### The stepdown procedure for complex predictor coefficients

J. Picone, D. Prezas, W. Hartwell, J. Locicero
1986 IEEE Transactions on Acoustics Speech and Signal Processing
is, all components are calculated in place in the same way as the ordinary DIT algorithm. In Fig. 1 , the first weight in each bunch of butterflies of each stage is unity: W o = 1, while it is not always so in Fig. 2 . Thus, the modified algorithm apparently seems to have more complex multiplications. The brief observation, however, reveals one of the weights in the bunch to be either W o = 1 or WNI2 = -1. Since a subtraction is equivalent in complexity to an addition, the modified algorithm
more » ... same load of calculations as the ordinary one. Pruning Pruning the modified algorithm is shown in Fig. 2. Here again, only calculations which correspond to bold solid lines are necessary. They have a repetitive pattern between adjacent stages, in contrast to the apparent random pattern in Fig. 1 . The repetitive pattern simplifies the modified algorithm. In order to obtain the components of the Kth to ( K + L -1)th frequencies, one has only to compute simply from the first to Lth or all butterflies in each bunch of the ith stage, depending on whether 2i-1 > L or not, respectively. Owing to the repetitive pattern, the bold solid lines exist always only in the first to Lth, or all butterflies in each bunch. This results in a very simple Fortran program as shown in Fig. 3 . This program requires shuffled data and calculates components within desired frequency band effectively. There are shortened butterflies to be calculated in pruned stages, as can be seen in Fig. 2; but this program calculates complete butterflies. For the example of Fig. 2 , therefore, it calculates not only the outputs K and K + 1 , but also K + 8 and K + 9. Addition (subtraction) time is negligible, and complete butterfly evaluation, which is also employed in [2], fairly simplifies the program. So far, only the case to compute narrow-band components has been considered. If time sequence has trailing zeros for high resolution of frequency, the algorithm given by [4] can be taken in as similarly as in [2]. Abstract-The stepdown procedure is an algorithm in which the predictor parameters of a direct form digital filter are converted to the