- Show that the minimum #multiplications required for an 11-point cyclic convolution is 20.

Show that the minimum #multiplications required for an 12-point cyclic convolution is 18.

Thus a*longer*convolution can require*fewer*multiplications, even for the lower bound!

- Stubborn, you still want to convolve (real) h
_{n}and u_{n}by multiplying their DFTs.

Show how to compute the DFTs of*both*h_{n}and u_{n}using*only one*complex DFT.

- Compute the cyclic convolution of complex constants (say that 5 times) of order 2
- Show that Winograd breaks down the cyclic convolution into a series of smaller convolutions:
Yr(z)+jYi(z)=(Hr(z)+jHi(z))(Ur(z)+jUi(z)) mod(z
^{2K}+1), 0 < K < N. - Show that Yr(z)+z
^{2K-1}Yi(z)=(Hr(z)+z^{2K-1}Hi(z)) (Ur(z)+z^{2K-1}Ui(z)) mod(z^{2K}+1). - Show computing Yr(z)+z
^{2K-1}Yi(z) and Yr(z)-z^{2K-1}Yi(z) recovers both Yr(z) and Yi(z). - What is the interpretation of the computation in (c) in terms of usual complex numbers?

^{N}:

Note that this computes a complex convolution using two real convolutions, doubling #mults,

while simply replacing each real mult with a complex mult would triple or quadruple #mults.

- Show that Winograd breaks down the cyclic convolution into a series of smaller convolutions:
Yr(z)+jYi(z)=(Hr(z)+jHi(z))(Ur(z)+jUi(z)) mod(z
- Show that any polynomial of degree N can be written in the following form:

X(z)= a+b(z-z_{1})+c(z-z_{1})(z-z_{2})+d(z-z_{1})(z-z_{2})(z-z_{3})+... - Show that the constants a,b,c,d... can be computed
*recursively*by setting z=z_{1}, etc. - Show the Kth recursion requires about 2K mults+adds, for a total of about N² mults+adds.
- Show that the standard Lagrange interpolation formula requires about N³ mults+adds.
Apply this to find the unique INTERPOLATE POINT 1 2 3 4 cubic polynomial satisfying POLYNOMIAL VALUE 2 7 18 41

*Recursive*implementation of the*Lagrange interpolation formula*:

- Show that any polynomial of degree N can be written in the following form:
- We use the Cooley-Tukey FFT to break down a large n
_{1}n_{2}...n_{N}-point DFT

into small n_{1}-point DFTs, n_{2}-point DFTs...n_{N}-point DFTs and twiddle mults.

Show that the Cooley-Tukey FFT maps input v_{i}to output V_{k}, where

input index i=i_{1}+n_{1}i_{2}+n_{1}n_{2}i_{3}+...and output index k=i_{N}+n_{N}i_{N-1}+n_{N}n_{N-1}i_{n-2}+... - Set n
_{1}=n_{2}=...=n_{N}=2 to derive*bit-reversal*for 2^{N}-point DFT. - Show how to compute i
_{1}, i_{2}, etc. quickly from i (and similarly for k).

You need*not*show uniqueness of this*mixed integer radix*representation.

*Multidimensional digit reversal*in the Cooley-Tukey n_{1}n_{2}...-point FFT:- We use the Cooley-Tukey FFT to break down a large n