Help calculating Fourier series coefficients
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I am trying to solve the following problem in my textbook.
Use the Fourier series analysis equation (3.39) to calculate the coefficients $a_k$ for the continuous-time periodic signal
$$
xleft(tright) = left{
begin{array}{lr}
1.5, & 0 le x lt 1\
-1.5, & 1 le x lt 2
end{array}
right.\
$$
with fundamental frequency $w_0 = pi$.
Equation 3.39: $$a_k = frac{1}{T}int_Tx(t)e^{-jkw_0t}dt$$ where $T$ is the fundamental period and $j$ is the imaginary unit.
The correct answer is
$$
a_k = left{
begin{array}{lr}
0, & k = 0\
e^{-jkpi/2}frac{3sin(frac{k}{pi/2})}{kpi}, & k ne 0
end{array}
right.\
$$
But this is not the answer I get. My answer is
$$
a_k = left{
begin{array}{lr}
0, & k = 0\
frac{3}{j4kw_0}(1 + e^{-j2kw_0}), & k ne 0
end{array}
right.\
$$
How do I get the correct answer?
fourier-series
asked Nov 17 at 0:43
PurpleMoonrise
234
add a comment |
up vote
-1
down vote
favorite
I am trying to solve the following problem in my textbook.
Use the Fourier series analysis equation (3.39) to calculate the coefficients $a_k$ for the continuous-time periodic signal
$$
xleft(tright) = left{
begin{array}{lr}
1.5, & 0 le x lt 1\
-1.5, & 1 le x lt 2
end{array}
right.\
$$
with fundamental frequency $w_0 = pi$.
Equation 3.39: $$a_k = frac{1}{T}int_Tx(t)e^{-jkw_0t}dt$$ where $T$ is the fundamental period and $j$ is the imaginary unit.
The correct answer is
$$
a_k = left{
begin{array}{lr}
0, & k = 0\
e^{-jkpi/2}frac{3sin(frac{k}{pi/2})}{kpi}, & k ne 0
end{array}
right.\
$$
But this is not the answer I get. My answer is
$$
a_k = left{
begin{array}{lr}
0, & k = 0\
frac{3}{j4kw_0}(1 + e^{-j2kw_0}), & k ne 0
end{array}
right.\
$$
How do I get the correct answer?
fourier-series
asked Nov 17 at 0:43
PurpleMoonrise
234
Sorry if this question seems too basic. I'm trying to teach myself about Fourier series, and as you can see, it's not going very well.
– PurpleMoonrise
Nov 17 at 1:22
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I am trying to solve the following problem in my textbook.
Use the Fourier series analysis equation (3.39) to calculate the coefficients $a_k$ for the continuous-time periodic signal
$$
xleft(tright) = left{
begin{array}{lr}
1.5, & 0 le x lt 1\
-1.5, & 1 le x lt 2
end{array}
right.\
$$
with fundamental frequency $w_0 = pi$.
Equation 3.39: $$a_k = frac{1}{T}int_Tx(t)e^{-jkw_0t}dt$$ where $T$ is the fundamental period and $j$ is the imaginary unit.
The correct answer is
$$
a_k = left{
begin{array}{lr}
0, & k = 0\
e^{-jkpi/2}frac{3sin(frac{k}{pi/2})}{kpi}, & k ne 0
end{array}
right.\
$$
But this is not the answer I get. My answer is
$$
a_k = left{
begin{array}{lr}
0, & k = 0\
frac{3}{j4kw_0}(1 + e^{-j2kw_0}), & k ne 0
end{array}
right.\
$$
How do I get the correct answer?
fourier-series
asked Nov 17 at 0:43
PurpleMoonrise
234
I am trying to solve the following problem in my textbook.
Use the Fourier series analysis equation (3.39) to calculate the coefficients $a_k$ for the continuous-time periodic signal
$$
xleft(tright) = left{
begin{array}{lr}
1.5, & 0 le x lt 1\
-1.5, & 1 le x lt 2
end{array}
right.\
$$
with fundamental frequency $w_0 = pi$.
Equation 3.39: $$a_k = frac{1}{T}int_Tx(t)e^{-jkw_0t}dt$$ where $T$ is the fundamental period and $j$ is the imaginary unit.
The correct answer is
$$
a_k = left{
begin{array}{lr}
0, & k = 0\
e^{-jkpi/2}frac{3sin(frac{k}{pi/2})}{kpi}, & k ne 0
end{array}
right.\
$$
But this is not the answer I get. My answer is
$$
a_k = left{
begin{array}{lr}
0, & k = 0\
frac{3}{j4kw_0}(1 + e^{-j2kw_0}), & k ne 0
end{array}
right.\
$$
How do I get the correct answer?
fourier-series
fourier-series
asked Nov 17 at 0:43
PurpleMoonrise
234
asked Nov 17 at 0:43
PurpleMoonrise
234
asked Nov 17 at 0:43
PurpleMoonrise
234
asked Nov 17 at 0:43
PurpleMoonrise
234
asked Nov 17 at 0:43
PurpleMoonrise
234
234
Sorry if this question seems too basic. I'm trying to teach myself about Fourier series, and as you can see, it's not going very well.
– PurpleMoonrise
Nov 17 at 1:22
add a comment |
Sorry if this question seems too basic. I'm trying to teach myself about Fourier series, and as you can see, it's not going very well.
– PurpleMoonrise
Nov 17 at 1:22
Sorry if this question seems too basic. I'm trying to teach myself about Fourier series, and as you can see, it's not going very well.
– PurpleMoonrise
Nov 17 at 1:22
Sorry if this question seems too basic. I'm trying to teach myself about Fourier series, and as you can see, it's not going very well.
– PurpleMoonrise
Nov 17 at 1:22
add a comment |
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Sorry if this question seems too basic. I'm trying to teach myself about Fourier series, and as you can see, it's not going very well.
– PurpleMoonrise
Nov 17 at 1:22