New answers tagged power-spectral-density
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Question 1: Yes you are correct, the power spectral density is the power distribution per unit frequency so is a continuous function of frequency.
Question 2: The single number as given can be an estimate of total power.
What they gave is completely incorrect starting with the formula as given:
$$P = lim_{T \rightarrow \infty}\frac{1}{T}\int_0^T|x(k)|^2dt$$
...
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If someone is quoting one number for power spectral density, then the underlying assumption is most often that the noise is white, with a constant density across the entire spectrum (or at least across the entire spectrum that we care about -- do a web search on "the ultraviolet catastrophe" for the underlying physical reason that true white noise ...
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We can't compute the PSD down to DC due to its divergence, but we can deduce the PSD everywhere else from knowing the PSD of a white noise process which is stationary (and the PSD is constant for all frequencies), and that the integration of white noise is a random walk process (Brownian), and the fact that the power response of integration goes as $1/f^2$ (...
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I currently work with the inner design details of Atomic Clocks so use the Allan Variance and Allan Deviation (ADEV) extensively. The primary point is that it can be used for non-stationary processes (which frequency noise is). For non-stationary signals where the autocorrelation or power spectral density can't be used with consistency, the Allan Deviation ...
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Given a signal y(t), the magnitude spectrum is
$$Y(f)=\left\lvert\int y(t)e^{-2\pi ft}\,\mathrm dt\right\rvert,$$
in other words the absolute value of the FT of the signal.
But that's only applicable for deterministic signals¹. If your signal itself is random, then you can't know $y(t)$ – you only know some stochastic properties of it. There's no $y(t)$ to ...
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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 stringent, and people can be interested in a wider range, like $1/f^\alpha$ with $1/2\le\alpha\le3/2$.
The plot above is from 1/f noise.
Since only looking ...
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time history of 1/f noise follows a power law proportional to t−α
If "time history" is supposed to mean "time domain waveform" that sentence is just non-sense.
1/f noise is just pink noise. There are many ways to generate it but in most cases it's a steady state signal with a flat envelope and it can be as long or as short as you want it ...
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should imply that also the time history has a power law
No. The PSD is an incomplete description of the signal since it has lost the phase information. There are infinitely many time sequences that have the same PSD. For example a unit impulse, white noise and a linear sweep have the same PSD (roughly speaking) but are completely different time-domain wave ...
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A digital filter doesn't have power spectral density, a signal has. I guess what you want is the transfer function of this filter, i.e., $H(z)=Y(z)/X(z)$.
Let's add two intermediate signal $u(n)$ and $v(n)$ in the diagram.
We have:
$$
y(n) = v(n-1) + du(n) \tag{1}
$$
$$
u(n)=v(n-1)+x(n)\tag{2}
$$
$$
v(n) = x(n) -du(n) \tag{3}
$$
Take Z-transform and we get
$...
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By Welch's method, you can calculate the power spectrum by averaging the magnitude of a bunch of FFT frames (or time frames of the STFT). If your signal is $x(t)$, and its STFT is $X(\omega, \tau)$, where $\omega$ is the frequency bin, $\tau$ is the time frame and $T$ is the total number of frames, the PSD is
$$P_x(\omega) = \frac{1}{T} \sum_{\tau=0}^{T-1} |...
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For linear-swept sinusoids (not exponentially-swept), there is a well-developed theory that is spelled out in this answer.
In general, if you squirt anything broadbanded at the two microphones, you can divide the FFT of the signal from the device under test (DUT) by the FFT of the signal from the reference microphone. The logarithm of the magnitude (dB) of ...
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If the values for each sample of $n(t)$ are given, then the mean and variance can be estimated using the equations for sample mean and sample variance, which is trivial so I assume the OP is only given that it is Gaussian-distributed with average power of 5 mW. From that alone, there is no way to know what the mean of the signal is since it hasn't been ...
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You can get the sound pressure level from PSD, and the velocity is related to the sound pressure. According to the equation of motion,
$$
\rho \frac{\mathrm{d}\vec{v}}{\mathrm{d}t} = -\nabla p \tag{1}
$$
where $\rho$ is the density of the medium, $\vec{v}$ is the particle velocity, $\nabla=\frac{\partial}{\partial x}\vec{i}+\frac{\partial}{\partial y}\vec{j}+...
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