Utilisateur:Sylvain d'Altaïr/Brouillon

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Théorème de l'élément primitif
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[modifier] Discrete Hilbert transform

If the signal s(t)\, is bandlimited, then \widehat s(t) is bandlimited in the same way. Consequently, both these signals can be sampled according to the sampling theorem, resulting in the discrete signals s[n]\, and \widehat{s}[n]. The relation between the two discrete signals is then given by the convolution:

\widehat{s}[n] = h[n] * s[n]\,

where

h[n]= \begin{cases} 0, & \mbox{for }n\mbox{ even},\\ \frac2{\pi n} & \mbox{for }n\mbox{ odd} \end{cases}

which is non-causal and has infinite duration. In practice, a shortened and time-shifted approximation is used. The usual filter design tradeoffs apply (e.g. filter-order and latency vs. frequency-response). Also notice, that h[n]\, is not just a sampled version of the Hilbert filter h(t)\,, defined above. Rather it is a sequence with this discrete-time Fourier transform:

H(e^{i\omega}) = \begin{cases} +i, & -\pi \leq \omega < 0 \\ -i, & 0 \leq \omega < \pi \end{cases}

We note that a sequence similar to h[n]\, can be generated by sampling H(e^{i\omega})\, and computing the inverse discrete Fourier transform. The larger the transform (i.e. more samples per 2π radians), the better the agreement (for a given value of the abscissa, n). The figure shows the comparison for a 512-point transform. (Due to odd-symmetry, only half the sequence is actually plotted.)
But that is not the actual point, because it is easier and more accurate to generate h[n]\, directly from the formula. The point is that many applications choose to avoid the convolution by doing the equivalent frequency-domain operation:  simple multiplication of the signal transform with the samples of H(e^{i\omega})\,, made even easier by the fact that the real and imaginary components are 0 and ±1 respectively. After transforming back to the time-domain, those applications have indirectly generated (and convolved with) not h[n]\,, but the DFT approximation to it. Frequency-domain filtering is often called fast convolution.

Fast convolution filters the signal piecewise, and the outputs are subsequently pieced back together. An important issue to understand about that approach is circular convolution. It is a type of distortion that can be avoided by overlapping the segments and choosing a segment size usually several times larger than the duration of the filter impulse response. But when the DFT approximation is used instead of a designed filter, the impulse response duration equals the segment length, and circular convolution cannot be avoided completely. However, it can be made arbitrarily small with an appropriate choice of the segment length.