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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: $$\mathrm{PSD}=\frac{2\cdot \hat{X}}{f_s\cdot S} $$ ...
jojeck's user avatar
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17 votes
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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 ...
hops's user avatar
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9 votes

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 ...
Peter K.'s user avatar
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8 votes

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 ...
leonbloy's user avatar
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7 votes
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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 ...
JorgeGT's user avatar
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7 votes
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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 ...
SleuthEye's user avatar
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7 votes
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Power Spectral Density From Variance

The power spectral density is NOT the Fourier transform of the variance. For wide-sense-stationary (WSS) signals, the power spectral density is the Fourier transform of the Autocorrelation function (...
Dan Boschen's user avatar
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7 votes

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 ...
robert bristow-johnson's user avatar
6 votes

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 ...
Matt L.'s user avatar
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6 votes
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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 ...
Robert L.'s user avatar
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6 votes

$|X(e^{jω})|^2$ - Power or Energy Density?

do you see anywhere in your book where this "DTFT" is defined for your w.s.s. process, $x[n]$? the DTFT is normally defined as $$ X(e^{j \omega}) \triangleq \sum\limits_{n=-\infty}^{+\infty} x[n] e^{...
robert bristow-johnson's user avatar
6 votes
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signal of $\frac{1}{f}$ Noise

Power-law behaviors in frequency can be found in several unrelated observations and systems. This is apparently the case for $1/f$ or flicker noise. Note that an exact $\alpha=1$ exponent might be too ...
Laurent Duval's user avatar
6 votes
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Allan Variance vs Autocorrelation - Advantages

My current work involves the design details of atomic clocks where we use the Allan Variance and Allan Deviation (ADEV) extensively. The primary point is that it can be used for non-stationary ...
Dan Boschen's user avatar
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6 votes

Periodogram (Welch) has different levels depending on length of segment/ resolution

The power spectrum measures the distribution of power vs frequency components, so its scaling preserves the correct power spectrum peak heights, while the PSD measures the distribution of power vs ...
Jdip's user avatar
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6 votes
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Modelling MEMS accelerometer noise

Yes, you can use the datasheet noise density. Model the accelerometer as a source of white noise, and compute its characteristics. That works to the extent that you can trust the data sheet. That's ...
TimWescott's user avatar
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5 votes
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How do you calculate spectral flatness from an FFT?

The most authoritative reference I can come up with is from Jayant & Noll, Digital Coding Of Waveforms, (c) Bell Telephone Laboratories, Incorporated 1984, published by Prentice-Hall, Inc. On ...
Peter K.'s user avatar
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5 votes
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Power spectral density of $\left(x(t)\right)^2$?

Since the question has been raised as to whether the hint that I had given to the OP in a comment on the original question was appropriate for a newcomer to signal processing, here goes. Stripped of ...
Dilip Sarwate's user avatar
5 votes

When to use cross spectral density versus cross correlation?

The cross-spectral density is in the frequency domain while the cross-correlation function is in the time domain. The two are Fourier Transform pairs, the FT of the cross correlation function is the ...
Dan Boschen's user avatar
  • 51.4k
5 votes

Interpretation of Clarke's Doppler power spectral density

In addition to Carlos answer, I want to correct your general understanding: What I understand of Doppler spread is that the relative motion between Transmitter (TX) and Receiver (RX) change the ...
Maximilian Matthé's user avatar
5 votes
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A query on Power spectral density (PSD)

Almost, but let's start at the beginning. If the random process $N(t)$ has a power spectrum then it is at least wide-sense stationary (WSS), i.e., its mean and its autocorrelation function do not ...
Matt L.'s user avatar
  • 90k
5 votes

PSDs and Parseval's theorem

That equation is indeed simple, but it's also wrong. So your doubts are completely justified. What is probably meant is something like $$S_X(f)=\lim_{T\rightarrow\infty}\frac{1}{T}\left|X_T(f)\right|^...
Matt L.'s user avatar
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5 votes
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variance in the time domain versus variance in frequency domain

Variance is never defined as power. For a wide-sense stationary random process $X(t)$ with zero mean $$\mu_X=E\{X(t)\}=0\tag{1}$$ the variance of $X(t)$ equals its power. The autocorrelation of $X(...
Matt L.'s user avatar
  • 90k
5 votes

Why do we need the power spectral density?

Power Spectral Density (PSD) is a theoretical construct, required to deal with random processes (signals). It's by definition the Fourier transform of the auto-correlation function (another ...
Fat32's user avatar
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5 votes
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Filter before or after multiplication of two signals?

A personal rule: in general, it can be better to perform non-linear operations before linear ones. One reason behind that is that a lot of practical concerns are related to outliers or suspect ...
Laurent Duval's user avatar
5 votes
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Am I supposed to normalize FFT in Python?

Your decision to normalize or not does not change the accuracy of your answer, as it is simply a scaling factor. If you use the common scaling of $1/N$, then the output for each DFT bin will represent ...
Dan Boschen's user avatar
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5 votes
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Is spectral density sometimes normalized by sampling rate rather than bin size?

The use of $rad/\sqrt{\text{Hz}}$ suggests that this is phase noise specifically (a spectral density due to phase fluctuations), and typically in my use this has been described as a power spectral ...
Dan Boschen's user avatar
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5 votes
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Is spectral density conserved after aliasing?

No. The aliased component will interfere with the non-aliased components and the interference can constructive or destructive. Trivial example: $$x[n] = \sin\left(\frac\pi2n\right)$$ If you down ...
Hilmar's user avatar
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5 votes
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Conversion of dBm/Hz into watt

First, multiply (-174dBm/Hz) by 180000, then convert the result into watt? Because the decibel is a logarithmic unit, you need to be careful here. $-174\mathrm{dBm}$ means $10^{-17.4} \mathrm{mW}$. ...
TimWescott's user avatar
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5 votes

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 ...
Hilmar's user avatar
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4 votes
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Why Fourier Transform doesn't show obvious periodicity?

An FFT doesn't necessarily show a fundamental periodicity, but instead spectral periodicities, which might be higher harmonics (or overtones) of the periodicity that you might perceive visually. Also,...
hotpaw2's user avatar
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