Foruier Series coefficients.
up vote
0
down vote
favorite
I am given a following signal and want to calculate it's Fourier series coefficients:
$$ x(t) = left{
begin{array}{ll}
0 & quad 0< t < 2 \
2 & quad 2< t < 3 \
0& quad 3< t < 4 \
-2 & quad 5< t < 6
end{array}
right. $$
Following is my method:
$$ a_k= frac{1}{6} ( int_{2}^{3} 2 e^{-jkw_0t}dt + int_{5}^{6} (-2) e^{-jkw_0t}dt) $$
$$ a_k= frac{2}{6} ( int_{2}^{3} e^{-jkw_0t}dt - int_{5}^{6} e^{-jkw_0t}dt) $$
After doing some simplification we get:
$$ a_k= frac{1}{jkpi}(e^{-2jfrac{kpi}{3}}-e^{-jkpi}+e^{-2jkpi}-e^{-5jfrac{kpi}{3}}) $$
I am stuck after this last step.
integration
add a comment |
up vote
0
down vote
favorite
I am given a following signal and want to calculate it's Fourier series coefficients:
$$ x(t) = left{
begin{array}{ll}
0 & quad 0< t < 2 \
2 & quad 2< t < 3 \
0& quad 3< t < 4 \
-2 & quad 5< t < 6
end{array}
right. $$
Following is my method:
$$ a_k= frac{1}{6} ( int_{2}^{3} 2 e^{-jkw_0t}dt + int_{5}^{6} (-2) e^{-jkw_0t}dt) $$
$$ a_k= frac{2}{6} ( int_{2}^{3} e^{-jkw_0t}dt - int_{5}^{6} e^{-jkw_0t}dt) $$
After doing some simplification we get:
$$ a_k= frac{1}{jkpi}(e^{-2jfrac{kpi}{3}}-e^{-jkpi}+e^{-2jkpi}-e^{-5jfrac{kpi}{3}}) $$
I am stuck after this last step.
integration
I forgot to say you can use Euler's formula $$e^{jx}=cos x+jsin x$$
– Nosrati
Nov 20 at 19:28
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am given a following signal and want to calculate it's Fourier series coefficients:
$$ x(t) = left{
begin{array}{ll}
0 & quad 0< t < 2 \
2 & quad 2< t < 3 \
0& quad 3< t < 4 \
-2 & quad 5< t < 6
end{array}
right. $$
Following is my method:
$$ a_k= frac{1}{6} ( int_{2}^{3} 2 e^{-jkw_0t}dt + int_{5}^{6} (-2) e^{-jkw_0t}dt) $$
$$ a_k= frac{2}{6} ( int_{2}^{3} e^{-jkw_0t}dt - int_{5}^{6} e^{-jkw_0t}dt) $$
After doing some simplification we get:
$$ a_k= frac{1}{jkpi}(e^{-2jfrac{kpi}{3}}-e^{-jkpi}+e^{-2jkpi}-e^{-5jfrac{kpi}{3}}) $$
I am stuck after this last step.
integration
I am given a following signal and want to calculate it's Fourier series coefficients:
$$ x(t) = left{
begin{array}{ll}
0 & quad 0< t < 2 \
2 & quad 2< t < 3 \
0& quad 3< t < 4 \
-2 & quad 5< t < 6
end{array}
right. $$
Following is my method:
$$ a_k= frac{1}{6} ( int_{2}^{3} 2 e^{-jkw_0t}dt + int_{5}^{6} (-2) e^{-jkw_0t}dt) $$
$$ a_k= frac{2}{6} ( int_{2}^{3} e^{-jkw_0t}dt - int_{5}^{6} e^{-jkw_0t}dt) $$
After doing some simplification we get:
$$ a_k= frac{1}{jkpi}(e^{-2jfrac{kpi}{3}}-e^{-jkpi}+e^{-2jkpi}-e^{-5jfrac{kpi}{3}}) $$
I am stuck after this last step.
integration
integration
asked Nov 20 at 16:08
Ahmad Qayyum
303
303
I forgot to say you can use Euler's formula $$e^{jx}=cos x+jsin x$$
– Nosrati
Nov 20 at 19:28
add a comment |
I forgot to say you can use Euler's formula $$e^{jx}=cos x+jsin x$$
– Nosrati
Nov 20 at 19:28
I forgot to say you can use Euler's formula $$e^{jx}=cos x+jsin x$$
– Nosrati
Nov 20 at 19:28
I forgot to say you can use Euler's formula $$e^{jx}=cos x+jsin x$$
– Nosrati
Nov 20 at 19:28
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Hint:
for these two terms
$$e^{-2jfrac{kpi}{3}}-e^{-5jfrac{kpi}{3}}$$
let
$$alpha-x=-2jfrac{kpi}{3}~~~;~~~alpha+x=-5jfrac{kpi}{3}$$
then
$$alpha=-7jfrac{kpi}{6}~~~;~~~x=-jfrac{kpi}{2}$$
and therefore
$$e^{-2jfrac{kpi}{3}}-e^{-5jfrac{kpi}{3}}=color{blue}{2je^{-7jkpi/6}sinfrac{kpi}{2}}$$
can you explain a bit more.
