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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:
Rx w = p
Here, w is a vector of beamformer
weights, which can be obtained with inversion
of the correlation matrix Rx as shown
in the equation
w = Rx-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.
Beamforming and Matrix Inversion
Figure 1 shows a basic narrowband beamformer
with K sensor elements arranged in a
uniform linear array (ULA); this also shows
a signal source sq(t) impinging on the array
at an angle of incidence q. The K beamformer
weights (w1, w2, …, wK) are used to
linearly combine the array data observation
samples (x1(n), x2(n), ..., xK(n)). These are
set to "steer" the response of the array for
optimum reception. The output of the
beamformer is the scalar y(n).
1.Narrowband beamformer
A generalized sidelobe canceller (GSC) is
a special beamformer structure that allows
the use of unconstrained optimization
methods in the design of the optimum
beamformer weights. The structure of the
GSC is shown in Figure 2. To find the optimum
weights wa using the LS criterion, the
following deterministic normal equation
must be solved:
Rx wa = b
Here, Rx is the correlation matrix of the
input to the unconstrained section of the
GSC and the vector b is the cross-correlation
of the input Xa and the ideal response.
2. Generalized sidelobe canceller (GSC)
One effective technique for the solution
of this equation is the recursive least-squares
(RLS) approximation with QR decomposition
of the input data matrix. This technique
finds the solution without explicit
inversion of a matrix and avoids constructing
the correlation matrix, explicitly reducing
the dynamic range requirements of
signals involved in the computations.
Figure 3 shows the diagram of an adaptive
GSC beamformer that uses a QRD-RLS
algorithm for a recursive solution of
the normal equation.
3. Adaptive GSC beamformer
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