Optimization of Number Representations - Signal Processing Systems - page 1303

Although each signal can have a separate integer word-length, one common
integer word-length is assigned to all the signals in the same group in order to lower
the use of shifters for implementation. All the signs and the integer word-lengths
are determined in just one simulation.

5.3.3 Determination of the Minimum Word-Length for Each Group

Assume a word-length vector w , whose component is the word-length in each group.

w =( w 1 , w 2 ,···, w N ) , (13)

where N is the number of groups.
The performance of a fixed-point system, such as SQNR or negative of mean
squared error, is represented by p

( w ) , and the hardware cost, usually the number

of gates, is c . Then, the optimum word-length vector, w opt , should have the

minimum value of c

( w )

is larger than p desired . We assume the following
relation between the word-length vector and the fixed-point performance.

( w ) while p ( w )

p (( w 1 , w 2 ,···, w i ,···, w N )) ≥ p (( w 1 , w 2 ,···, w i − 1 ,···, w N )) , (14)

where 1 N . The above equation represents that reducing a word-length of

a group decreases, or at least does not increase, the fixed-point performance of a
system. Then, the number of simulations required for the search can be reduced
greatly by decomposing the procedure into two steps: the minimum word-length
and the cost optimum word-length determination.
The minimum word-length for a group, w i , min , is the smallest word-length that
satisfies the fixed-point performance of a system when the word-lengths of all
other groups are very large, typically a 64 bit fixed-point or the floating-point
type. By the assumption shown in Eq. ( 14 ) , this minimum word-length is not
larger than the optimum word-length for the group. The minimum word-length
determination procedure for the first order digital filter is illustrated in Table 6 . The
word-length vector for this filter consists of three components, which are “ADC,”
“DAC” and filter word-lengths. The uniform word-length that satisfies the fixed-
point performance is first determined. The uniform word-length is not only useful
for some architecture, such as bit-serial implementations, but also the upper limit
of the minimum word-length according to the assumption in Eq. ( 14 ) . Thus, the
search for finding the minimum word-length for each group goes downward from
the determined uniform word-length. The number of simulations required for the
minimum word-length determination procedure is typically two to four times the
number of groups. The minimum word-length determination does not require the
hardware cost model.

≤ i ≤