Dimension Of Signal Space Research Articles

Fault dictionary based upon stimulus design

A method of fault signature generation is presented that is based upon state space analysis of linear circults. An input control sequence is designed to reduce a nontrivial initial state of the circuit under test to the zero state in finite time. The realization of this stimulus as a piecewise constant waveform has step amplitudes that are exponential functions of the poles of the circuit under test. Perturbations of these amplitudes, engendered by element drift failure, constitute a fault signature. Single element value perturbations engender fault signature trajectories in signal space, and the fault dictionary is constructed by defining disjoint decision regions (hypervolumes) around each fault signature trajectory in the signal space. Circuit zeros of transmission allow the dimension of the signal space to be augmented with perturbation of such response waveform parameters as zero crossings. The theory of stimulus design for fault isolation in linear networks and a generalized matrix inverse method for computing the stimulus amplitudes from the pulse response of strictly proper circuits are presented. Examples of response waveforms and fault signature trajectories are given for several circuits.

Center-of-gravity information feedback

We consider <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> signals in a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</tex> -dimensional signal space. These <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> signals are used to communicate over an additive Gaussian white noise channel. It is shown that if a noiseless feedback channel is available one could use this feedback channel to inform the transmitter of the location of the center of gravity of the signal structure and thus obtain a very efficient signaling scheme. Each signal point is assigned a mass proportional to its posterior probability at the particular instant. The center of gravity is used by the transmitter as a new origin for the signal space. It is shown that some previously considered coding schemes for channels with feedback are particular cases of center-of-gravity feedback. The probability of error decreases as a double exponential function of the coding delay as opposed to an exponential decrease for one-way systems. The effect of noise in the feedback path is briefly considered.

On the Optimum Performance of N-ary Systems Having Two Degrees of Freedom

A digital transmission system with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> possible transmitted symbols is considered. If the time between transmitted symbols is T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> seconds and the bandwidth is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</tex> , the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> possible symbols correspond to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> vectors in a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2WT_{0}</tex> dimensional signal space. This paper considers the theoretical properties of a class of digital systems where the signal space is two-dimensional. Such systems are both amplitude-and phase-modulated. Approximate expressions are derived for the average probability of error for these systems as a function of the placement of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> symbol vectors in the twodimensional signal space. Optimum placements are then given which minimize this probability of error for a given average or peak power SNR constraint. It is shown that the optimum channel structure is a function of the alphabet size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , and the type of power constraint, as well as the SNR. In general the optimum system is a phase-modulated system for low SNR's and for alphabet sizes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \leq 16</tex> in the high SNR region. The performance of this optimum system in terms of channel capacity and probability of error is then compared with the performance of one-dimensional systems, AM-only and PM-only, in a complete set of curves for both peak and average power.