A note on the convergence analysis of LMS adaptive filters with Gaussian data

1087 puts of each module in a special way to cause the output of the total algorithm to be in proper order 111, 191. This results in the fxtest algorithm, but the program must be written for a specific length. A fourth method is similar, but achieves the unscrambling by properly choosing the mulitplier constants in the modules [IO]. A program based on this method could transform many different lengths by using a stored table of appropriate multipliers. The fifth method uses a separate indexing method for the input and output of each module [ I ] , [ I l l. This results in a slightly slower program than methods three or four, but it is compact and can transform all lengths possible from the combination of modules. ALGORITHMS FOR N = 2 " ' A Cooley-Tukey radix-2 FFT that is calculated in place produces the output in bit-reversed order. The unscrambler usually uses a bit-reverse counter [I]. 1121 or nested loops 131. A particularly efficient form has been programmed by Morris [131. A fast unscrambler has been developed by Evans [ 14). but it requires a precalculated " seed " table. The normal in-place Cooley-Tukey FFT does not allow the un-scrambling to be done in the modules (butterflies) as is possible with the PFA. That is an intrinsic property of the type of index map used and the fact that the module lengths are not relatively prime. It is possible to design an in-place, in-order radix-2 FFT if the butterflies are calculated two at a time and a special indexing used. This is described in [15], [I61 and a structure is shown in The important observation contained in this correspondence is that a radix-4 or any radix-2 " ' FFT can be modified so the output is in bit-reversed order. If a normal radix-4 FFT with the output of each butterfly in normal order is used, the output occurs in base-4 reversed order, and similarly for radix-8 and others. However, if for the radix-4 FFT, the short length-4 butterflies are modified to have their outputs in bit-reversed order, the output of the total ra-dix-4 FFT will be in bit-reversed order, not base-4 reversed order. Likewise, if the output of the length-8 butterflies in a radix-8 FFT are placed in bit-reversed order, the output of the total radix-8 FFT will be in bit-reversed order. This allows converting the output …