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Switching from the definition (2.7) to the frequency domain, one obtains
![$\displaystyle s_1\left(x\right)
=\sum_{k \in I_n} \hat g_k \, c_k\left(\tilde...
...(\tilde \varphi\right) \,
{\rm e}^{-2\pi{\mbox{\scriptsize {i}}} (k + n r)x }$](img142.png) |
(2.8) |
with the discrete Fourier coefficients
![$\displaystyle \hat g_k := \sum_{l \in I_n} g_l \, {\rm e}^{ 2\pi{\mbox{\scriptsize {i}}} \frac{k l}{n}}.$](img143.png) |
(2.9) |
Comparing (2.6) to (2.7) and assuming
small for
suggests to set
![$\displaystyle \hat g_k := \left\{
\begin{array}{ll}
\frac{\hat f_k}{c_k \left...
...\\ [1ex]
0 & \text{for } k \in I_n \backslash I_N . \\
\end{array}
\right.$](img146.png) |
(2.10) |
Then the values
can be obtained from (2.9) by
a FFT of size
.
This approximation causes an aliasing error.
Stefan Kunis
2004-09-03