Allen G. Lindgren

Optimal synthesis of second-order state-space structures for digital filters

Sufficient conditions are derived for a second-order statespace digital filter with L_2 scaling to be optimal with respect to output roundoff noise; and from these, a simple synthesis procedure is developed. Parallel-form designs produced by this method are equivalent to the block-optimal designs of Mullis and Roberts. The corresponding cascadeform designs are not equivalent, but they are shown, by example, to be quite close in performance. It is also shown that the coefficient sensitivities of this structure are closely related to its noise performance. Hence, the optimal design has low-coefficient sensitivity properties, and any other low-sensitivity design is a good candidate for near-optimal noise performance. The uniform-grid structure of Rader and Gold is an interesting and useful case in point.

The Inverse Discrete Radon Transform with Applications to Tomographic Imaging Using Projection Data

Publisher Summary This chapter deals with the situation when the Radon transform samples are on a regular grid. By viewing the Radon transform as a bivariate function, rather than as a parameterized univariate function as is usually done, the power of two-dimensional signal theory is applied to Radon transform theory. This approach permits the story of tomographic imaging, told by the varied and vast body of literature, to be unified and simplified. The chapter reviews and clarifies the requirements for a set of regularly spaced measurements to specify adequately the Radon transform. In order to provide insight into potential areas of applications of the radon transform in imaging, several areas where imaging is achieved from projection data are reviewed. The chapter discusses the historical background and review of reconstruction algorithms of tomography. Selected properties of the θu-coordinatized Radon transform are summarized and illustrated. The chapter elaborates on sampling the radon transform with parallel-beam projections and fan-beam projections. The existing questions regarding the amount of information in a finite number of projections are addressed and the processing required to extract this information is identified. It is shown to, within the standard approximations employed in sampling theory, that for Radon transforms adequately sampled on a rectangular grid an exact inverse discrete Radon transform exists. The effects of finite detector (source) size and motion are reviewed, and a systematic approach is again shown to be available through the unifying viewpoint provided by the application of two-dimensional signal processing techniques. The ultimate purpose of this chapter is to establish a firm theoretical foundation for tomographic imaging systems where the measurements are regularly spaced.

Adaptive step-size control in blind source separation

Abstract The behavior of the classic algorithm for blind source separation is reviewed for a fixed step-size. This analysis is extended to the case where the step-size decreases proportionally to 1/t. Although such a step-size sequence guarantees error-free convergence, mismatches and perturbations make it unrealistic for most practical implementations. To ameliorate these difficulties an error-dependent step-size must be employed. The coefficients of the estimating function provide an appropriate “measure of error” and serve as the basis for a self-adjusting time-varying step-size. Extensive simulations show the proposed approach tracks a time-varying mixing environment and performs error-free convergence in a time-invariant environment.

Transpose properties in the stability and performance of the classic adaptive algorithms for blind source separation and deconvolution

Abstract This paper presents a tutorial review of the problem of Blind Source Separation (BSS) and the properties of the classic adaptive algorithms when either the score-function or a general (nonscore) nonlinearity is employed in the algorithm. In new findings it is shown that the separating solution for both sub- and super-Gaussian signals can be stabilized by an algorithm employing any given nonlinearity. For these separating solutions the steady-state error levels are also given in terms of the nonlinearity and the pdfs of the source signals. These results show that a transpose symmetry exists between the nonlinear algorithms for sub- and super-Gaussian signals. The behavior of the algorithm is then detailed when the ideal score-function nonlinearity is replaced by a general (hard saturation or u3) nonlinearity. The phases of convergence to decorrelated output signals and then to recovery of the source signals are explained. The results are then extended to single- and multi-channel deconvolution and shown by analysis and extensive simulation to hold for mixed and convolved source signals. The results allow the design of stable algorithms for multichannel blind deconvolution with a general nonlinearity when sub- and super-Gaussian source signals are present.

