Problem of finding the null-space arises times and often in many important science and engineering applications. A few of them are bioinformatics, gene expression analysis, structural analysis, computation fluid dynamics, electromagnetics, and optimization theory. Many of the existing methods rely heavily on the structure, size, and sparsity of the matrices in question and are therefore, tailored only for particular applications. Besides these, many hybrid methods have also been tried out. Instead of being helpful, they turn out to be even more complex. In this paper, we propose a novel method for computing the nullspace of a matrix. The method makes no apriori assumptions as such and is applicable to purely random matrices of arbitrary size. In addition, the method is recursive in nature which provides the flexibility of finding an approximate solution whenever the cost of increase in accuracy is unjustifiable by the corresponding increase in computation time. And yet, despite all this, the method is simple, stable, and has excellent convergence properties.
Finite Impulse Response (FIR) filter model emulates the Inter Symbol Interference (ISI) in a wireless communication channel. An equalizer, typically an Infinite Impulse Response (IIR) filter, behaves as an inverse filter to the FIR filter to remove the effects of the ISI. IIR filters are generally avoided due to tractability issues, and an FIR filter, with an adaptive signal processing algorithm to minimize the error due to the ISI, is deployed at the receiver. However, the filter is observed to quickly reach a steady state where further iterations do not yield a reduction in the error. This can be attributed to relatively slow variations in the steady state error which prevent further reduction of the errors. This work focuses on converting the low frequency error variations to high frequency variations by the use of multirate signal processing. As such, the steady state error can be damped as well, providing further reduction in the error and an enhanced adaptive filter performance.
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