New answers tagged fourier-series
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The amplitude you “see” in a spectrum is the result of a narrow band filter, a bin of a DFT, or a pixel at some finite DPI. All of those cover some non-zero bandwidth. So what you see does not have infinitesimal bandwidth (or less).
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You can see the non-zero magnitude of the spectrum, but the interpretation that the signal contains a sinusoidal component at a specific frequency where the magnitude is non-zero is wrong. If there is a sinusoidal component present, then there is a Dirac impulse at the respective frequency. A non-zero Fourier transform at a certain frequency is not ...
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Let me summarize my understanding of what you're trying to do. You have a real-valued sequence $x[n]$, obtained by sampling a real-valued continuous function, and you computed its DFT $X[k]$. The sequence can be expressed in terms of its DFT coefficients:
$$x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{j2\pi nk/N},\qquad n\in[0,N-1]\tag{1}$$
where $N$ is the ...
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I have looked at your image (and didn't read the post) and its explanation is as follows.
Let the data in the blue curve be $x[n]$, and the data in the red curve be $y[n]$; then it can be seen and shown that:
$$y[n] = \tfrac12 ( x[n] + (-1)^n x[n] ) $$
In the DTFT domain this relationship becomes :
$$ Y(e^{j\omega}) = \tfrac12 \big( X(e^{j\omega}) + X(...
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