In deriving formulas for the noise-to-signal ratio for the fixed-point radix-2 decimatio...

In deriving formulas for the noise-to-signal ratio for the fixed-point radix-2 decimation-in-time FFT algorithm, we assumed that each output node was connected to (N −1) butterfly computations, each of which contributed an amount to the output noise variance. However, when the multiplications can in fact be done without error. Thus, if the results derived in Section 9.7 are modified to account for this fact, we obtain a less pessimistic estimate of quantization noise effects.

(a) For the decimation-in-time algorithm discussed in Section 9.7, determine, for each stage, the number of butterflies that involve multiplication by either ±1 or ±j.

(b) Use the result of part (a) to find improved estimates of the output noise variance, Eq. (9.58), and noise-to-signal ratio, Eq. (9.68), for odd values of k. Discuss how these estimates are different for even values of k. Do not attempt to find a closed form expression of these quantities for even values of k.

(c) Repeat parts (a) and (b) for the case where the output of each stage is attenuated by a factor of 1 2 ; i.e., derive modified expressions corresponding to Eq. (9.71) for the output noise variance and Eq. (9.72) for the output noise-to-signal ratio, assuming that multiplications by ±1 and ±j do not introduce error.