Page 414 - DSP Integrated Circuits

9.5 Implementations Based on Complex PEs 399

Figure 9.13 Implementation of eighth-order bandpass filter using vector-multiplier PEs

9.5.2 Numerically Equivalent Implementation

In this section we will show how to derive a numerically equivalent algorithm that
is equivalent to, or in some respects even better than, numerical properties com-
pared to the original algorithm. The new algorithm is suitable for implementation
using vector-multipliers [31,37].
Consider the recursive
digital filter depicted in
Figure 9.14. Arithmetic
operations are represented
by the network N. We
assume that all values are
computed with full precision
and that the quantizations
are done only in front of the
delay elements in the
original algorithm. Hence,
Figure 9.14 Recursive digital filter
no overflow or rounding
errors occur, and the outputs
and Vi(n\ i = 1,..., M are
computed exactly before
they are quantized. The
node values, which must be computed explicitly, are the outputs and the values
that shall be stored in the delay elements—i.e., vi(n), i - I,..., M.
These values can be computed in either of two numerically equivalent ways:
First, as just mentioned, by doing the multiplications using extended precision so
that no errors occur; and second, by precomputing new equivalent coefficients we
get a set of expressions with fewer multiplications, but usually more than in the
original algorithm. Thus, the new algorithm is numerically equivalent to the
original since the quantization errors that occur in the two algorithms are the same.