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We consider the problem of estimating mutual information between dependent data, an important problem in many science and engineering applications. We propose a data-driven estimator of mutual information in this paper. The main novelty of our solution lies in transforming the data to frequency domain to make the problem tractable. We define a novel metric-mutual information in frequency (Ml-in-frequency)-to detect and quantify the dependence between two random processes across frequency using Cramer's spectral representation. Our solution calculates mutual information as a function of frequency to estimate the mutual information between the dependent data over time and validate its performance on linear and nonlinear models. We then use our MI-in-frequency metric to infer the cross-frequency coupling during epileptic seizures, by analyzing electrocorticographic recordings from a total of eleven seizures in four medial temporal lobe epilepsy patients.

The capacity defines the ultimate fidelity limits of information transmission by any system. We derive the capacity of parallel Poisson process channels to judge the relative effectiveness of neural population structures. Because the Poisson process is equivalent to a Bernoulli process having small event probabilities, we infer the capacity of multi-channel Poisson models from their Bernoulli surrogates. For neural populations wherein each neuron has individual innervation, inter-neuron dependencies increase capacity, the opposite behavior of populations that share a single input. We use Shannon's rate-distortion theory to show that for Gaussian stimuli, the mean-squared error of the decoded stimulus decreases exponentially in both the population size and the maximal discharge rate. Detailed analysis shows that population coding is essential for accurate stimulus reconstruction. By modeling multi-neuron recordings as a sum of a neural population, we show that the resulting capacity is much less than the population's, reducing it to a level that can be less than provided with two separated neural responses. This result suggests that attempting neural control without spike sorting greatly reduces the achievable fidelity. In contrast, single-electrode neural stimulation does not incur any capacity deficit in comparison to stimulating individual neurons.

We consider the problem of digital communication in a Rayleigh flat-fading environment using a multiple-antenna system, when the channel state information is available neither at the transmitter nor at the receiver. It is known that at high signal-to-noise ratio (SNR), or when the coherence interval is much larger than the number of transmit antennas, a constellation of unitary matrices can achieve the capacity of the noncoherent system. However, at low SNR, high spectral efficiencies, or for small values of coherence interval, the unitary constellations lose their optimality and fail to provide an acceptable performance. In this work, inspired by the Stein's lemma, we propose to use the Kullback-Leibler (KL) distance between conditional distributions to design space-time constellations for noncoherent communication. In fast fading, i.e., when the coherence interval is equal to one symbol period and the unitary construction provides only one signal point, the new design criterion results in pulse amplitude modulation (PAM)-type constellations with unequal spacing between constellation points. We also show that in this case, the new design criterion is equivalent to design criteria based on the exact pairwise error probability and the Chernoff information. When the coherence interval is larger than the number of transmit antennas, the resulting constellations overlap with the unitary constellations at high SNR, but at low SNR they have a multilevel structure and show significant performance improvement over unitary constellations of the same size. The performance improvement becomes especially more significant when an appropriately designed outer code or multiple receive antennas are used. This property, together with the facts that the proposed constellations eliminate the need for training sequences and are most suitable for low SNR, makes them a good candidate for uplink communication in wireless systems.

Signals that represent information may be classified into two forms: numeric and symbolic. Symbolic signals are discrete-time sequences that at, any particular index, have a value that is a member of a finite set of symbols. Set membership defines the only mathematical structure that symbolic sequences satisfy. Consequently, symbolic signals cannot be directly processed with existing signal processing algorithms designed for signals having values that are elements of a field (numeric signals) or a group. Generalizing an approach due to Stoffer (see Biometrika, vol.85, p.201-213, 1998), we extend time-frequency and time-scale analysis techniques to symbolic signals and describe a general linear approach to developing processing algorithms for symbolic signals. We illustrate our techniques by considering spectral and wavelet analyses of DNA sequences.

The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA). It was accepted in 1999 as an ANSI standard and in 2000 as IEEE and NIST standards. It was also accepted in 1998 as an ISO standard and is under consideration for inclusion in some other ISO standards. Unlike the ordinary discrete logarithm problem and the integer factorization problem, no subexponential-time algorithm is known for the elliptic curve discrete logarithm problem. For this reason, the strength-per-key-bit is substantially greater in an algorithm that uses elliptic curves. This paper describes the ANSI X9.62 ECDSA, and discusses related security, implementation, and interoperability issues.

