Questions tagged [gaussian]
The gaussian function, an exponential function with a negative square of the argument in the exponent, is interesting in signal processing because the Fourier Transform of a gaussian function is also a gaussian function.
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Gaussian blur with variable standard deviation
I'm trying to find fast ways to evaluate the following integral defined over points $\mathbf{x} \in \mathbb{R}^2$:
$$\Phi(\mathbf{x}) = \frac{1}{2\pi\sigma^2(\mathbf{x})}\int d^2y\, \exp\left(-\frac{|\...
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Amplitude error of STFT using gaussian window
I am implementing the STFT algorithm and I also need to find the amplitude accuracy and Probability of Intercept (POI).
The only reference I have been given for these factors is this formula
$$ E_{ZD,...
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How to determine the cut-off frequency for a Gaussian filter in a reasonable way using frequency spectrum analysis?
I have 100Hz data, which is upsampled to 500 Hz using Cubic spline interpolation.
Subsequently, I would like to smooth this upsampled data using filtering, but I struggle to find an objective basis ...
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The variance of a band-limited Gaussian process out of correlation filter
Let $X(t)$ be a real Gaussian process of bandwidth $W$ and spectral density $N_0/2$. Let $\phi(t)$ be a orthonormal (in $L_2$ sense) function with bandwidth $W$.
Consider the correlator output $\int_{-...
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Determining when a time series switches which of two Gaussians it samples from
I'm sampling from what should be an elliptical 2D Gaussian. However, one of the instruments in the setup seems to be switching between two values for the phase between synthesizers, which results in ...
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Analytical Proof of LoG Filter Separability
I've been studying the Laplacian of Gaussian (LoG) filter, and I understand its importance in edge and blob detection. However, I’ve come across some references that mention the LoG filter being ...
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Time delay estimation for superimposed random signals sampled from a multivariate Gaussian mixture?
Suppose my signal model is:
$$
\mathbf{y} = D(\tau_1) \mathbf{x}_1 + D(\tau_2) \mathbf{x}_2
$$
where $D(\tau)$ is a delay matrix that shifts a signal by $\tau$ samples.
$\mathbf{x}_1$ and $\mathbf{x}...
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Why isn't the Gaussian blur kernel defined via the cdf?
A 1-D Gaussian blur kernel of size $2k + 1$ is defined as:
$$K = \text{Normalize}([G(-k), \ldots, G(0), \ldots, G(k)])$$
where
$${\displaystyle G(r) = {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {...
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Minimum Number of Base Accelerations Needed to Simulate Gaussian White Noise in Structural Dynamics?
I am currently working on a scientific paper where I subject a structure to base accelerations modeled as Gaussian white noise. I am relatively new to signal processing and would appreciate some ...
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Autocorrelating output signal which sampled at rates higher than signalling rate
A bpsk signal(a$_{n}$) with d$_{min}$ = 1, passes through a pulse shaping filter p(t), the resultant pulse shaped signal ($\sum_{n=0}^{k-1} a_{n}p(t-nT_{s})$) passes through an AWGN channel which adds ...
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GMSK Modulator Large sample Buffer size and high Bandwidth
I found good answers to my queries here Gaussian Pulse shaping filter and suggestion for NCO based GMSK modulator seems to be undersstandable. Thanks to Dan Boschen
, but when moving ahead to design ...
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Gaussian Pulse shaping filter
I need expert advice on implementation of Gaussian Pulse Shaping filter for generating GMSK signal in further steps. I have implemented as follows, Can anyone confirm that the Pulse would be shaped as ...
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Autocorrelation function of a Gaussian random signal
Im a beginner at signal processing and I've gotten the following question in an exercise:
"Write down the equation for a Gaussian probability density distribution and relate the different
...
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Regarding SINR (Signal to Interference Plus Noise Ratio) in wireless communication
I am working on a research paper related to wireless communication, wherein I am facing some doubt while writing expression of SINR (which is a ratio of Signal variance in numerator to Interference ...
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Narrowband Gaussian Noise
Could anyone lead me on the right path on where I could get more information in regards to a narrowband Gaussian signal can be modeled as:
$$
X(t)\cos(2\pi f t)+Y(t)\sin(2\pi f t)
$$
where $X(t)$ and $...
