Multiresolution wavelet analysis of evoked potentials

Neurological injury, such as from cerebral hypoxia, appears to cause complex changes in the shape of evoked potential (EP) signals. To characterize such changes we analyze EP signals with the aid of scaling functions called wavelets. In particular, we consider multiresolution wavelets that are a family of orthonormal functions. In the time domain, the multiresolution wavelets analyze EP signals at coarse or successively greater levels of temporal detail. In the frequency domain, the multiresolution wavelets resolve the EP signal into independent spectral bands. In an experimental demonstration of the method, somatosensory EP signals recorded during cerebral hypoxia in anesthetized cats are analyzed. Results obtained by multiresolution wavelet analysis are compared with conventional time-domain analysis and Fourier series expansions of the same signals. Multiresolution wavelet analysis appears to be a different, sensitive way to analyze EP signal features and to follow the EP signal trends in neurologic injury. Two characteristics appear to be of diagnostic value: the detail component of the MRW displays an early and a more rapid decline in response to hypoxic injury while the coarse component displays an earlier recovery upon reoxygenation     

Walsh Functions and Gray Code

Gray code is a natural way of ordering binary vectors in dyadic space, hence it appears frequently in connetion with Walsh functions. In Paley's definition of Walsh functions their sequencies are arranged in Gray code. Gray code also appears in a new Walsh function generation algorithm which obtains a function by locating all its sign changes. There are certain computational advantages in using Gray code rather than sequency ordering. Examples in fast Walsh transform, dyadic convolution and digital filtering are given. Methods of Gray code to binary conversion are discussed.     

Ordering of Walsh Functions

The construction of Walsh functions is derived by means of the concepts of "symmetric copy" and "shift copy." Recursive relations based on the Kronecker product of matrices are also deduced from these concepts. There is a fourth useful ordering of the Walsh functions, called here X-ordering, in addition to the three known orderings usually referred to as Walsh-, Paley-, and Hadamardordering. An X-ordering function has the following features: lower order numbers of X-ordering correspond to even functions; higher order numbers of X-ordering correspond to odd functions; even order numbers of X-ordering correspond to lower sequencies; odd number of X-ordering correspond to higher sequencies. Finally, relations between the four orderings are given.     

Evolutionary periodogram for nonstationary signals

Presents a novel estimator for the time-dependent spectrum of a nonstationary signal. By modeling the signal, at any given frequency, as having a time-varying amplitude accurately represented by an orthonormal basis expansion, the authors are able to compute a minimum mean-squared error estimate of this time-varying amplitude. Repeating the process over all frequencies, they obtain a power distribution as a function of time and frequency that is consistent with the Wold-Cramer evolutionary spectrum. Based on the model assumptions, the authors develop the evolutionary periodogram (EP) for nonstationary signals, an estimator analogous to the periodogram used in the stationary case. They also derive the time-frequency resolution of the new estimator. The approach is free of some of the drawbacks of the bilinear distributions and of the short-time Fourier transform spectral estimates. It is guaranteed to produce nonnegative spectra without the cross-term behavior of the bilinear distributions, and it does not require windowing of data in the time domain. Examples illustrating the new estimator are given     

Walsh transform applied to digital filtering

The filtering process can also be implemented by other transformations than the well-known Fourier transformation. The paper is devoted to the one-dimensional (1-D) and two-dimensional (2-D) digital filtering process by the use of 1-D and 2-D discrete Walsh transforms (DWTs). Particularly, the optimum Walsh filter problem with one- and two-dimensional discrete data with emphasis on reduction of the computational requirements is considered. The optimum Walsh filters which are determined are based upon some modified minimum-mean-square-error criterion. Two basic classes of signals, which are to be filtered, have been taken into account: periodic signals and stationary random processes. Two computational examples are given.     

Walsh Spectral Estimates with Applications to the Classification of EEG Signals

A basis for the processing of EEG signals using the discrete, orthogonal set of Walsh functions is presented. The Walsh power spectrum is examined from the point of view of its statistical properties, especially as it relates to spectral resolution. Features, selected from the spectrum of sleep EEG data are compared to corresponding Fourier features. Each feature set is used to classify the data using a minimum-distance clustering algorithm. The results show that the Walsh spectral features classify the data in much the same way as the Fourier spectral features. This provides sufficient justification for usage ofWalsh spectral features in place of Fourier spectral features, enabling one to take advantage of the vast computational superiority of the fast Walsh transform over the fast Fourier transform.     