Often in extracellular recordings the activity of multiple units is recorded. “Spike sorting” is the process of determining the time of occurrence of each unit's action potentials. In this work, we suggest a framework for bounding the performance of spike sorting algorithms in terms of the mean squared estimation error. We examine the suitability of two theoretical bounds, the Cramer–Rao bound and the Ziv–Zakai bound. We compare the empirical performance of the optimal spike sorting system and a suboptimal scheme to the theoretical limits in two examples.

In an attempt to gain insight into the design of linear detectors for additive white noise channels (discrete-time case), we describe several procedures, both optimal and suboptimal. Using the Wiener representation for nonlinear systems, we derive an ad hoc suboptimal design procedure. Exact designs are found when the noise amplitude's probability distribution is stable and when the noise is Laplacian. Considering all the linear detectors thus derived, no general form for the optimal linear detector's unit-sample response becomes apparent. Performance analyses and simulations indicate substantial performance losses occur when linear detectors are used instead of optimal (likelihood ratio) ones.

Weave analysis of artist canvas examines x-ray images taken of the paintings. Algorithms assume an underlying regularity of the canvas weave over short distances and exploits shortspace spectral analysis to determine the fundamental frequency of the horizontal and vertical thread regularity. However, many paintings are too large to be covered by a single x-ray. Feature point analysis exploits brushstrokes and composition to merge several x-ray images into a single one, taking into account and removing both spatial and amplitude distortions. Certain master artists used low-quality canvas that is more irregular than the norm. A theoretical study of quasiperiodic signals shows that while the expected spectrum is a peak that broadens as period irregularity increases, sample function spectra have distinct peaks having an envelope equal to the expected spectrum.

In the context of the narrowband array processing problem, robust methods for accurately estimating the spatial correlation matrix using a priori information about the matrix structure are developed. By minimizing the worse case asymptotic variance, robust, structured, maximum-likelihood-type estimates of the spatial correlation matrix in the presence of noises with probability density functions in the in -contamination and Kolmogorov classes are obtained. These estimates are robust against variations in the noise's amplitude distribution. The Kolmogorov class is demonstrated to be the natural class to use for array processing applications, and a technique is developed to determine exactly the size of this class. Performance of bearing estimation algorithms improves substantially when the robust estimates are used, especially when nonGaussian noise is present. A parametric structured estimate of the spatial correlation matrix that allows direct estimation of the arrival angles is also demonstrated. >

The problem of determining the maximum number of narrowband signals whose parameters can be estimated with a linear array of M equally spaced sensors is examined. While this number previously has been taken to be M-1 when the signals are mutually uncorrelated, it is shown how to estimate directions and amplitudes for as many as 2M-1 signals. This over twofold increase is accomplished by retaining snapshot-to-snapshot phase information usually lost in algorithms based on spatial correlation matrices. The approach uses length 2M real signal vectors rather than the usual M complex vectors. It is shown that 2M of these real vectors are linearly independent with probability one, and, thus, in the presence of additive white noise, the parameters of 2M-1 signals can be estimated. An algorithm for determining directions and amplitudes is presented. Because of the algorithm's computational complexity, its application is limited to small M and low time-bandwidth products. >

Thread counting algorithms seek to determine the vertical and horizontal weave densities of the canvas underlying a painting. X-ray images of a painting serve as the raw data. Canvas texture can be modeled as a sum of two sinusoids having nearly orthogonal spatial frequencies. The actual painting serves as an additive signal that only distracts from the thread-counting process. The algorithm employs a variant of short-time Fourier analysis to the image domain. In simple cases, two-dimensional Fourier transforms of small areas reveal isolated peaks at the proper vertical and horizontal frequencies. In more challenging cases, even within the same painting, additional peaks appear in the expected frequency range, requiring heuristics to extract from the spectral measurements representative thread-count estimates for a painting. The resulting algorithm, while still under development, can analyze weave density for entire paintings with an accuracy comparable to human measurements.

The task of determining the weave density in the canvas support of Old Master paintings is introduced as a period extraction problem. Because of the way paintings were commonly prepared and preserved, the threads in the horizontal and vertical directions in the canvas support can be counted from x-ray images of the painting. Current procedures are tedious, time-consuming, and (usually) insufficiently documented. This paper describes the design of an algorithm for counting threads from x-rays that uses the Fourier Transform of the Radon Transform of a portion of the image with some crude, but appropriate, decision-making. The algorithm is presented as a sequence of refinements based on a simple mathematical model of the available image data: high resolution x-rays of paintings by Vincent van Gogh from the collection of the Van Gogh Museum. Over 900 spot counts were performed manually by a student team at Cornell using a graphical user interface created for this project. These manual counts provide a dataset for evaluating performance of the algorithm. A major goal is to convince art historians of the viability of automated (and semi-automated) counting procedures.