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Kalman filter for multiple data sources, measurements from which have different characteristics of Gaussian noise
I am trying to use the Kalman filter for my task:
During the time, I receive data from different sensors. The state of the model may change over time according to the Const Velocity model, or the ...
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How to implement scale-dependent Gaussian averaging using Morlet wavelet envelope in Python?
I'm trying to reproduce the scale-dependent Gaussian averaging of a time series as described in this paper: https://arxiv.org/pdf/1706.01126.pdf
The process involves performing a continuous wavelet ...
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How to mask part of signal?
I am trying to implement masking in 1D signal data,
I saw in one paper that they are masking 70% of the signal as in the figure below:
In another study, they have mentioned that the mask part is ...
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$i^{\text{th}}$-dimensional autocorrelation function
I am referring to the work of Stephen A. Billings on "Identification of a class of nonlinear systems using correlation analysis" from the year 1978, where it is mentioned that the $i^{\text{...
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ADC response inversion to Gaussian noise
Assuming the input to an ADC is a Gaussian white noise signal, and being a bit idealistic in all senses, is there a theoretical expression that links input power to output power which can be inverted, ...
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ADC bias, noise and number of bits under Gaussian signals
I'm digitizing a zero-mean complex Gaussian white noise signal with certain variance, through independent I/Q baseband sampling (two ADCs).
The noise variance (power) depends on the thermal emission/...
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Discrete Fourier Transform of the Gaussian
Cross-posted from here
I encountered the following question in a Digital Image Processing examination:
Find the 2D DFT of $\frac{1}{2 \pi \sigma^2} e^{-\frac{(x - x_0)^2 + (y - y_0)^2}{2 \sigma^2}}$ ...
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Finding mean, autocorrelation, and power spectrum of $y[n]=(x∗h)[n]$ where $x[n]$ is zero mean white gaussian noise
If given $x[n]$ is zero mean white Gaussian noise with variance $\sigma_x^2$, and a filter with a known impulse response $h[n]$, how would I find the mean, autocorrelation, and power spectrum of $y[n] ...
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What's the distribution of the DFT of a real-valued, zero-mean, normally distributed random vector?
Suppose $X$ is a real-valued N-dimensional Gaussian vector, $X \sim \mathcal{N}(\mathbf{0}, C_X)$. The discrete Fourier transform can be obtained by left-multiplying with the unitary DFT matrix, i.e. $...
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Is it possible to reformulate a Kalman Filter as a Gaussian Markov Random Field?
The generic formulation of a KF uses a set of transitition equations, while the GMRF is typically specified through its mean and precision. However, a simple KF involves Gaussianity and Markov ...
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Generating time-domain noise from PSD
I have a question regarding generating a time-domain noise from the power spectral density (PSD), this was addressed in this question (How to generate time-series from a given one-sided PSD?) but I ...
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Can the standard deviation of the Gaussian window in a Gabor filter be made infinitesimally small?
My understanding is that the standard deviation of the Gaussian window in a Gabor filter dictates the temporal resolution.
Wouldn't it always be better then to make the window smaller, thus achieving ...
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Which probability density functions to use in a likelihood ratio test for the radar detection problem?
I am trying to understand the radar detection problem in the form of the generalized likelihood ratio test and am having a little trouble with understanding the noise distributions. Perhaps this will ...
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How to evaluate pairwise error probability and detection in presence of gaussian noise?
I am reading Digital Communication Systems by Simon Haykin and I am stuck at one point.
Consider a two-dimensional signal space that has a message constellation of four points, given by $s_1,s_2,s_3,...
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Spectral power density of thermal noise
In telecommunications the received signal $y(t)$ of a receiver is equal to $x(t)S+n(t)$ where $x(t)$ is the sent signal of the sender , $S$ is a constant which describes the total attenuation of the ...
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Contradiction when simulating WGN
To my knowledge, white Gaussian noise (WGN) is defined as a process with a correlation function:
$$
R[k]=\sigma^2 \delta[k]
$$
and whose symbols are distributed according to $N(0,\sigma^2)$.
Naturally,...
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Gaussian filter: Plotting DTFT and DFT (by hand) from the continuous-time impulsive response
I am trying to make an algorithm that plots out the Discrete-Time Fourier Transform (DTFT) and the Discrete Fourier Transform (DFT) of the Gaussian filter. The impulsive response and its transfer ...
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Can a time signal be modeled as a multivariate Gaussian distribution?