Sinusoids versus Walsh functions

In most of the applications contemplated for Walsh functions these binary waveforms would replace the more usual sinusoids, as the fast-Walsh-transform algorithm appears to make them very attractive for many kinds of signal processing. This paper begins with a brief review of the characteristics of Walsh functions and of their applications. Some old and some new interrelations are presented between sinusoids and Walsh functions, but the principal aim of the paper is to investigate the truncation and roundoff errors associated with the use of Fourier and of Walsh series. By employing simplifying approximations it is found that, for long samples of smooth signals, far more terms are required in the Walsh-series representation and greater accuracy is required of their coefficients for a given rms total error. Even for discontinuous signals the Walsh series may require substantially more terms, thus counterbalancing the computational advantage of the fast Walsh transform. This relative inefficiency of the Walsh-series representation of long waveforms may explain why it has not proven particularly effective in applications.     

Using an image database to constrain the acquisition and reconstruction of MR images of the human head

A training set of MR images of normal and abnormal heads was used to derive a complete set of orthonormal basis functions which converged to head-like images more rapidly than Fourier basis functions. The new image representation was used to reconstruct MR images of other heads from a relatively small number of phase-encoded signal measurements. The training images also determined exactly which phase-encoded signals should be measured to minimize image reconstruction error. These signals were nonuniformly scattered throughout k-space. Experiments showed that head images reconstructed with the new method had less serious truncation artifacts than conventional Fourier images reconstructed from the same number of signals. The resulting images were characterized by spatially variable spatial resolution and were particularly well-resolved in regions where the training images had structural detail.     

Walsh power spectrum for wide-sense stationary stochastic processes (Corresp.)

Existing concepts of Walsh power spectra for wide-sense stationary stochastic processes are restricted to the case of autopower spectra because they are based on real Walsh functions. In this paper a Walsh power spectrum is developed which is based on a system of complex Walsh functions and thus applies to auto and cross power spectra as well. It is shown that this Walsh power spectrum is related to the Fourier power spectrum by a linear transformation. This fact makes it possible to calculate Fourier power spectrum estimates from corresponding Walsh power spectrum estimates. An estimation algorithm for the Walsh power spectrum is given in the paper.     

LFM信号参数估计的插值FrFT修正算法

针对线性调频（LFM）信号参数估计插值FrFT算法在信噪比较低时性能下降而且针对不同参数估计性能不稳定的问题,提出了一种修正的插值FrFT算法。首先分析了现有插值FrFT算法问题出现的原因,然后定义了分数阶域量化频率,指出当信噪比较低时,若LFM信号初始频率接近分数阶域量化频率点,插值FrFT算法出现反向补偿的概率增大,性能下降。修正的插值FrFT算法改进了插值方向的判决条件以提高噪声免疫力,并通过频移LFM信号初始频率使其不在分数阶域量化频率点附近。最后,对不同初始频率的LFM信号进行仿真,结果表明,修正的插值FrFT算法提高了LFM信号参数估计精度,性能稳定,而计算量并没有明显增加。      

Some Comments Concerning Walsh Functions

The use of explicit forms for Walsh functions removes much of the confusion surrounding these interesting functions and permits simple proofs of their properties. Thus, for example, their period is far easier to determine than Alexandridis found, but their Fourier spectra are more complex than Schreiber's approximation suggests. For wave-form analysis, certain Walsh functions--the regular symmetric square waves--are more useful than are the others.     

Adaptive estimation of latency changes in evoked potentials

Changes in latency of evoked potentials (EP) may indicate clinically and diagnostically important changes in the status of the nervous system. A low signal-to-noise ratio of the EP signal makes it difficult to estimate small, transient, time-varying changes in latency, or delays. Here, the authors present an adaptive algorithm that estimates small delay (latency change) values even when EP signal amplitudes are time-varying. When the delay is time invariant, the adaptive algorithm produces an unbiased estimate with delay estimation error less than half of the sampling interval. A lower estimation error variance is obtained when, in a pair of signals, the adaptive algorithm delays the signal with the higher SNR. The adaptive delay estimation algorithm was tested on intra-operative recordings of somatosensory EP, and analysis of those recordings reveals that the anesthetic etomidate produces a step change in the amplitude and latency of the EP signals     

Walsh function current patterns and data synthesis for electrical impedance tomography

A data collection method which uses Walsh functions as injection current patterns is presented. This method can satisfy two conditions: the optimality of current patterns in every iteration and the single-time data measurement condition. The use of Walsh functions simplifies the design of current sources since only two levels of current (+1 and -1) are required, whereas sinusoidal injection requires a digital-to-analog converter to produce many different values of currents. Compared to diagonal or neighboring type of pulses as injection current patterns, Walsh injection current patterns provide more information about the interior of the subject since Walsh function simulate low and high spatial frequency patterns. Therefore, Walsh function injection uses the simplicity of pulse type injection and yields the better distinguishability or SNR of sinusoidal injection.     

