Page 386 - Matrix Analysis & Applied Linear Algebra

382 Chapter 5 Norms, Inner Products, and Orthogonality

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5.8.16. For p(x)= α 0 + α 1 x + α 2 x + ··· + α n−1 x n−1 , prove that
n−1
1 2 2 2 2
p(ξ ) = |α 0 | + |α 1 | + ··· + |α n−1 | ,
k
n
k=0
2 n−1 th
where 1,ξ,ξ ,. . . ,ξ are the n roots of unity.
5.8.17. Consider a waveform that is given by the finite sum

x(τ)= (α k cos 2πf k τ + β k sin 2πf k τ)
k

in which the f k ’s are distinct integers, and let

x = (α k cos 2πf k t + β k sin 2πf k t)
k
be the vector containing the values of x(τ)at n> 2 max{f k } equally
spaced points between τ =0 and τ =1 as described in Example 5.8.3.
Use the discrete Fourier transform to prove that
n
2 2 2
x = α + β .
2 k k
2
k
5.8.18. Let η be an arbitrary scalar, and let

 
1  
α 0
η
 
 2   α 1 
c =  η  and a =  .  .
. .
   . 
.
 . 
η 2n−1 α n−1

T
T
Prove that c (a a)= c ˆ a 2 .
 
x 0
x 1

5.8.19. Apply the FFT algorithm to the vector x 8 =  . , and then verify

.
.
x 7
that your answer agrees with the result obtained by computing F 8 x 8
directly.