Alan V. Oppenheim

The importance of phase in signals

In the Fourier representation of signals, spectral magnitude and phase tend to play different roles and in some situations many of the important features of a signal are preserved if only the phase is retained. Furthermore, under a variety of conditions, such as when a signal is of finite length, phase information alone is sufficient to completely reconstruct a signal to within a scale factor. In this paper, we review and discuss these observations and results in a number of different contexts and applications. Specifically, the intelligibility of phase-only reconstruction for images, speech, and crystallographic structures are illustrated. Several approaches to justifying the relative importance of phase through statistical arguments are presented, along with a number of informal arguments suggesting reasons for the importance of phase. Specific conditions under which a sequence can be exactly reconstructed from phase are reviewed, both for one-dimensional and multi-dimensional sequences, and algorithms for both approximate and exact reconstruction of signals from phase information are presented. A number of applications of the observations and results in this paper are suggested.

Synchronization of Lorenz-based chaotic circuits with applications to communications

A circuit implementation of the chaotic Lorenz system is described. The chaotic behavior of the circuit closely matches the results predicted by numerical experiments. Using the concept of synchronized chaotic systems (SCSs), two possible approaches to secure communications are demonstrated with the Lorenz circuit implemented in both the transmitter and receiver. In the first approach, a chaotic masking signal is added at the transmitter to the message, and at the receiver, the masking is regenerated and subtracted from the received signal. The second approach utilizes modulation of the coefficients of the chaotic system in the transmitter and corresponding detection of synchronization error in the receiver to transmit binary-valued bit streams. The use of SCSs for communications relies on the robustness of the synchronization to perturbations in the drive signal. As a step toward further understanding the inherent robustness, we establish an analogy between synchronization in chaotic systems, nonlinear observers for deterministic systems, and state estimation in probabilistic systems. This analogy exists because SCSs can be viewed as performing the role of a nonlinear state space observer. To calibrate the robustness of the Lorenz SCS as a nonlinear state estimator, we compare the performance of the Lorenz SCS to an extended Kalman filter for providing state estimates when the measurement consists of a single noisy transmitter component. >

Signal reconstruction from phase or magnitude

In this paper, we develop a set of conditions under which a sequence is uniquely specified by the phase or samples of the phase of its Fourier transform, and a similar set of conditions under which a sequence is uniquely specified by the magnitude of its Fourier transform. These conditions are distinctly different from the minimum or maximum phase conditions, and are applicable to both one-dimensional and multidimensional sequences. Under the specified conditions, we also develop several algorithms which may be used to reconstruct a sequence from its phase or magnitude.

Estimation of fractal signals from noisy measurements using wavelets

The role of the wavelet transformation as a whitening filter for 1/f processes is exploited to address problems of parameter and signal estimations for 1/f processes embedded in white background noise. Robust, computationally efficient, and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood, and Cramer-Rao bounds are obtained. Included among these algorithms are optimal fractal dimension estimators for noisy data. Algorithms for obtaining Bayesian minimum-mean-square signal estimates are also derived together with an explicit formula for the resulting error. These smoothing algorithms find application in signal enhancement and restoration. The parameter estimation algorithms find application in signal enhancement and restoration. The parameter estimation algorithms, in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process, are useful in problems of signal detection and classification. Results from simulations are presented to demonstrated the viability of the algorithms. >

Multi-channel signal separation by decorrelation

Identification of an unknown system and recovery of the input signals from observations of the outputs of an unknown multiple-input, multiple-output linear system are considered. Attention is focused on the two-channel case, in which the outputs of a 2*2 linear time invariant system are observed. The approach consists of reconstructing the input signals by assuming that they are statistically uncorrelated and imposing this constraint on the signal estimates. In order to restrict the set of solutions, additional information on the true signal generation and/or on the form of the coupling systems is incorporated. Specific algorithms are developed and tested. As a special case, these algorithms suggest a potentially interesting modification of Widrows (1975) least-squares method for noise cancellation, where the reference signal contains a component of the desired signal. >

Effects of finite register length in digital filtering and the fast Fourier transform

When digital signal processing operations are implemented on a computer or with special-purpose hardware, errors and constraints due to finite word length are unavoidable. The main categories of finite register length effects are errors due to A/D conversion, errors due to roundoffs in the arithmetic, constraints on signal levels imposed by the need to prevent overflow, and quantization of system coefficients. The effects of finite register length on implementations of linear recursive difference equation digital filters, and the fast Fourier transform (FFT), are discussed in some detail. For these algorithms, the differing quantization effects of fixed point, floating point, and block floating point arithmetic are examined and compared. The paper is intended primarily as a tutorial review of a subject which has received considerable attention over the past few years. The groundwork is set through a discussion of the relationship between the binary representation of numbers and truncation or rounding, and a formulation of a statistical model for arithmetic roundoff. The analyses presented here are intended to illustrate techniques of working with particular models. Results of previous work are discussed and summarized when appropriate. Some examples are presented to indicate how the results developed for simple digital filters and the FFT can be applied to the analysis of more complicated systems which use these algorithms as building blocks.

Signal processing with fractals: a wavelet-based approach

Wavelet transmission statistically self-similar signals detection and estimation with 1/processes deterministically self-similar signals fractal modulation linear self-similar signals.

Nonlinear filtering of multiplied and convolved signals

An approach to some nonlinear filtering problems through a generalized notion of superposition has proven useful In this paper this approach is investigated for the nonlinear filtering of signals which can be expressed as products or as convolutions of components. The applications of this approach in audio dynamic range compression and expansion, image enhancement with applications to bandwidth reduction, echo removal, and speech waveform processing are presented.

Sequential signal encoding from noisy measurements using quantizers with dynamic bias control

Signal estimation from a sequential encoding in the form of quantized noisy measurements is considered. As an example context, this problem arises in a number of remote sensing applications, where a central site estimates an information-bearing signal from low-bandwidth digitized information received from remote sensors, and may or may not broadcast feedback information to the sensors. We demonstrate that the use of an appropriately designed and often easily implemented additive control input before signal quantization at the sensor can significantly enhance overall system performance. In particular, we develop efficient estimators in conjunction with optimized random, deterministic, and feedback-based control inputs, resulting in a hierarchy of systems that trade performance for complexity.

Discrete representation of signals

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.