# Tag Info

Accepted

### 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
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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

### 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

### 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

### 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
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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

### 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 ...
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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

### 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
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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 ...
• 2,222

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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 ...
• 32.1k
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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 ...
• 53.9k
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### 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
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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 ...
• 25.9k
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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 ...
• 20.7k

### 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

### 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
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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 ...
• 91.4k
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• 12.9k

### 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
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)...