o

your dft code

Discrete Fourier Transform: Create your onw code

outline

equations

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$$\tag{1} X(\omega) = \int_{-\infty}^{+\infty} x(t) e^{-j \omega t} \ dt $$

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$$\tag{2} X(\omega) \approx \sum_{n=0}^{\infty} x(t_n) e^{-j \omega t_n} \ \Delta t $$

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$$\tag{0} e^{j \omega t} = \cos \omega t + j \sin \omega t $$

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$$\tag{4} x(t) = 3 \sin 20\pi t + 4 \cos 5\pi t + 1 $$

matrix

Is it possible to formulate all in matrix form?

$$\tag{5} y_k^{\rm Re} = \sum_{n=0}^{N-1} x_n \cos \left( \frac{2\pi kn}{N} \right) $$

$$\tag{6} y_k^{\rm Im} = -\sum_{n=0}^{N-1} x_n \sin \left( \frac{2\pi kn}{N} \right) $$

$$ \left[ \begin{array}{cccccc} 1 & 1 & 1 & \dots & 1 & 1 \newline \newline 1 & \cos \left( \frac{2\pi}{N} \cdot 1 \right) & \cos \left( \frac{2\pi}{N} \cdot 2 \right) & \dots & \cos \left( \frac{2\pi}{N} \cdot (N-2) \right) & \cos \left( \frac{2\pi}{N} \cdot (N-1) \right) \newline \newline 1 & \cos \left( \frac{2\pi k}{N} \cdot 1 \right) & \cos \left( \frac{2\pi k}{N} \cdot 2 \right) & \dots & \cos \left( \frac{2\pi k}{N} \cdot (N-2) \right) & \cos \left( \frac{2\pi k}{N} \cdot (N-1) \right) \newline \newline \vdots & \vdots &\vdots & \ddots & \vdots & \vdots \newline \newline 1 & \cos \left( \frac{2\pi(N-2)}{N} \cdot 1 \right) & \cos \left( \frac{2\pi(N-2)}{N} \cdot 2 \right) & \dots & \cos \left( \frac{2\pi (N-2)}{N} \cdot (N-2) \right) & \cos \left( \frac{2\pi (N-2)}{N} \cdot (N-1) \right) \newline \newline 1 & \cos \left( \frac{2\pi(N-1)}{N} \cdot 1 \right) & \cos \left( \frac{2\pi(N-1)}{N} \cdot 2 \right) & \dots & \cos \left( \frac{2\pi(N-1)}{N} \cdot (N-2) \right) & \cos \left( \frac{2\pi(N-1)}{N} \cdot (N-1) \right) \end{array} \right] \left[ \begin{array}{c} x_0 \newline \newline x_1 \newline \newline x_2 \newline \newline \vdots \newline \newline x_{N-2} \newline \newline x_{N-1} \end{array} \right] = \left[ \begin{array}{c} y_0^{\rm Re} \newline \newline y_1^{\rm Re} \newline \newline y_2^{\rm Re} \newline \newline \vdots \newline \newline y_{N-2}^{\rm Re} \newline \newline y_{N-1}^{\rm Re} \end{array} \right] $$

$$ \left[ \begin{array}{cccccc} 0 & 0 & 0 & \dots & 0 & 0 \newline \newline 0 & \sin \left( \frac{2\pi}{N} \cdot 1 \right) & \sin \left( \frac{2\pi}{N} \cdot 2 \right) & \dots & \sin \left( \frac{2\pi}{N} \cdot (N-2) \right) & \sin \left( \frac{2\pi}{N} \cdot (N-1) \right) \newline \newline 0 & \sin \left( \frac{2\pi k}{N} \cdot 1 \right) & \sin \left( \frac{2\pi k}{N} \cdot 2 \right) & \dots & \sin \left( \frac{2\pi k}{N} \cdot (N-2) \right) & \sin \left( \frac{2\pi k}{N} \cdot (N-1) \right) \newline \newline \vdots & \vdots &\vdots & \ddots & \vdots & \vdots \newline \newline 0 & \sin \left( \frac{2\pi(N-2)}{N} \cdot 1 \right) & \sin \left( \frac{2\pi(N-2)}{N} \cdot 2 \right) & \dots & \sin \left( \frac{2\pi (N-2)}{N} \cdot (N-2) \right) & \sin \left( \frac{2\pi (N-2)}{N} \cdot (N-1) \right) \newline \newline 0 & \sin \left( \frac{2\pi(N-1)}{N} \cdot 1 \right) & \sin \left( \frac{2\pi(N-1)}{N} \cdot 2 \right) & \dots & \sin \left( \frac{2\pi(N-1)}{N} \cdot (N-2) \right) & \sin \left( \frac{2\pi(N-1)}{N} \cdot (N-1) \right) \end{array} \right] \left[ \begin{array}{c} x_0 \newline \newline x_1 \newline \newline x_2 \newline \newline \vdots \newline \newline x_{N-2} \newline \newline x_{N-1} \end{array} \right] = \left[ \begin{array}{c} y_0^{\rm Re} \newline \newline y_1^{\rm Re} \newline \newline y_2^{\rm Re} \newline \newline \vdots \newline \newline y_{N-2}^{\rm Re} \newline \newline y_{N-1}^{\rm Re} \end{array} \right] $$

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