Ideal Spectral Interpolation
Ideally, the spectrum of any signal
at any frequency
is obtained by projecting the signal
onto the
zero-phase, unit-amplitude, complex sinusoid at frequency
[264]:
![]() |
(3.43) |
where
![\begin{eqnarray*}
s_\omega(t) &\isdef & e^{j\omega t}\qquad\qquad\qquad\quad\;\,\mbox{(Fourier Transform)}\\
s_\omega(t_n) &\isdef & e^{j\omega_k t_n} \isdefs e^{j2\pi nk/N} \quad\mbox{(DFT)} \\
s_\omega(t_n) &\isdef & e^{j\omega t_n} \isdefs e^{j\omega n} \quad\qquad\;\mbox{(DTFT)}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img256.png)
Thus, for signals in the DTFT domain which are time limited to
,
we obtain
![]() |
(3.44) |
This can be thought of as a zero-centered DFT evaluated at
![$ \omega\in[-\pi,\pi)$](http://www.dsprelated.com/josimages_new/sasp2/img94.png)
![$ \omega_k =
2\pi k/N$](http://www.dsprelated.com/josimages_new/sasp2/img102.png)
![$ k\in[0,N-1]$](http://www.dsprelated.com/josimages_new/sasp2/img101.png)
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Interpolating a DFT
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Spectral Roll-Off