Trajectory Estimation with Uncertain and Nonassociated Data

The local behavior of a maximum likelihood estimator that adaptively weights data of uncertain origin to make a probabilistic measurement-to-track assignment is examined. The results are placed in the framework of the classic estimation theory of Fisher and Cramer. The Cramer-Rao bound is derived and the MLS error level is compared with this lower bound and the level achieved with known data association. Deterioration in performance was found to depend solely on the false-detection to valid-contact ratio, and the actual (measured) covariance matrix is a scalar multiple of the covariance matrix computed for associated data.

Simplified adaptive IIR filters based on optimized orthogonal prefiltering

In order to reduce the circuit complexity associated with the estimation of echoes coming from systems with a long impulse response, we consider an estimator which is based on prefiltered input data. We propose a design of this prefilter which is optimal for a given system environment. In doing so, we represent the unknown discrete-time system by a set of characteristic impulse responses, which adequately describe the variety of the system. For such an environment we determine the optimum poles of a recursive prefilter. These poles are assumed to be fixed during the on-line LMS estimation process, which estimates the unknown echo by linearly weighting the prefilter states. An echo canceler for a typical European telephone subscriber-loop environment is used as a practical example. For this example the prefilter is optimized and realized with an orthogonal-state (lattice) filter. This not only reduces the computational costs-if compared to a conventional FIR filter design-but also permits a substantial speed-up of the on-line LMS adaptation process. >

Theory and Noise Dynamics of the Delay-Locked Loop

The delay-locked loop, with its applications to radar, sonar, seismic propagation studies and interferometry, is a basic geoscience instrument. This paper presents the theory and investigates the performance of the delay-locked loop with and without clipping in a noisy (Gaussian) environment. The signal processing required to produce an appropriate error characteristic is derived and the significant factors affecting system operation are presented in the form of a normalized model. When signal clipping is introduced, a delay tracking system is shown to result that is self-adjusting and capable of providing near optimum performance in varying signal/ noise environments. Application is made to the tracking of a research submersible randomly exploring the ocean floor. The dual problem of continuously tracking the peak spectral frequency of a random process exhibiting a slowly changing resonance is also considered.

Linear echo cancellation using optimized recursive prefiltering

In order to reduce the circuit complexity associated with the estimation of echos coming from systems with long impulse responses, an estimation is considered which is based on prefiltered input data. The design of a prefilter which is optimal for a given system environment is proposed. In doing so, the unknown discrete-time system is represented by a set of characteristic unit-sample responses which adequately describe the variation of the system. For such an environment, the optimal poles of a recursive prefilter are determined, and an orthogonal realization is suggested. These poles are assumed to be fixed during the on-line estimation process, which estimates the unknown system output by linearly weighting a set of optimally prefiltered input data. A 70-dB echo canceller for the North American Telephone subscriber loop environment is used as an example.<<ETX>>

Analysis of partitioned frequency-domain LMS adaptive algorithm with application to a hands-free telephone system echo canceller

For adaptive filters with long impulse responses and requiring modelling by an FIR filter, algorithms incorporating computationally efficient DFT-based adaptive block filters are the design of choice. The reduction in computational complexity is very significant in applications such as active noise control and the hands-free telephone system echo cancellation problem where impulse responses of more than a 1000 samples are common. The performance of frequency-domain adaptive algorithms is analysed and, when properly designed, shown to be equivalent to time-domain algorithms with uncorrelated input signals. The design parameters include: block length, DFT length and partitioning of the impulse response. The study includes both constrained and unconstrained parameter adaptation. Guidelines to the design of partitioned frequency-domain LMS (PFLMS) algorithms are given. Copyright

Noise Dynamics of the Phase-Locked Loop with Signal Clipping

Employing techniques similar to the averaging methods of Krylov and Bogoliubov, an approximate noise analysis of the phase-locked loop with signal clipping is presented. The validity of the method is demonstrated by comparing the stationary probability density function for the phase error, generated by a system simulation, with the derived theoretical results. The latter portion of the paper discusses the relation of the phase-locked loop to Kalman-Bucy filter theory and presents a demodulator design that illustrates the self-adaptive properties attainable in phase-locked loops with signal clipping.