phase response differences between the two components is avoided by a combination of mixing to an IF frequency, sampling and digitizing, and digital filtering. The novelty of the method is in the design of the digital filter, which is realized as a pair of 900 phase splitting networks with several symmetries which are exploited to save computation.

Effects of parameter quantization on the poles of a digital filter

A configuration is proposed which leads to a digital filter that is less sensitive to parameter quantization than are the standard configurations. A few examples illustrate the differences between the various realizations.

VLSI systolic arrays for adaptive nulling [radar]

Presents a case study of the design of a computationally intensive system to do adaptive nulling of interfering signals for a phased-array radar with many antenna elements. The goal of the design was to increase the computational horsepower available for this problem by about three orders of magnitude under the tight constraints of size, weight and power which are typical of an orbiting satellite. By combining the CORDIC rotation algorithm, systolic array concepts, Givens transformations, and restructurable VLSI, we built a system as small as a package of cigarettes, but capable of the equivalent of almost three billion operations per second. Our work was motivated by the severe limitations of size, weight and power which apply to computation aboard a spacecraft, although the same factors impose costs which are worth reducing in other circumstances. For an array of N antennas, the cost of the adaptive nulling computation grows as N/sup 3/, so simply using more resources when N is large is not practical. The architecture developed, called MUSE (matrix update systolic experiment) determines the nulling weights for N=64 antenna elements in a sidelobe cancelling configuration. After explaining the antenna nulling system, we discuss another DSP computation that might benefit from similar architecture, technology, or algorithms: the solution of Toeplitz linear equations.