– Ahmad Qayyum
Nov 20 at 17:03
We want to make an expression like $$e^{alpha-x}pm e^{alpha+x}=e^{alpha}left(e^{-x}pm e^{x}right)$$
– Nosrati
Nov 20 at 17:07
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Hint:
for these two terms
$$e^{-2jfrac{kpi}{3}}-e^{-5jfrac{kpi}{3}}$$
let
$$alpha-x=-2jfrac{kpi}{3}~~~;~~~alpha+x=-5jfrac{kpi}{3}$$
then
$$alpha=-7jfrac{kpi}{6}~~~;~~~x=-jfrac{kpi}{2}$$
and therefore
$$e^{-2jfrac{kpi}{3}}-e^{-5jfrac{kpi}{3}}=color{blue}{2je^{-7jkpi/6}sinfrac{kpi}{2}}$$
can you explain a bit more.
– Ahmad Qayyum
Nov 20 at 17:03
We want to make an expression like $$e^{alpha-x}pm e^{alpha+x}=e^{alpha}left(e^{-x}pm e^{x}right)$$
– Nosrati
Nov 20 at 17:07
add a comment |
up vote
1
down vote
accepted
Hint:
for these two terms
$$e^{-2jfrac{kpi}{3}}-e^{-5jfrac{kpi}{3}}$$
let
$$alpha-x=-2jfrac{kpi}{3}~~~;~~~alpha+x=-5jfrac{kpi}{3}$$
then
$$alpha=-7jfrac{kpi}{6}~~~;~~~x=-jfrac{kpi}{2}$$
and therefore
$$e^{-2jfrac{kpi}{3}}-e^{-5jfrac{kpi}{3}}=color{blue}{2je^{-7jkpi/6}sinfrac{kpi}{2}}$$
can you explain a bit more.
– Ahmad Qayyum
Nov 20 at 17:03
We want to make an expression like $$e^{alpha-x}pm e^{alpha+x}=e^{alpha}left(e^{-x}pm e^{x}right)$$
– Nosrati
Nov 20 at 17:07
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Hint:
for these two terms
$$e^{-2jfrac{kpi}{3}}-e^{-5jfrac{kpi}{3}}$$
let
$$alpha-x=-2jfrac{kpi}{3}~~~;~~~alpha+x=-5jfrac{kpi}{3}$$
then
$$alpha=-7jfrac{kpi}{6}~~~;~~~x=-jfrac{kpi}{2}$$
and therefore
$$e^{-2jfrac{kpi}{3}}-e^{-5jfrac{kpi}{3}}=color{blue}{2je^{-7jkpi/6}sinfrac{kpi}{2}}$$
Hint:
for these two terms
$$e^{-2jfrac{kpi}{3}}-e^{-5jfrac{kpi}{3}}$$
let
$$alpha-x=-2jfrac{kpi}{3}~~~;~~~alpha+x=-5jfrac{kpi}{3}$$
then
$$alpha=-7jfrac{kpi}{6}~~~;~~~x=-jfrac{kpi}{2}$$
and therefore
$$e^{-2jfrac{kpi}{3}}-e^{-5jfrac{kpi}{3}}=color{blue}{2je^{-7jkpi/6}sinfrac{kpi}{2}}$$
answered Nov 20 at 16:41
Nosrati
26.2k62353
26.2k62353
can you explain a bit more.
– Ahmad Qayyum
Nov 20 at 17:03
We want to make an expression like $$e^{alpha-x}pm e^{alpha+x}=e^{alpha}left(e^{-x}pm e^{x}right)$$
– Nosrati
Nov 20 at 17:07
add a comment |
can you explain a bit more.
– Ahmad Qayyum
Nov 20 at 17:03
We want to make an expression like $$e^{alpha-x}pm e^{alpha+x}=e^{alpha}left(e^{-x}pm e^{x}right)$$
– Nosrati
Nov 20 at 17:07
can you explain a bit more.
– Ahmad Qayyum
Nov 20 at 17:03
can you explain a bit more.
– Ahmad Qayyum
Nov 20 at 17:03
We want to make an expression like $$e^{alpha-x}pm e^{alpha+x}=e^{alpha}left(e^{-x}pm e^{x}right)$$
– Nosrati
Nov 20 at 17:07
We want to make an expression like $$e^{alpha-x}pm e^{alpha+x}=e^{alpha}left(e^{-x}pm e^{x}right)$$
– Nosrati
Nov 20 at 17:07
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3006508%2fforuier-series-coefficients%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
I forgot to say you can use Euler's formula $$e^{jx}=cos x+jsin x$$
– Nosrati
Nov 20 at 19:28