Page 533 - DSP Integrated Circuits

518 Chapter 11 Processing Elements

Figure 11.53 Layout of the butterfly PE

relatively small. If only one ROM were used, it would have been necessary to use
long bit-parallel buses, which are costly in terms of area, to distribute the twiddle
factors to the butterfly PEs.
The values of the twiddle factors, W, are spaced uniformly around the unit cir-
cle. Generally, there are N twiddle factors, but it is possible to reduce the number
of unique values by exploring symmetries in the trigonometric functions. In fact, it
can be shown that only N/8 +1 coefficient values need be stored in a table [1].
Instead of storing WP, we will store the values

where, a = 2np/N. Table 11.6 shows that the twiddle factors in the eight octants
can be expressed in terms of the twiddle factors in the range 0 to rc/4.
Notice that the three most significant bits in the binary representation of a
(i.e., aya-[a<2) can be used to select the appropriate entry in the table. For example,
0,2 is used to select the argument—i.e., either 6 or 7C/4-&. Let the seven least sig-
nificant bits in the binary representation of a be a^a^a^asajasaQ then the binary
representation for 71/4-6 is

Table 11.6 Address generation for the modified twiddle factors