21
votes
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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:
$$\mathrm{PSD}=\frac{2\cdot \hat{X}}{f_s\cdot S} $$
...
- 11.1k
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 ...
- 1,422
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 ...
- 25.9k
9
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 ...
- 254
8
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 ...
- 6,644
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 (...
- 53.9k
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 ...
- 21.1k
7
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 ...
- 12.9k
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 ...
- 91.4k
6
votes
Accepted
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 ...
- 2,222
6
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
$$S_X(f)=\lim_{T\rightarrow\infty}\frac{1}{T}\left|X_T(f)\right|^...
- 91.4k
6
votes
$|X(e^{jω})|^2$ - Power or Energy Density?
do you see anywhere in your book where this "DTFT" is defined for your w.s.s. process, $x[n]$? the DTFT is normally defined as
$$ X(e^{j \omega}) \triangleq \sum\limits_{n=-\infty}^{+\infty} x[n] e^{...
- 21.1k
6
votes
Accepted
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 ...
- 32.1k
6
votes
Accepted
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 ...
- 53.9k
6
votes
Accepted
Hard time figuring out whether the following random process is wide sense stationary
Let $\{\mathcal Y(t): -\infty < t < \infty\}$ denote a random process defined by
$$\mathcal Y(t) = \sum_{k=-\infty}^\infty \mathcal B_k p\left({t-kT}\right), \ -\infty < t < \infty\tag{1}$$...
- 20.7k
5
votes
Accepted
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 ...
- 25.9k
5
votes
Accepted
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 ...
- 20.7k
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 ...
- 53.9k
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 ...
- 6,248
5
votes
Accepted
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 ...
- 91.4k
5
votes
Accepted
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(...
- 91.4k
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 ...
- 28.4k
5
votes
Accepted
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 ...
- 32.1k
5
votes
Accepted
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 ...
- 53.9k
5
votes
Accepted
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 ...
- 53.9k
5
votes
Accepted
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 ...
- 47.1k
5
votes
Accepted
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 $10^{-17.4} \mathrm{mW}$. ...
- 12.9k
5
votes
Trying to understand the nperseg effect of Welch method
A good reference for the performance of Welch's method for Power Spectral Density (PSD) estimation can be found in this report by Solomon.
Welch's method involves the averaging of multiple ...
- 970
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 ...
- 47.1k
4
votes
Where does the delta function come from if we derive autocorrelation directly?
The book is indeed not consistent. Continuous-time (zero-mean) white Gaussian noise has an autocorrelation function
$$R(\tau)=\sigma^2\delta(\tau)\tag{1}$$
and a constant power spectral density (PSD)...
- 91.4k
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