Implementing matrix inversions in fixed-point hardware

By Ramon Uribe and Tom Cesear, Xilinx     
October 12, 2006 -- dspdesignline.com

We implement fixed-point matrix inversion on a Virtex-4 FPGA using a synthesizable QR-decomposition MATLAB model and the AccelDSP Synthesis tool. The resulting function occupies 12% of a XC4VSX55 device and has a 1.7 MSPS data rate.

Matrix inversion is an important operation in many state-of-the-art DSP algorithms and implementations, including radar, sonar, and multiple antenna systems for communications. A common component of these algorithms is a beamformer or spatial filter, whose function is to steer (in some optimal fashion) the response of an array of sensors for the reception of signal sources.

When using the least-squares (LS) criterion, the computation of optimum weights is based on the solution of a system of linear equations known as the deterministic normal equation. This is shown in the equation:

R_x w = p

Here, w is a vector of beamformer weights, which can be obtained with inversion of the correlation matrix R_x as shown in the equation

w = R_x^-1 p

From a numerical point of view, the best approach to matrix inversion is to not do it explicitly, whenever possible. Instead, it is better to solve the system of equations using an adequate solution technique.

Traditionally, implementations like this have been done with general-purpose DSP devices using floating-point arithmetic to minimize round-off error. A disadvantage of these implementations, however, is the limited processing power because of the small number of floating-point processing units commonly available per device. An appealing alternative for implementation is to use the Xilinx Virtex-4 FPGA family, which offers large amounts of parallelism. One complication with these silicon fabrics is that they are tailored for fixed-point arithmetic, and implementation in these is inherently challenging because of sensitivity to round-off error.

In this article, we'lI present an efficient methodology that enables the implementation of algorithms involving matrix-inversion operations in hardware with fixed-point arithmetic. This methodology includes three essential steps to follow in the development process:

• Capturing the DSP algorithm description in the MATLAB language
• Definition of the fixed-point parameters directly coupled to the MATLAB algorithm description
• Automated generation of a hardware implementation that is bit-accurate to the fixed-point arithmetic model and that meets area/speed requirements for a particular application

Using this methodology, you can fully exploit the benefits of the processing power offered by implementations in FPGA or ASIC fixed-point hardware.

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