While evidence indicates that neural systems may be employing sparse approximations to represent sensed stimuli, the mechanisms underlying this ability are not understood. We describe a locally competitive algorithm (LCA) that solves a collection of sparse coding principles minimizing a weighted combination of mean-squared error and a coefficient cost function. LCAs are designed to be implemented in a dynamical system composed of many neuron-like elements operating in parallel. These algorithms use thresholding functions to induce local (usually one-way) inhibitory competitions between nodes to produce sparse representations. LCAs produce coefficients with sparsity levels comparable to the most popular centralized sparse coding algorithms while being readily suited for neural implementation. Additionally, LCA coefficients for video sequences demonstrate inertial properties that are both qualitatively and quantitatively more regular (i.e., smoother and more predictable) than the coefficients produced by greedy algorithms.

We develop neurally plausible local competitive algorithms (LCAs) for reconstructing compressively sensed images. Reconstruction requires solving a sparse approximation problem. Our solution technique uses a neural network that emulates how the brain may actually solve such sparse approximation problems. This method could ultimately be implemented in analog electronics, which would not only significantly diminish processing time, but also enable analog implementations for both acquisition and reconstruction.

A new sphericity test statistic is proposed for the complex case with a distribution that approaches the asymptotic chi-squared distribution much faster than previously proposed statistics, especially for situations in which the sources are very difficult to detect, such as closely spaced or weak sources. Consequently, the probability of a correct estimate is superior to those resulting from other sphericity tests. The capability of this new sphericity test to detect closely space sources is examined and compared to the resolution capabilities of bearing estimation algorithms as well as other well-known source detection algorithms. >

We develop a simulated annealing technique to jointly optimize a distributed quantization structure meant to maximize the asymptotic error exponent of a downstream classifier or detector. This distributed structure sequentially processes an input vector and exploits broadcasts to improve the best possible error exponents. The annealing approach is a robust technique that avoids local maxima and is easily tailored to a broadcast quantizer's structural constraints

Practical sparse approximation algorithms (particularly greedy algorithms) suffer two significant drawbacks: they are difficult to implement in hardware, and they are inefficient for time-varying stimuli (e.g., video) because they produce erratic temporal coefficient sequences. We present a class of locally competitive algorithms (LCAs) that correspond to a collection of sparse approximation principles minimizing a weighted combination of reconstruction MSE and a coefficient cost function. These systems use thresholding functions to induce local nonlinear competitions in a dynamical system. Simple analog hardware can implement the required nonlinearities and competitions. We show that our LCAs are stable under normal operating conditions and can produce sparsity levels comparable to existing methods. Additionally, these LCAs can produce coefficients for video sequences that are more regular (i.e., smoother and more predictable) than the coefficients produced by greedy algorithms.

This paper applies the concepts of information processing [S. Sinanovic et al., Jun. 2002] to the study of binary detectors and block decoders in a single user digital communication system. We quantify performance in terms of the information transfer ratio /spl gamma/ which measures how well systems preserve discrimination information between two stochastic signals. We investigate hard decision detectors and minimum distance decoders in various additive noise environments. We show that likelihood ratio digital demodulators maximize /spl gamma/.

The exact equations governing the narrow-band resolution and detection capabilities of conventional (CONV) and minimum energy (ME) adaptive beamforming [1], are the basis for a class of linear arrays which offers simultaneously optimum resolution and detection capacities. Signal processing considerations indicate that, in this class, linear minimum redundancy (LMR) arrays [2] are best suited to CONV beamforming, and linear minimum hole (LMH) arrays [3] are best suited to minimum energy (ME) beamforming. The effect of coarray redundancies and holes on sidelobe level is examined analytically, and lower bounds are established on the number of coarray redundancies and holes. A simple recursive algorithm for the design of LMH arrays is presented.

