Page 266 - DSP Integrated Circuits

6.7 Equivalence Transformations 251

shimming delays (D flip-flops) are needed, but 11 of these are taken from the
delay elements. The minimum delay between input and output is 11 clock cycles.
Altogether, 51 D flip-flops are required. The final wave-flow graph, shown in Fig-
ure 6.37, with proper timing can be readily transformed into a computation
graph.

6.7.3 Minimizing the Amount of Shimming Delay

Minimizing the required shimming delay in the computation graph is important in
the isomorphic implementation approach based on bit-serial arithmetic, which will
be discussed in Chapter 9. The main reason is that the shimming delays in bit-serial
arithmetic represent moving signal values (shift registers) that consume significant
amounts of power. The amount of shimming delay is of less importance in bit-paral-
lel implementations since data can be stored for arbitrarily long times in static regis-
ters, i.e., the power consumption does not depend on how long the values are stored.
The amount of shimming delay can be minimized by using Theorem 6.4,
which is based on the fact that the ordering of additions (subtractions) is irrele-
vant if two's-complement representation is used and if the final sum is properly
scaled, as was discussed in Chapter 5.

Theorem 6.4
For a multi-input
adder (subtracter)
that has inputs that
arrive at times t\ <
t2 <••• < t-tf the mini-
mum latency and the
minimum amount of
shimming delay are
attained if, recur-
sively, the two earliest
inputs or a previously
computed partial sum
are added (subtracted)
using two's-complement
representation, as illus-
trated in Figure 6.38. Figure 6.38 Adder tree with minimum latency and
amount of shimming delay
The required amount
ol smmmmg delay can olten be reduced by simple cnanges to me computation
graph that do not affect the numerical properties of the algorithm [5]—for exam-
ple, reordering the additions.

6.7.4 Maximally Fast Critical Loops
In many cases, the order of arithmetic operations can be changed without affecting
the numerical properties of the algorithm. Generally, the associative, commutative,