Prime Numbers

9.7 Exercises 529
9.49. In the FFT literature there exists an especially efficient real-signal
transform called the Sorenson FFT. This is a split-radix transform that
uses √ 2 and a special decimation scheme to achieve essentially the lowestcomplexity
FFT known for real signals; although in modern times the issues
of memory, machine cache, and processor features are so overwhelming that
sheer complexity counts have fallen to a lesser status. Now, for the ring Zn
with n =2 m + 1 and m a multiple of 4, show that a square root of 2 is given
by √ 2=2 3m/4 − 2 m/4 .
Then, determine whether a Sorenson transform modulo n canbedonesimply
by using what is now the standard Sorenson routine but with √ 2 interpreted
as above. (Detailed coding for a Sorenson real-signal FFT is found in [Crandall
1994b].)
9.50. Study the transform that has the usual DFT form
Xk =
N−1
j=0
xjh −jk ,
except that the signal elements xj and the root h of order N exist in the field
Q √ 5 . This has been called a number-theoretical transform (NTT) over the
“golden section quadratic field,” because the golden mean φ = √ 5 − 1 /2
is in the field. Assume that we restrict further to the ring Z[φ] so that the
signal elements and the root are of the form a + bφ with a, b integers. Argue
first that a multiplication √ in the domain takes three integer multiplies. Then
consider the field Fp 5 and work out a theory for the possible length of
such transforms over that field, when the root is taken to be a power of the
golden mean φ. Then, consider the transform (N is even)
Xk =
N/2−1
j=0
H −jk xj
where the new signal vector is xj =(aj,bj) and where the original signal
component was xj = aj + bjφ in the field. Here, the matrix H is
1
H =
1
1
.
0
Describe in what sense this matrix transform is equivalent to the DFT
definition preceding, that the powers of H are given conveniently in terms
of Fibonacci numbers
H n
Fn+1 Fn
=
,
Fn
Fn−1
and that this n-th power can be computed in divide-and-conquer fashion in
O(ln n) matrix multiplications. In conclusion, derive the complexity of this
matrix-based number-theoretical transform.