The geometry of linearly and quadratically constrained optimization problems for signal processing and communications

Constrained minimization problems considered in this paper arise in the design of beamformers for radar, sonar, and wireless communications, and in the design of precoders and equalizers for digital communications. The problem is to minimize a quadratic form under a set of linear or quadratic constraints. We present solutions to these problems and establish a connection between them. A majorization result for matrix trace and Poincare's separation theorem play key roles in establishing the connection. We show that our solutions can be formulated as generalized sidelobe cancellers (GSCs), which tie our constrained minimizations to linear minimum mean-squared error (LMMSE) estimations. We then express our solutions in terms of oblique projection matrices and establish the geometry of our constrained minimizations.     

Can observational practice facilitate error recognition and movement production?

Two experiments are reported that examined the usefulness of observational learning for acquiring both error detection and movement production capabilities. In both experiments, individuals were assigned to a no-practice, physical-practice, or observational practice condition. Those assigned to a physical-practice condition acted as models for those assigned as observers. In both experiments, models were administered a random practice of three serial key-press tasks that had the same spatial pattern and same relative timing requirement but differed in the overall time goal. During the retention test, individuals provided estimates of their overall time after each trial. Data from these experiments revealed that error detection and overall time specification were similar following observation and physical practice. However, data from Experiment 2 indicated that physical practice offered an advantage beyond that afforded via observation, with regard to acquiring the appropriate relative time pattern. These data are discussed with respect to the role of observation for learning movement recognition and production processes.     

Programming and reprogramming sequence timing following high and low contextual interference practice

Individuals practiced two unique discrete sequence production tasks that differed in their relative time profile in either a blocked or random practice schedule. Each participant was subsequently administered a "precuing" protocol to examine the cost of initially compiling or modifying the plan for an upcoming movement's relative timing. The findings indicated that, in general, random practice facilitated the programming of the required movement timing, and this was accomplished while exhibiting greater accuracy in movement production. Participants exposed to random practice exhibited the greatest motor programming benefit, when a modification to an already prepared movement timing profile was required. When movement timing was only partially constructed prior to the imperative signal, the individuals who were trained in blocked and random practice formats accrued a similar cost to complete the programming process. These data provide additional support for the recent claim of Immink & Wright (2001) that at least some of the benefit from experience in a random as opposed to blocked training context can be localized to superior development and implementation of the motor programming process before executing the movement.