(Solution) - a For the decimation in time algorithm discussed in Section 9 -(2025 Original AI-Free Solution)

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(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. (9.55), and noise-to-signal ratio, Eq. (9.65), 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 ofk.