The effect of source coherence on the capabilities of classical, minimum energy (ME), and linear predictive (LP) array processing algorithms to estimate the bearings of two equal-energy sources is examined. Source coherence is shown to adversely affect the resolution and detection capabilities, as well as the bias characteristics, of all three algorithms. For linear arrays of equally-spaced sensors (LES arrays), the superior resolution capability of the LP algorithm is demonstrated. However, the LP algorithm is least capable of detecting highly coherent sources. The estimates of source bearing produced by each algorithm are shown to be asymptotically biased.

A signal-processing model for the generation of point processes is derived. In the context of this model, the interval between successive events is treated as a time series. The input-output relationship of a system generating inter-event intervals is shown to be derivable from the intensity of the desired point process. Both stationary and non-stationary processes can be generated with this technique. The input-output characteristic of this system is shown to be nonlinear for most interesting cases.

The use of eigenanalysis to aid in the determination of bearing in passive sonar systems is shown to fall into two classes, each of which has been applied to minimum energy and linear prediction algorithms. The application of eigenanalysis is extended to non-white ambient noise environments in a particular class. The performances of these various methods is compared to some extent in this situation.

When transmitting a sampled signal digitally, data and error correction bits must be transmitted at least as fast as the sampling rate. Typically, each bit is allocated the same transmission time interval, which means the optimal detector yields the same error probability for each bit. An alternative is to vary the bit interval duration according to the bit's contribution to the reconstructed sample. The optimal solution yields significant gains in mean-squared error (several dB) over that provided by equal-duration bit intervals. These gains occurred over a wide range of signal-to-noise ratios. When block error correction is performed, we derive the optimal decoder from a Bayesian viewpoint and show that gains obtain here as well.

Array processing algorithms generally assume that the received signal, composed of both narrowband signals and noise, is Gaussian. This is not true in general. In the context of the narrowband array processing problem, the authors develop robust methods to estimate accurately the spatial correlation matrix. This estimate is robust against amplitude distribution of the noise, be it Gaussian or not, and takes into account sensor placement. Given this estimate, existing array processing algorithms designed for Gaussian circumstances can be used on non-Gaussian problems. The resulting algorithms can be used in the presence of any type of second-order noise process and perform nearly as well as existing algorithms do with Gaussian noise. >

In many situations some information about the structure of the covariance matrix of a random process is known beyond the fact that it is symmetric and positive definite; for instance, the matrix is frequently Toeplitz. Many people have considered the structured covariance matrix estimation problem for Gaussian processes. However, in actual practice, random signals are seldom, if ever, Gaussian. By using a generalization to processes with known non-Gaussian densities, the authors demonstrate how to find the maximum-likelihood estimate of complex Toeplitz covariance matrices and then evaluate the use of this estimate in some passive array beamforming algorithms. There is substantial improvement in the performance of these bearing estimation algorithms when the authors' estimate is used, especially when non-Gaussian noise is present. >

Averaging of the spatial correlation function over the spatial variable-spatial averaging- is performed on the estimated spatial correlation matrix prior to beamforming calculations. Generally speaking, this preprocessing operation reduces the intersignal coherence in the spatial correlation matrix and the variance of the spatial correlation function estimate. Two types of spatial averaging are considered: subaperture averaging and redundancy averaging. Subaperture averaging is shown to significantly reduce coherence only when the source propagation directions are widely spaced while variance reduction is accompanied by a reduction in apparent aperture. In contrast, the reduction of coherence resulting from redundancy averaging is relatively insensitive to source bearing separations. While the aperture is not affected, redundancy averaging may induce bias in the estimated source bearings. >

The authors examine the maximum number of signals whose parameters can be estimated with a linear, equally spaced array of M sensors. The conventional view is that, when none of the signals is fully correlated, this number is one less than the number of sensors; however, the authors show how to make the extension to one less than twice the number of sensors. This increase in the number of signals is accomplished by using length 2M real signal vectors rather than the usual length M complex vectors. It is shown that 2M of these real vectors are linearly independent with probability one, and, thus, angles of arrival can be unambiguously estimated for 2M-1 signals. >

Doppler-time images (DTIs) of a rotating object were formed from the returns of high resolution radars by expressing the Doppler values as a function of time over a span of ranges. Returns from scatterers on the object yielded a characteristic signature of the object's rotation: a section of a sinusoid centered about the negatively sloped zero-crossing. As several scatterers are usually present, these signatures may overlap. The return from each scatterer is noisy and, more importantly, may be missing (falling below the detection threshold) or may be multi-valued (yielding several values exceeding the detection threshold). The Hough transform-the extraction of parametrically expressible features-was used to extract straight line approximations to the return signatures. This algorithm is shown to be insensitive to missing or extraneous values. Quantization in Hough transform computation is shown to determine the sensitivity of the calculation to noisy values. Parameters of the signatures were extracted using a data-dependent clustering algorithm designed to be insensitive to quantization and feature parameterization. >