Deconvolution by thresholding in mirror wavelet bases

The deconvolution of signals is studied with thresholding estimators that decompose signals in an orthonormal basis and threshold the resulting coefficients. A general criterion is established to choose the orthonormal basis in order to minimize the estimation risk. Wavelet bases are highly sub-optimal to restore signals and images blurred by a low-pass filter whose transfer function vanishes at high frequencies. A new orthonormal basis called mirror wavelet basis is constructed to minimize the risk for such deconvolutions. An application to the restoration of satellite images is shown.     

A beamformer for the acquisition of evoked potentials

Evoked potentials (EP) contain information about various physiological parameters and the estimation and detection of these signals can aid in the diagnosis of many pathological conditions. However, the signal-to-noise ratio (SNR) for EP measurement is often very low, and thus signal processing techniques must be employed to enhance the SNR. A delay and sum beamformer acquisition system has the potential for significant SNR improvement in EP measurements. In this communication it is shown that an electrode array acquisition system implements a uniform coherent delay and sum beamformer. The performance of the beamformer is characterized in terms of the number of electrodes, and cross-channel correlation. When compared to conventional ensemble averaging, the beamformer reduces the number of response repetitions required to achieve a given SNR by a factor which approaches the number of channels in the acquisition system.     

Walsh-to-Fourier Spectral Conversion for Periodic Waves

Relations are developed for the determination of the Fourier spectra of frequency-limited periodic waves from truncated Walsh spectra. The matrix conversion process is simplest if the highest-order Walsh coefficient in the spectru to be converted is 2n, where n is an integer. For such cases, compensation for truncation consists of a diagonal matrix that premultiplies the Walsh to Fourier conversion matrix and the elements of which are [(sinx)/x]-2 terms. Element values range between unity and less than ?2/4. The same compensation matrix is used for determning the Walsh spectra. of sequency-limited waves from 2n Fourier expansion terms. Examples are included which demonstrate the spectral conversion processes, Walsh to Fourier and Fourier to Walsh.     

Performance analysis for impulse radio and direct-sequence impulse radio in narrowband interference

The impact of narrowband interference (NBI) on two ultra wideband (UWB) systems is analyzed. The two systems are impulse radio (IR) and a variation of it, termed direct-sequence IR (DS-IR). The signal-to-noise ratio (SNR) at the decision device of a correlation receiver is computed for both systems, assuming that the NBI is wide sense stationary and that the channel is frequency-selective. The SNR is expressed by means of a simple equation involving the signal and the interference spectrum. Next, a statistical model for the interference is introduced, considering the interference as the sum of a given number of sinusoidal signals with random powers and frequencies. The bit-error rate of IR and DS-IR is derived. The results are then specialized and compared with simulations in three case studies of practical interest, where the NBI is a single jammer with deterministic power and frequency, a multitone signal with random frequencies, or a grid of interfering signals with random powers.     

Walsh analysis of power-law systems (Corresp.)

Walsh-type signals exciting memoryless power-law systems are considered. The input is assumed to consist of a sum of weighted Walsh functions. A method is given for finding the weights of the Walsh functions in the output when the input is a finite Walsh series. It is shown that the fast Walsh transform can be used to facilitate the necessary computation.     

Presence detection of binary-phase-shift-keyed and direct-sequencespread-spectrum signals using a prefilter-delay-and-multiply device

The specific problem of detecting the presence of either binary-phase-shift-keyed (BPSK) signals or BPSK direct-sequence spread-spectrum (DS/SS) signals with a prefilter-delay-and-multiply (PFDM) device is considered. Using stationary process theory and Fourier analysis, the optimum PFDM structures for signal presence detection of BPSK signals with known bit rates and carrier frequencies and BPSK DS/SS signals with known chip rates and carriers in additive colored Gaussian noise are derived. The structures are optimum in the sense that they maximize the spectral signal-to-noise ratio (SNR) of an output periodic waveform which has fundamental frequency equal to the bit or chip rate of the signal. Two of the optimum structures that are derived and analyzed are the optimal prefilter-square device and the optimal PFDM with delay set to one half of the signal's bit or chip duration     

A new Walsh domain technique of harmonic elimination and voltagecontrol in pulse-width modulated inverters

A method for selective harmonic elimination in pulse-width-modulated (PWM) inverter waveforms by the use of Walsh functions is presented. The Walsh operational matrix of PWM is introduced as a means of obtaining the Walsh spectral equations of PWM waveforms. The slope and intercept Fourier operational matrices of PWM are also introduced as a means of obtaining Fourier spectral equations of PWM waveforms. A noniterative algorithm that produces piecewise-linear, global solutions between angles and for the angles is proposed. The algorithm also produces the full range of variation of fundamental voltage for given harmonic elimination constraints. The set of systems of linear equations obtained replaces the system of nonlinear transcendental equations used in the Fourier series harmonic elimination approach. In general, the algorithm makes possible the synthesis of two-state PWM inverter waveforms with specified old harmonic content