# Tag Info

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### What should be the correct scaling for PSD calculation using $\tt fft$

There is only one correct way of scaling DFT when calculating PSD with RMS values. Given input signal $x$ and its DFT $X$, the exact formula is: $$\mathrm{PSD}=\frac{2\cdot \hat{X}}{f_s\cdot S}$$ ...
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### What is the difference between the PSD and the Power Spectrum?

The power spectrum is a general term that describes the distribution of power contained in a signal as a function of frequency. From this perspective, we can have a power spectrum that is defined over ...

### When is it true that "the Fourier transform of the autocorrelation is the spectral density"?

The FIRST PROBLEM you ask about is how does $$R_x(\tau) = E[x(t)x(t+\tau)]$$ equal $$R_x(\tau) = \lim_{T\rightarrow \infty} \frac{1}{2T} \int_{-T}^{+T} x(t)x(t+\tau) dt$$ because (7) requires ...

### When is it true that "the Fourier transform of the autocorrelation is the spectral density"?

To add to Peter K.'s answer: The correct defition of the autocorrelation is $R_x(\tau) = E[x(t)x(t+\tau)]$. If the process is stationary, that simplifes a little to $R_x(\tau) = E[x(0) x(\tau)]$. If ...
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### How to interpret spectrogram correctly?

The problem is not the spectrogram parameters, these are correct since they only depend on what resolution you want in time and frequency domain. Also, the spectrogram interpretation is correct, there ...
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### Numerically generating noise time series from spectral density

As mentioned by @MBaz in comments, you could use the fact that filtering white noise with unitary power spectral density (for example from independent Gaussian samples zero mean and unitary standard ...

### When is it true that "the Fourier transform of the autocorrelation is the spectral density"?

The magnitude-square $\Big| X(\omega) \Big|^2$ of the Fourier Transform of energy signals is the energy spectral density. For power signals, you gotta do a little bit with normalization, so that ...

### What should be the correct scaling for PSD calculation using $\tt fft$

I'll try to explain how one arrives at the correct scaling factor. In order to keep things simple I will assume a rectangular window. Other windows can be taken into account by an additional factor ...
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### Interpretation of Clarke's Doppler power spectral density

A simple, "non-technical" way of thinking of it is the fact that the Doppler frequency is proportional to $\cos\theta$. The amplitudes of cosine, however, are not uniformly distributed, but are ...

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### How does SciPy's Welch function change the shape of the data?

Assuming that "6041" is a typo and it's actually "6401" that would be expected behavior. The result of welch() is a frequency domain vector the ...