Traditional change detection strategies are limited in cases where the data are nonstationary and the distributions are unknown. We present an algorithm for change detection problems in which we do not know the form of the distribution. Our algorithm uses distributed detection with a bank of type-based front end detectors that achieve asymptotically optimal type I error performance. Our simulations indicate that this algorithm performs much better than traditional methods.

A type is the histogram estimate of a stationary sequence's amplitude distribution. From training data, types are computed and use to determine which training data best describe subsequently obtained observations. Such type-based detectors are asymptotically optimal in the sense that the maximal exponential error rate is achieved. For digital communication systems using direct-sequence signaling, type-based detectors are shown to be effective. Training data are obtained from preambles, and then used to make individual bit decisions. Simulations show that in this communications scenario, type-based detectors yield nearly optimal performance without any a priori channel information.

Optimal detectors for non-Gaussian channels are nonlinear, and are therefore sensitive to problem parameters. We describe a technique for designing linear detectors for arbitrary additive noise channels that takes into account dependence structure, amplitude distribution, and transmitted signals. This technique is based on Wiener's theory of nonlinear systems, and amounts to cross-correlating the optimal detector's output with its input when it is driven by white Gaussian noise. We have not proven this linear detector's optimality; simulations demonstrate that it has some of the properties required of optimal detectors in several cases. >

The universal classifier using empirically observed statistics, developed by Gutman (1989), is applied to the spread-spectrum communications problem. We call the resulting classifier a type-based detector and show that it has close to optimal performance over a wide range of unknown channel environments, including static and multipath fading channels in additive Gaussian and non-Gaussian noise. In a multiple-access system, the type-based detector is capable of rejecting high levels of multiple-access interference. The type-based detector uses a training sequence to form its knowledge about the bit hypotheses, and in a communications system, the training sequence corresponds to a periodically transmitted preamble or "sync" sequence.

Type-based receivers assume no a priori channel model, and were previously shown to be effective in direct-sequence spread spectrum communications. In this paper, we investigate the macro-diversity combining of multiple type-based receivers in single-mobile-user reception (BPSK). We present a method for the exact calculation of the bit error rate of type-based receivers in this scenario. Maximal ratio combining of type-based receivers gives near optimal performance in Gaussian noise and better performance than combining matched filter outputs in Laplacian noise. The performance gain achieved by diversity reception is significant regardless of noise statistics, especially in the presence of frequency non-selective Rayleigh fading.

Information processing theory endeavors to quantify how well signals encode information and how well systems, by acting on signals, process information. We use information-theoretic distance measures, the Kullback-Leibler distance in particular, to quantify how well signals represent information. The ratio of distances calculated between two informationally different signals at a system's output and input quantifies the system's information processing properties. Using this approach, we derive the fundamental processing capabilities of simple system architectures that apply universally: the systems and the kinds of signals they process and produce do not affect our general results. Applications in array signal processing and in neural signal analysis illustrate how to apply the theory.

Information processing is performed when a system preserves aspects of the input related to what the input represents while it removes other aspects. To describe a system's information processing capability, input and output need to be compared in a way invariant to the way signals represent information, Kullback-Leibler distance, information-theoretic measure that reflects the data processing theorem, is calculated on the input and output separately and compared to obtain information transfer ratio. We consider the special case where input serves several parallel systems and show that this configuration has the capability to represent the input information without loss. We also derive bounds for asymptotic rates at which the loss decreases as more parallel systems are added and show that the rate depends on the input distribution.

Symbolic signals are, in discrete-time, sequences of quantities that do not assume numeric values. In the most general case, these quantities have no mathematical structure other than that they are members of some set, but they can have a sequential structure. The authors show that processing such signals does not entail mapping them directly to the integers, which would impose more structure-ordering and arithmetic-than present in the data. The authors describe how linear estimation and prediction can be performed on symbolic sequences. They show how spectrograms can be computed from neural population responses and from DNA sequences.

