The discrete Fourier transform (DFT) of a periodic array f₁, for j = 0, 1,..., N - 1 (corresponding to data at equally spaced points, starting at the left end point of the interval of periodicity) is evaluated via the fast Fourier transform (FFT) algorithm. Use an FFT package, i.e. an already coded FFT (e.g. scipy.fftpack or numpy.fft) for the problems below that require the DFT. 1. Let SN/2(x) be the trigonometric polynomial of lowest order that interpolates the periodic array fj, j = 0, 1, ..., N – 1 at the equidistributed nodes x; = j(2π/N), j= 0, 1, ..., N - 1, i.e SN/2(x) = for x = [0, 2π], where N/2-1 ao + Σ (ak cos kx + b₁ sin kæ) + 2 k=1 aN/2 2 COS N (+/+) 2 (1)
The discrete Fourier transform (DFT) of a periodic array f₁, for j = 0, 1,..., N - 1 (corresponding to data at equally spaced points, starting at the left end point of the interval of periodicity) is evaluated via the fast Fourier transform (FFT) algorithm. Use an FFT package, i.e. an already coded FFT (e.g. scipy.fftpack or numpy.fft) for the problems below that require the DFT. 1. Let SN/2(x) be the trigonometric polynomial of lowest order that interpolates the periodic array fj, j = 0, 1, ..., N – 1 at the equidistributed nodes x; = j(2π/N), j= 0, 1, ..., N - 1, i.e SN/2(x) = for x = [0, 2π], where N/2-1 ao + Σ (ak cos kx + b₁ sin kæ) + 2 k=1 aN/2 2 COS N (+/+) 2 (1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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math —— Numerical analysis problem
![The discrete Fourier transform (DFT) of a periodic array fj, for j = 0, 1, . . . ,
.., N 1
(corresponding to data at equally spaced points, starting at the left end point of the
interval of periodicity) is evaluated via the fast Fourier transform (FFT) algorithm. Use
an FFT package, i.e. an already coded FFT (e.g. scipy.fftpack or numpy.fft) for the
problems below that require the DFT.
1. Let SN/2(x) be the trigonometric polynomial of lowest order that interpolates the
periodic array fj, j = 0, 1, .. ., N − 1 at the equidistributed nodes xj = j(2π/N),
j = 0, 1,
N – 1, i.e
SN/2(x)
for x = [0, 2π], where
• 9
-
=
-
N/2-1
ao
+ Σ (ak cos kx + b₁ sin kæ) +
2
k=1
ak =
bk
ZN
=
array:
fo= 6.000000000000000
10.242640687119284
aN/2
2
N-1
Σ f; cos kx; for k = 0, 1, ..., N/2,
f₁
f2 2.000000000000000
f3 = -2.585786437626905
f4 = 2.000000000000000
f5 = 1.757359312880716
f6= -6.000000000000000
f = -5.414213562373098
COS
(3)
(a) Write a formula that relates the complex Fourier coefficients computed by your
fft package to the real Fourier coefficients, ak and b that define sÃ/2(x).
(b) Using your fft package and (a) find s4(x) on [0, 2π] for the following periodic
N-1
Σf, sin kr, for k= 1,..., N/2 – 1.
N
N
(2₂)
(1)
(2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F979d0aba-5428-414f-a3ba-5510f0301082%2F1af74af1-5398-4b23-abf0-3584dc103cf8%2Fnlimu6_processed.png&w=3840&q=75)
Transcribed Image Text:The discrete Fourier transform (DFT) of a periodic array fj, for j = 0, 1, . . . ,
.., N 1
(corresponding to data at equally spaced points, starting at the left end point of the
interval of periodicity) is evaluated via the fast Fourier transform (FFT) algorithm. Use
an FFT package, i.e. an already coded FFT (e.g. scipy.fftpack or numpy.fft) for the
problems below that require the DFT.
1. Let SN/2(x) be the trigonometric polynomial of lowest order that interpolates the
periodic array fj, j = 0, 1, .. ., N − 1 at the equidistributed nodes xj = j(2π/N),
j = 0, 1,
N – 1, i.e
SN/2(x)
for x = [0, 2π], where
• 9
-
=
-
N/2-1
ao
+ Σ (ak cos kx + b₁ sin kæ) +
2
k=1
ak =
bk
ZN
=
array:
fo= 6.000000000000000
10.242640687119284
aN/2
2
N-1
Σ f; cos kx; for k = 0, 1, ..., N/2,
f₁
f2 2.000000000000000
f3 = -2.585786437626905
f4 = 2.000000000000000
f5 = 1.757359312880716
f6= -6.000000000000000
f = -5.414213562373098
COS
(3)
(a) Write a formula that relates the complex Fourier coefficients computed by your
fft package to the real Fourier coefficients, ak and b that define sÃ/2(x).
(b) Using your fft package and (a) find s4(x) on [0, 2π] for the following periodic
N-1
Σf, sin kr, for k= 1,..., N/2 – 1.
N
N
(2₂)
(1)
(2)
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