HW Problem 13
Let
V =
v
(0) v
(1) v
(2) v
(3) v
(4) v
(5) v
(6) v
(7)
be the matrix of Fourier sinusoids of length N = 8.
(i) (6 pts.) If
x =
3 1 −5 3 3 1 −5 3 T
,
use projections to represent x in t
...
HW Problem 13
Let
V =
v
(0) v
(1) v
(2) v
(3) v
(4) v
(5) v
(6) v
(7)
be the matrix of Fourier sinusoids of length N = 8.
(i) (6 pts.) If
x =
3 1 −5 3 3 1 −5 3 T
,
use projections to represent x in the form x = Vc. Verify that x is a linear combination of four
columns of V (only).
(ii) (6 pts.) Repeat for
y =
0 0 2 0 0 0 −2 0 T
,
expressing it as y = Vd. Verify that y is a linear combination of four columns of V.
(iii) (2 pts.) Verify your results in (i) and (ii) using the FFT command in MATLAB (which will
generate the vectors 8c and 8d).
(iv) (2 pts.) Verify that the vectors x and y are orthogonal by computing their inner product.
Do your answers to (i) and (ii) above also support this conclusion?
(v) (4 pts.) If s = x + y, use your results from (i) and (ii) above to obtain the least squares
approximation ˆs of s in terms of v
(0)
, v
(1) and v
(7). Display the entries of ˆs. Also, compute the
squared error norm ks − ˆsk
2
.
HW Problem 14
All vectors have length N = 9.
(i) (4 pts.) The entries of the time-domain vector
x
(1) =
2 −1 −1 2 −1 −1 2 −1 −1
T
are given by 2 cos ωn, where n = 0 : 8. What is the value of ω? Express x
(1) as the sum of two
Fourier sinusoids. By considering the appropriate column of the Fourier matrix V, determine and
display the DFT X(1)
.
(ii) (4 pts.) Similarly, express the time-domain vector
x
(2) =
0 1 −1 0 1 −1 0 1 −1
T
as a linear combination of the same two Fourier sinusoids as in part (i). Hence determine and
display the DFT X(2)
.
(iii) (4 pts.) Determine and display the DFT X(3) of
x
(3)[n] = cos(2πn/9) + 3 cos(4πn/9) , n = 0, . . . , 8
(iv) (4 pts.) Determine and display the DFT X(4) of
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