Page 520 - DSP Integrated Circuits

11.14 Distributed Arithmetic 505

time is W^ clock cycles. The word length in the ROM, WRQM> depends on the F^
with the largest magnitude and the coefficient word length,^, and

EXAMPLE 11.11

Determine the values that should be stored in ROM for the inner product
y = ai xi + a 2 x 2 + a 3 #3

where ai = 33/128 = (0.0100001)20, «2 = 85/128 = (0.1010101) 2 c, and a 3 = -11/128 =
so
(1.1110101)2C- Al determine the necessary word length in the shift-accumulator.
We shall precompute all possible linear combinations of coefficients according
to Equation (11.48). These values shall then be stored in ROM at the addresses
given by the binary weights in the linear combination. Table 11.3 shows the values
both in binary and decimal form with the corresponding addresses.

*1*2 *3 Fk ?k F_k
00 0 0 0.0000000 0.0000000
00 1 03 1.1110101 0.0859375
01 0 02 0.1010101 0.6640625
01 1 02 + 03 0.1001010 0.5781250
10 0 01 0.0100001 0.2578125
10 1 01 + 03 0.0010110 0.1718750
11 0 oi + a 2 0.1110110 0.9218750
11 1 ai + 02 + 03 0.1101011 0.8359375
Table 11.3 ROM contents

The shift-accumulator must be able to add correctly the largest possible value
obtained in the accumulator register and in the ROM. The largest value in the
accumulator register is obtained when the largest (magnitude) value stored in the
ROM is repeatedly accumulated. Thus, at the last clock cycle, corresponding to the
sign bit, the value in REG according to Equation (11.49) is

Hence, the shift-accumulator must be able to add two numbers of magnitude
- Fmax- The necessary number range is ± 1. The word length in the shift-accumula-
tor must be 1 guard bit for overflow detection + 1 sign bit + 0 integer bits + 7 frac-
tional bits = 9 bits.

Notice the similarity between Equations (11.22) and (11.47). In Equation
(11.22), the partial products are summed while in Equation (11.47) the values of
Ffc are summed. In both cases, the same type of shift-accumulator can be used.
Hence, the distributed arithmetic unit essentially consists of a serial/parallell
multiplier augmented by a ROM.