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258 chapter 10Figure 10.9 The basic butterfly operation in which elements y p and y q are transformed intoy p + Zy q and y p − Zy q .Y 1 = Z 0 (y 0 − y 4 )+Z 1 (y 1 − y 5 )+Z 2 (y 2 − y 6 )+Z 3 (y 3 − y 7 ),Y 2 = Z 0 (y 0 + y 4 )+Z 2 (y 1 + y 5 ) − Z 0 (y 2 + y 6 ) − Z 2 (y 3 + y 7 ),Y 3 = Z 0 (y 0 − y 4 )+Z 3 (y 1 − y 5 ) − Z 2 (y 2 − y 6 )+Z 1 (y 3 − y 7 ),Y 4 = Z 0 (y 0 + y 4 ) − Z 0 (y 1 + y 5 )+Z 0 (y 2 + y 6 ) − Z 0 (y 3 + y 7 ),Y 5 = Z 0 (y 0 − y 4 ) − Z 1 (y 1 − y 5 )+Z 2 (y 2 − y 6 ) − Z 3 (y 3 − y 7 ),Y 6 = Z 0 (y 0 + y 4 ) − Z 2 (y 1 + y 5 ) − Z 0 (y 2 + y 6 )+Z 2 (y 3 + y 7 ),Y 7 = Z 0 (y 0 − y 4 ) − Z 3 (y 1 − y 5 ) − Z 2 (y 2 − y 6 ) − Z 1 (y 3 − y 7 ),Y 8 = Y 0 .Note the repeating factors inside the parentheses, with combinations of the formy p ± y q . These symmetries are systematized by introducing the butterfly operation(Figure 10.9). This operation takes the y p and y q data elements from the left wingand converts them to the y p + Zy q elements in the upper- and lower-right wings.In Figure 10.10 we show what happens when we apply the butterfly operations toan entire FFT process, specifically to the pairs (y 0 ,y 4 ), (y 1 ,y 5 ), (y 2 ,y 6 ), and (y 3 ,y 7 ).Notice how the number of multiplications of complex numbers has been reduced:For the first butterfly operation there are 8 multiplications by Z 0 ; for the secondbutterfly operation there are 8 multiplications, and so forth, until a total of 24 multiplicationsis made in four butterflies. In contrast, 64 multiplications are requiredin the original DFT (10.8).10.8.1 Bit ReversalThe reader may have observed that in Figure 10.10 we started with 8 data elementsin the order 0–7 and that after three butterfly operators we obtained transforms inthe order 0, 4, 2, 6, 1, 5, 3, 7. The astute reader may may also have observed thatthese numbers correspond to the bit-reversed order of 0–7.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 258