# Polynomial Matrix Decompositions

The interest and dramatic increase in demand for high-speed data transmission has exploded because the introduction of triple play multimedia services. Among the pioneering contributions is the increasingly higher data rates achievable by using multiple-input multiple output (MIMO) channels. A further increase in data rates can be done by having a well-coordinated multiple channels.

Precoding and equalization transmission blocks represent an average coordinated strategy that improve the channel capacity of any deterministic narrowband MIMO channel. This strategy can be derived by performing a singular value decomposition (SVD) of the channel matrix.

These blocks allow a MIMO channel matrix to resolve a host of special multicarrier problems using the orthogonal property of the machine. It can even be exploited to approximate low-rank channel matrix by reducing the dimensionality of high-dimensional data sets. The technique referred to here as an orthogonal spatial multiplexing (OSM) method.

OSM allows multiple users to employ a given bandwidth simultaneously by dividing the available channel into multiple narrow orthogonal bands that are spectrally spaced. Each band is then divided into numerous subcarriers, which can be structures so as not to hinder one another. The transmit signal is then modulated onto these subcarriers.

The technique exploits physical separation methods that let the sharing of channel resources simultaneously. Every users becomes spatially spaced far enough from each other to counteract interference. In addition to spacing, dual polarizations are introduced to help expand avoid interference.

When signalling over MIMO channels using an orthogonal spatial multiplexing, an SVD may be used to derive every sub-carrier/tone. In the traditional method, an increase in the amount of tones also increase the complex computational load. It is therefore the focus of the study to investigate an alternative opportinity for obtaining a competent decomposition.

A MIMO channel can be modelled as a weighted sum of the past and present samples of transmit data. The channel considers a finite impulse response (FIR) filters that may be represented by the covariance matrix whose elements are polynomials. This study targets investigating algorithms that decompose the covariance matrix directly.

Approximation factors may then be introduced to obtain the Precoding and equalization transmission blocks. Existing polynomial singular value decomposition algorithm is utilized and studied in the context of channel quality and computational complexity settings. The decomposition algorithms were shown to give decompositions of good channel quality, if the goal is to acquire Precoding and equalization transmission blocks, the computational load is fixed with higher multidimensional channels.

An algorithm for approximating direct decomposition of covariance matrices is investigated. Although we discuss simple cases resulted in excellent decompositions but analyse with numerical stability of an spectral factorization steps for large-case decompositions.

For high frequency selective MIMO channels, the performance attained by utilizing the polynomial SVD algorithm were set alongside the channel capacity.

It was shown that if the transmit sequences are approximated individually at the receiver, as done in the original approach, the performance more likely to be sensitive to errors in the decomposition. An equalizer with a spatially joint detector seems promising to achieve an improved performance near to the single-user transmission. With this equalizer, the reduced complexity property of the traditional approach is compromised with performance.

Summarizing, this study shows a MIMO channel can be diagonalized in space and frequency using spatial multiplexing method in conjunction with a polynomial SVD algorithm. To be able to reach better performance near to the achivable of an single-user, the computational load becomes restraining set alongside the traditional approach, for channels with higher multidirectional channels.

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