Discrete Fourier transform of a time series extended by a few extra points
$begingroup$
With the time series $y_t = {y_1, ldots, y_n}$, we define the extended time series as $y^*_t = {y_1, ldots, y_n, ldots, y_{n+k}}$, with $k ll n$. We take the discrete Fourier transform of $y_t$ as $$ F[y_t](omega) = sum_{t=1}^n y_t exp(iomega t).$$ For this question, we take $y_t$ and $omega$ real.
Now assume that $F[y_t](omega)$ is given, e.g. we know the values of $F[y_t](omega)$ at the $n$ Nyquist frequencies $0, frac{2pi}{n}, ldots, frac{2pi(n-1)}{n}$. What can we say about $F[y_t^*](omega)$?
approximation fourier-transform
asked Dec 20 '18 at 10:35
marnixmarnix
12
$endgroup$
add a comment |
$begingroup$
With the time series $y_t = {y_1, ldots, y_n}$, we define the extended time series as $y^*_t = {y_1, ldots, y_n, ldots, y_{n+k}}$, with $k ll n$. We take the discrete Fourier transform of $y_t$ as $$ F[y_t](omega) = sum_{t=1}^n y_t exp(iomega t).$$ For this question, we take $y_t$ and $omega$ real.
Now assume that $F[y_t](omega)$ is given, e.g. we know the values of $F[y_t](omega)$ at the $n$ Nyquist frequencies $0, frac{2pi}{n}, ldots, frac{2pi(n-1)}{n}$. What can we say about $F[y_t^*](omega)$?
approximation fourier-transform
asked Dec 20 '18 at 10:35
marnixmarnix
12
$endgroup$
add a comment |
$begingroup$
With the time series $y_t = {y_1, ldots, y_n}$, we define the extended time series as $y^*_t = {y_1, ldots, y_n, ldots, y_{n+k}}$, with $k ll n$. We take the discrete Fourier transform of $y_t$ as $$ F[y_t](omega) = sum_{t=1}^n y_t exp(iomega t).$$ For this question, we take $y_t$ and $omega$ real.
Now assume that $F[y_t](omega)$ is given, e.g. we know the values of $F[y_t](omega)$ at the $n$ Nyquist frequencies $0, frac{2pi}{n}, ldots, frac{2pi(n-1)}{n}$. What can we say about $F[y_t^*](omega)$?
approximation fourier-transform
asked Dec 20 '18 at 10:35
marnixmarnix
12
$endgroup$
With the time series $y_t = {y_1, ldots, y_n}$, we define the extended time series as $y^*_t = {y_1, ldots, y_n, ldots, y_{n+k}}$, with $k ll n$. We take the discrete Fourier transform of $y_t$ as $$ F[y_t](omega) = sum_{t=1}^n y_t exp(iomega t).$$ For this question, we take $y_t$ and $omega$ real.
Now assume that $F[y_t](omega)$ is given, e.g. we know the values of $F[y_t](omega)$ at the $n$ Nyquist frequencies $0, frac{2pi}{n}, ldots, frac{2pi(n-1)}{n}$. What can we say about $F[y_t^*](omega)$?
approximation fourier-transform
approximation fourier-transform
asked Dec 20 '18 at 10:35
marnixmarnix
12
asked Dec 20 '18 at 10:35
marnixmarnix
12
asked Dec 20 '18 at 10:35
marnixmarnix
12
asked Dec 20 '18 at 10:35
marnixmarnix
12
asked Dec 20 '18 at 10:35
marnixmarnix
12
12
add a comment |
add a comment |
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