When applied to continuous-time observations, type-based detection strategies are limited by the necessity to crudely quantize each sample. To alleviate this problem, we smooth the types for both the training and observation data with a linear filter. This post-processing improves the detector performance significantly (error probabilities decrease by over a factor of three) without incurring a significant computational penalty. However this improvement depends on the amplitude distribution and on the quantizer's characteristics.

On the basis of a geometric theory of detection, the authors extend the notion of a signal constellation, a concept deeply rooted in Gaussian problems, to the non-Gaussian case. Significant differences between optimal designs for Gaussian and non-Gaussian situations are shown. In particular, square-wave signals are much more important in heavy-tailed, non-Gaussian noise situations than in Gaussian ones. Furthermore, design guidelines for non-Gaussian problems can vary with the number of signal set members and can depend on SNR. The extent to which suboptimal designs affect performance (using Gaussian-based designs in non-Gaussian situations, for example) can be predicted from calculations of the Kullback information, but only in the sense of determining how the logarithmic error probability rates differ. >

The capabilities of classical, minimum energy, and linear predictive array processing algorithms to estimate the bearings of two equal-energy sources is examined. Signal coherence is shown to affect adversely the resolution and detection capabilities, as well as the bias characteristics, of all three algorithms. For linear arrays of equally spaced sensors, the superior resolution capability of the linear predictive algorithm is demonstrated. The value of utilizing prediction elements in the center of the array to resolve very closely spaced source bearings is demonstrated. However, the linear predictive algorithm is least capable of detecting highly coherent sources. A tradeoff is established between resolving capability and sensitivity to finite averaging. Conditions are established which indicate which algorithm is best suited to anticipated levels of signal coherence and averaging. The estimates of source bearing produced by each algorithm are shown to be asymptotically biased. The bias produced by the classical beamformer is most severe, while the minimum energy beamformer produces the least bias.

In processing continuous-time signals by digitalmeans, it is necessary to represent the signal by a digital sequence. There are many ways other than periodic sampling for obtaining such a sequence. The requirements for such representations and some examples are discussed within the framework of simulating linear time-invariant systems. The representation of digital sequences by other digital sequences is also discussed, with particular emphasis on the use of such representations to implement a nonlinear warping of the digital frequency axis. Some applications and hardware implementation of this digital-frequency warping are described.

It is shown that non-Gaussian time series require new analysis methods to extract their structure. Spectral analysis does not seem to provide the precise information required to analyze important aspects of a time series. The conditional expected value can, in simple cases, be related to Components of the Barrett-Lampard expansion, which provides a mathematical tool for determining the system which can generate the time series.

Variable digital filters allow the frequency characteristics of a filter to be manipulated. Frequency transformations have been successfully used to obtain a variable filter from a nonrecursive prototype. However, this technique cannot be applied directly to filters having a recursive structure. A variation of this method is presented which allows variable filters to be obtained from both recursive and nonrecursive prototypes. This method has special advantages when the prototype has an equal number of poles and zeros (e.g., digital filters obtained by the bilinear transformation from an analog filter).

Many high‐resolution bearing estimation algorithms involve the computations of the spatial correlation matrix. The utility of applying an eigenanalysis to the spatial correlation matrix has been described by many workers for a variety of algorithms. A cohesive theory for analyzing the properties of these algorithms is presented. The accuracy of bearing estimates is shown to be directly related to the time‐bandwidth product of the computations. However, the various algorithms depend on this quantity in different ways. Adaptive beamforming is shown to yield somewhat better estimates than linear predictive methods when eigenanalysis is used. [Work supported by ONR.]

The composition and pressure of breathing mixtures used by divers translate frequency components of their speech nonlinearly. Recently, a method of forming a new sampled time function from a given finite‐duration sampled time function in which the spectra of the two are related through a nonlinear warping in frequency has been discussed [A. Oppenheim, D. Johnson, and K. Steiglitz, “Computation of Spectra with Unequal Resolution Using the Fast Fourier Transform,” Proc. IEEE 59, 299 (1971)]. The technique is based on the expansion of the original time function in terms of a set of functions related to Laguerre functions. The coefficients in the expansion, when interpreted as samples of a new time function, have a Fourier transform that is equal to that of the original function with a nonlinear transformation in frequency. These coefficients are obtained by processing the original signal with a second‐order network in cascade with a chain of first‐order all‐pass networks. To apply the technique to translation of divers' speech, individual pitch periods are processed by the above network. These modified pitch periods are than reconvolved with a train of pitch pulses to synthesize the translated speech.