Stephen A. Dyer

Corrections to, and comments on, "an improved approximation for the Gaussian Q-Function"

In a recent letter, Karagiannidis and Lioumpas (see ibid., vol.11, no.8, p.644-6, Aug. 2007) proposed a new approximation for the Gaussian Q-function. We provide corrections to two equations, corrected entries to a table, and an alternative to a figure in the letter, as well as a few comments.

A Fast Spectrum-Recovery Method for Hadamard Transform Spectrometers Having Nonideal Masks

A computationally inexpensive method is presented for the recovery of spectra from measurements obtained with Hadamard transform spectrometers having nonideal masks. Normally, N measurements are required in order to recover an N-point spectrum; this method requires N + N0 measurements to be taken, where, typically, N0 ≤ 10. Once the additional measurements have been taken, only O(N[log2N + 2]) arithmetic operations—mostly additions or subtractions—are needed in order to recover the spectrum; a conventional procedure requires O(2N2) operations. Preliminary work for this method is minimal, requiring O(N) operations as opposed to O(N3) for a conventional procedure; this work needs to be done only once for a given spectrometer. The spectrum-estimate obtained is unbiased.

Approximations to Error Functions

This paper addresses approximations to error functions and points out three representative approximations, each with its own merits. Codys approximation is the most computationally intensive of the three, it is not overly so, and there is no arguing over its accuracy. The other two approximations are much simpler computationally, and they both yield accuracies that would be considered more than sufficient in most practical situations. Absolute relative error provides an effective measure of goodness, and, for approximations to the Q-function, it also places a loose bound on the absolute error in the approximation. Codys approximation is an effective surrogate for the true error function; the values provided by that approximation match the actual values of the error function to within roughly the precision of double-precision floating point arithmetic.

An Efficient Method for Recovering the Optimal Unbiased Linear Spectrum-Estimate from Hadamard Transform Spectrometers Having Nonideal Masks

A spectrum-recovery method is presented which efficiently computes an optimal unbiased linear spectrum-estimate for measurements obtained with Hadamard transform (HT) spectrometers having nonideal masks. This method has the following advantages over other spectrum-recovery techniques: it is computationally efficient, it requires no additional measurements, and it computes an optimal spectrum-estimate. In the method presented, after the mask of the HT spectrometer has been characterized, approximately 3N preliminary arithmetic operations are performed once for a given spectrometer, where N is both the number of spectral resolution-elements desired and the number of measurements required. Each spectrum-estimate to be recovered then requires only an additional O[N(log2N + 4)] arithmetic operations. In contrast, conventional methods for obtaining an optimal unbiased linear spectrum-estimate require O(N3) preliminary operations, and O(2N2) operations during each spectrum-recovery.

A senior-level RF design course combining traditional lectures with an open laboratory format

In Kansas State Universitys Design of Communication Circuits course, 10 to 15 students each semester are introduced to the theory behind wireless communications hardware used in modern products such as pagers, wireless LANs, and cellular telephones. In contrast to typical senior-design courses that have separate laboratory and lecture sections, the class combines lecture and laboratory work, with the instructor managing and grading both. This allows scheduling a series of projects that can be combined at the middle and end of the semester to produce relatively sophisticated products, such as working FM broadcast transmitters and receivers. An additional feature of the course is the use of an open laboratory where students can work at any time during normal business hours to build and test their circuits. This allows a class of 10 or more to share a single copy of expensive equipment such as a spectrum or network analyzer, while providing a studio-type environment in which students can share experiences more effectively with others.

Hadamard transform spectrometry

Abstract Dyer, S.A., 1991. Hadamard transform spectrometry. Chemometrics and Intelligent Laboratory Systems, 12: 101–115. Hadamard transform spectrometry (HTS) provides a means of obtaining a multiplex advantage with a dispersive spectrometric technique. Two recent advances, stationary electro-optic encoding masks and efficient spectrum-recovery techniques to compensate for nonidealities in the masks, have combined to revive interest in HTS. The concepts of HTS are presented, followed by a discussion of electro-optic masks and available methods for spectrum-recovery.

Implementation problems in Hadamard transform spectrometry

The multiplex advantage offered by Hadamard transform spectrometry (HTS) can improve the signal-to-noise ratio (SNR) at the output of a spectrometer. However, additional processing of the spectrometer output is required to recover the individual spectral components. A block diagram description of HTS and the spectrum-recovery process is presented. A computer simulation of this model has been developed and can be used to examine the effects of certain nonidealities that may typically be encountered. Traditionally, the inverse Hadamard transform (IHT) has been used as the spectrum-recovery method, but the IHT does not take into account the nonidealities associated with the multiplexing process. Two spectrum-recovery methods that address the problems of nonidealities are presented. The relative performance of all three methods is compared with regard to mean square error (MSE) and the computational efficiency of the algorithms necessary to implement the schemes. An example application is described, and the performance of the three spectrum-recovery schemes is discussed. >

Enhancement of Raman spectra obtained at low signal-to-noise ratios: matched filtering and adaptive peak detection

Two methods of Raman spectral peak enhancement are described. The matched-filter approach employs an optimal filter to maximize the signal-to-noise ratio at its output. The impulse response of the matched filter is the reversed version of the profile of the spectral peak to be enhanced; therefore, the effectiveness of the matched filter depends on a priori knowledge of the peaks shape. The adaptive peak detector (APD) utilizes a filter having a time-varying impulse response. APD performance is relatively insensitive to peak shape, but is instead contingent on a number of user-specified parameters. Background is provided to acquaint the reader with signal processing terminology, and practical guidelines are given for computationally inexpensive computer implementations of both techniques. Examples are shown which demonstrate the effects of these two enhancement methods on simulated Raman spectra.

A Hadamard-multiplexed spectrometer based on an acousto-optic tunable filter

This paper presents a Hadamard-multiplexed spectrometer based on an acousto-optic tunable filter (AOTF). The multifrequency capability of the AOTF enables it to be operated in a multiplexed mode. The AOTF can then be operated at lower power to obtain improved passband characteristics without a reduction in the signal-to-noise ratio as compared to a higher-power, nonmultiplexing instrument. An expandable, PC-based spectrometer has been developed to illustrate this technique. The current implementation provides seven-frequency Hadamard multiplexing and collects the spectra in multiplexed blocks. The AOTF is controlled via IEEE-485 bussed VHF synthesizer modules based on a hybrid DDS/PLL technique. The modular design allows convenient system expansion. The design of and specifications for this instrument are discussed.

On the Mean-Square Error of Various Spectrum-Recovery Techniques in Hadamard Transform Spectrometry

The multiplexing inherent in the Hadamard transform (HT) spectrometer can result in an improved spectrum-estimate when the detector is the major source of noise. A spectrum-estimate may be further improved by taking into account any nonidealities in the system. In this paper, observations concerning the errors associated with such estimates are presented, with the use of results obtained from computer simulations. Three spectrum-recovery techniques for an HT spectrometer having a nonideal electro-optic mask are considered in terms of the mean-square error (MSE) associated with a given estimate. The discussion of the MSE is with respect to the input spectrum to be estimated, the detector noise, the transmittances of the nonideal mask, and the use of coaddition. Included is a review of the computational efficiency and the statistical bias of each method. The relative performances of the spectrum-recovery methods are presented with examples to help identify the sources of error for each of the techniques.