Mixing
Determining $N$ when calculating a fourier series
$begingroup$
calculate the Fourier series of the $2pi$-periodic continuation of
$$f(t):=t, quad tin[0,2pi)$$
Now we have in general:
$$hat{hat{f}}(t)=sum_{k=-N} ^Nc_k e^{-ikt},quad c_k=frac{1}{2pi}int_{-pi}^pi f(x) e^{-ikt}dt$$
So we get
$int_{-pi}^pi te^{-ikt}dt=frac{-1}{ik}t|_{-pi}^pi-int_{-pi}^pi frac{-1}{ik}e^{-ikt}dt=frac{-2pi}{ik}-underbrace{(frac{-1}{ik})^2e^{-ikt}|_{-pi}^pi}_{=0}=frac{-2pi}{ik}=frac{2pi i}{k}$
so we get
$$c_k=frac{i}{k},quad forall k$$
So we get
$$hat{hat{f}}=sum_{k=-N}^N frac{i}{k}e^{-ikt}$$
Now I am a bit confused to what $N$ now exactly is and how I can "choose" it, like when is it infinity? Sometimes I also see that the sum goes from $k=1$ to $infty$ or from $-infty$ to $+infty$. How do I know which case it is?
Also: Shouldn't I have a vertical scaling term? Something like $hat{hat{f}}=frac{a_0}{2}+sum_{k=-N}^Nc_ke^{-ikt}$?
fourier-series
asked Jan 30 at 18:35
xotixxotix
291411
$endgroup$
add a comment |
$begingroup$
calculate the Fourier series of the $2pi$-periodic continuation of
$$f(t):=t, quad tin[0,2pi)$$
Now we have in general:
$$hat{hat{f}}(t)=sum_{k=-N} ^Nc_k e^{-ikt},quad c_k=frac{1}{2pi}int_{-pi}^pi f(x) e^{-ikt}dt$$
So we get
$int_{-pi}^pi te^{-ikt}dt=frac{-1}{ik}t|_{-pi}^pi-int_{-pi}^pi frac{-1}{ik}e^{-ikt}dt=frac{-2pi}{ik}-underbrace{(frac{-1}{ik})^2e^{-ikt}|_{-pi}^pi}_{=0}=frac{-2pi}{ik}=frac{2pi i}{k}$
so we get
$$c_k=frac{i}{k},quad forall k$$
So we get
$$hat{hat{f}}=sum_{k=-N}^N frac{i}{k}e^{-ikt}$$
Now I am a bit confused to what $N$ now exactly is and how I can "choose" it, like when is it infinity? Sometimes I also see that the sum goes from $k=1$ to $infty$ or from $-infty$ to $+infty$. How do I know which case it is?
Also: Shouldn't I have a vertical scaling term? Something like $hat{hat{f}}=frac{a_0}{2}+sum_{k=-N}^Nc_ke^{-ikt}$?
fourier-series
asked Jan 30 at 18:35
xotixxotix
291411
$endgroup$
add a comment |
$begingroup$
calculate the Fourier series of the $2pi$-periodic continuation of
$$f(t):=t, quad tin[0,2pi)$$
Now we have in general:
$$hat{hat{f}}(t)=sum_{k=-N} ^Nc_k e^{-ikt},quad c_k=frac{1}{2pi}int_{-pi}^pi f(x) e^{-ikt}dt$$
So we get
$int_{-pi}^pi te^{-ikt}dt=frac{-1}{ik}t|_{-pi}^pi-int_{-pi}^pi frac{-1}{ik}e^{-ikt}dt=frac{-2pi}{ik}-underbrace{(frac{-1}{ik})^2e^{-ikt}|_{-pi}^pi}_{=0}=frac{-2pi}{ik}=frac{2pi i}{k}$
so we get
$$c_k=frac{i}{k},quad forall k$$
So we get
$$hat{hat{f}}=sum_{k=-N}^N frac{i}{k}e^{-ikt}$$
Now I am a bit confused to what $N$ now exactly is and how I can "choose" it, like when is it infinity? Sometimes I also see that the sum goes from $k=1$ to $infty$ or from $-infty$ to $+infty$. How do I know which case it is?
Also: Shouldn't I have a vertical scaling term? Something like $hat{hat{f}}=frac{a_0}{2}+sum_{k=-N}^Nc_ke^{-ikt}$?
fourier-series
asked Jan 30 at 18:35
xotixxotix
291411
$endgroup$
calculate the Fourier series of the $2pi$-periodic continuation of
$$f(t):=t, quad tin[0,2pi)$$
Now we have in general:
$$hat{hat{f}}(t)=sum_{k=-N} ^Nc_k e^{-ikt},quad c_k=frac{1}{2pi}int_{-pi}^pi f(x) e^{-ikt}dt$$
So we get
$int_{-pi}^pi te^{-ikt}dt=frac{-1}{ik}t|_{-pi}^pi-int_{-pi}^pi frac{-1}{ik}e^{-ikt}dt=frac{-2pi}{ik}-underbrace{(frac{-1}{ik})^2e^{-ikt}|_{-pi}^pi}_{=0}=frac{-2pi}{ik}=frac{2pi i}{k}$
so we get
$$c_k=frac{i}{k},quad forall k$$
So we get
$$hat{hat{f}}=sum_{k=-N}^N frac{i}{k}e^{-ikt}$$
Now I am a bit confused to what $N$ now exactly is and how I can "choose" it, like when is it infinity? Sometimes I also see that the sum goes from $k=1$ to $infty$ or from $-infty$ to $+infty$. How do I know which case it is?
Also: Shouldn't I have a vertical scaling term? Something like $hat{hat{f}}=frac{a_0}{2}+sum_{k=-N}^Nc_ke^{-ikt}$?
fourier-series
fourier-series
asked Jan 30 at 18:35
xotixxotix
291411
asked Jan 30 at 18:35
xotixxotix
291411
asked Jan 30 at 18:35
xotixxotix
291411
asked Jan 30 at 18:35
xotixxotix
291411
asked Jan 30 at 18:35
xotixxotix
291411
291411
add a comment |
add a comment |
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