AMATH 341 / CM 271 / CS 371
Assignment 8
Due : Monday April 5, 2010
Instructor: K. D. Papoulia
1. Moler exercise 8.7. The El Nino dataset is available on Moler’s NCM
website.
2. Develop an alternative version of the FFT that uses recursion based
on powers of 4 instead of 2. In other words, the original FFT summa-
tion is broken up into four terms rather than two, and each of the four
terms leads to an FFT on a smaller sequence. This alternative version
is applicable only if
n
, the number of samples, is an exact power of
4. (It is sometimes more eﬃcient, depending on the computer’s archi-
tecture.) Code this alternative version in Matlab, and then carry out
some sample runs to demonstrate that it computes the same (or very
close) answer as the built-in
fft
.
3. Suppose you are given two vectors
a
ˆ
a
0
, . . . , a
n
and
b
ˆ
b
0
, . . . , b
n
which are the coeﬃcents of two polynomials
p
ˆ
x
a
n
x
n
a
n
1
x
n
1
a
0
and
q
ˆ
x
b
n
x
n
b
n
1
x
n
1
b
0
. The problem is to ﬁnd the coeﬃcients
of the product polynomial
r
ˆ
x
p
ˆ
x
q
ˆ
x
; call these
c
ˆ
c
0
, . . . , c
2
n
.
(a) Write down the formula for
c
i
in terms of the entries of
a
and
b
.
(b) Count the number of ﬂoating point operations necessary to compute
ˆ
c
0
, . . . , c
2
n
using the algorithm implicit in the formula in part (a). It
should come out to be quadratic in
n
.
(c) Develop an eﬃcient
O
ˆ
n
log
n
algorithm to solve this problem.
Suggested approach: evaluate
p
at some roots of unity; evaluate also
q
at the same roots of unity, multiply (since for any
ω
,
r
ˆ
ω
p
ˆ
ω
q
ˆ
ω
),
and then interpolate to obtain the coeﬃcients of
r
. Be sure to clearly
state which roots of unity are being used, and in particular, what order
DFT and IDFT you are using.
4. Prove Parseval’s equality: If
y
0
, . . . , y
n
1
is a complex vector and
Y
0
, . . . , Y
n
1
is its DFT, then
n
1
Q
j
0
S
Y
j
S
2
n
n
1
Q
j
0
S
y
j
S
2
.
1