Unverified Commit 2ed9928d authored by Guillaume Anciaux's avatar Guillaume Anciaux
Browse files

update the lecture

parent dfaea2a1
......@@ -1729,7 +1729,7 @@ $$[DFT^{-1}(\hat{f})]_j = f_j = \frac{1}{N}\sum_{i=0}^{N-1} \hat{f}_k \cdot \exp
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<h1 id="Fourier-series-and-derivations">Fourier series and derivations<a class="anchor-link" href="#Fourier-series-and-derivations">&#182;</a></h1>$$f'(x) = \frac{d}{dx} \sum_{k=-\infty}^{\infty} c_k \cdot \exp\left(i \frac{2\pi kx}{T}\right) \simeq \sum_{k=-N/2}^{N/2} c_k i \frac{2\pi k}{T}\cdot \exp\left(i \frac{2\pi kx}{T}\right)$$<p>So that:</p>
<h1 id="Fourier-series-and-derivations">Fourier series and derivations<a class="anchor-link" href="#Fourier-series-and-derivations">&#182;</a></h1>$$f'(x) = \frac{d}{dx} \sum_{k=-\infty}^{\infty} c_k \cdot \exp\left(i \frac{2\pi kx}{T}\right) \simeq \sum_{k=-N/2}^{N/2-1} c_k i \frac{2\pi k}{T}\cdot \exp\left(i \frac{2\pi kx}{T}\right)$$<p>So that:</p>
$$f'(x) = \sum_{k=-N/2}^{-1} c_k i \frac{2\pi k}{T}\cdot \exp\left(i \frac{2\pi kx}{T}\right) + \sum_{k=0}^{N/2-1} c_k i \frac{2\pi k}{T}\cdot \exp\left(i \frac{2\pi kx}{T}\right)$$$$f'(x) = \sum_{k=N/2}^{N-1} c_{k-N} i \frac{2\pi (k-N)}{T}\cdot \exp\left(i \frac{2\pi x(k-N)}{T}\right) + \sum_{k=0}^{N/2-1} c_k i \frac{2\pi k}{T}\cdot \exp\left(i \frac{2\pi kx}{T}\right)$$$$f'(x) = \sum_{k=0}^{N-1} c_k i \frac{2\pi Freq(k)}{T}\cdot \exp\left(i \frac{2\pi kx}{T}\right)$$<p>with $Freq$ the shifted wave numbers, also given by numpy.fft.fftfreq:</p>
$$Freq = [0, 1, ..., N/2-1, -N/2, ..., -1]$$
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......@@ -1745,7 +1745,7 @@ $$Freq = [0, 1, ..., N/2-1, -N/2, ..., -1]$$
<li>$f'(x_j) = f_j'$</li>
<li>$c_{k+N} = c_k$</li>
</ul>
$$f_j' = \sum_{k=0}^{N-1} c_k i \frac{2\pi Freq(k)}{T}\cdot \exp\left(i \frac{2\pi kx}{T}\right)$$<p>So that:</p>
$$f_j' = \sum_{k=0}^{N-1} c_k i \frac{2\pi Freq(k)}{T}\cdot \exp\left(i \frac{2\pi k j}{N}\right)$$<p>So that:</p>
$$DFT\left[f'\right]_k = c_k i \frac{2\pi Freq(k)}{T}$$
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