Suppose I have several groups of signal measurements, each containing multiple replicates, and I know that within each group the signal "shape" is approximately the same but with variance/...
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Kalman Filter Under Non-Gaussian Noise
I know that Kalman filter is optimal filter under some assumption like process and measurement noise are Gaussian. But if the process and measurement noise is non-Gaussian, the estimation of the ...
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Real-time convolution with Gaussian noise
I have a brain activity simulator that is capable of receiving various stimuli. Both generated signals and input stimulus are causal: a single sample is created every time step. I use a sinusoidal ...
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Choosing sigma values for Gaussian blurring on an anisotropic image
I have an anisotropic image that is anisotropic both in terms of number of voxels and also in terms of voxel resolution.
...
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The Effect of the Finite Radius of Gaussian Kernel
Page 168 of Digital Image Processing, Global Edition says:
we know that the values of a Gaussian function at a
distance larger than 3𝜎 from the mean are small enough that they can be ignored.
If we ...
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Is f/fL a high pass filter where fL is a low-pass version of f?
Let there be a signal f and its low pass filtered signal be fL. Then what can we say about the spectrum of f/fL ?
To be specific I am obtaining fL simply using gaussian blurring in the spatial domain.
...
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Inverting the normalized Gaussian to get a kernel window radius
I am seeking to compute a kernel radius to use with my gaussian convolution filter, and inspired by https://stackoverflow.com/a/68050503/, I came up with:
$$r=\sqrt{-2\sigma^2\ln\left( \epsilon\sigma\...
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One-sided bandwidth of the Gaussian filter
As defined in the CCSDS (section 3.1.2), the impulse response of the Gaussian filter is given by
$$
h(t) = \frac{1}{\sigma T \sqrt{2\pi}}e^{-\frac{t^2}{2\sigma^2T^2}}
$$
where
$$
\sigma = \frac{\sqrt{\...
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Downsample (aggregate) raster by a non-integer factor, using a Gaussian filter kernel
The task is to downsample (aggregate) a raster from 100m pixel size to 460m. The aggregation should be performed using a Gaussian filter. To better understand the task, I am following the paper ‘The ...
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Finding the average probability of error for a transmission system using this signal constellation [duplicate]
We have a signal constellation with decision regions given in the figure below and symbols are equally likely to be transmitted.
Now we have to find the average probability of error for a ...
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ICA and Gaussianity: A Misleading Example in the Book Konstantinos Koutroumbas, Sergios Theodoridis - Pattern Recognition
A book reports that ICA cannot be used if the independent components of the analyzed data are Gaussian (at most one can be Gaussian, but no other). However, in the same book, the following example is ...
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Detector for vector-valued signals
I'm trying to find the optimal probability of detection/false alarm for the following detection task. Given $N$ signal samples with $d$ dimensions (independent channels) each, assign the samples to ...
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Remove additive-zero-mean-Gaussian noise
I have 5 values, and each value is corrupted by Additive-Zero-Mean-Gaussian Noise with variance = 0.05.
Each value is in the range from 0 to 1. Is there anyway for me to remove the noise?
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Where does the following expression for stationary Gaussian Noise come from: $\langle \tilde{n}(f)\tilde{n}(f')\rangle = \delta(f-f')\frac{1}{2}S_n$?
First, the definitions:
Definition (Gaussian process)
A random process $X(t)$ is a Gaussian process if for all time points, $t_1,\ldots,t_n$ the random variables $X(t_1 ),\ldots,X(t_n)$ have a ...
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Classifying samples from complex Gaussian distributions
Assume we are trying to classify a sample $X$ as coming from one of two distributions:
$$
\mathcal{CN}(\mu, \sigma^2) \\
\mathcal{CN}(\nu, \sigma^2),
$$
where $\mathcal{CN}$ denotes a (circularly ...
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Confusion about KL divergence between complex Gaussians
The KL divergence between two real-valued Gaussian distributions with means $\mu_1$ and $\mu_2$ and common variance $\sigma^2$ is well known to be:
$$
D_{\text{KL}}\left(\mathcal{N}(\mu_1, \sigma^2) \...
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Understanding y=Hx+n equation in detail?
Consider a wireless communication system having $t$ transmitting antennas and $r$ receiving antennas. Then, the received signal is given by
$y = \mathbb{H}x+n \tag{1}$
where $\mathbb{H}$ is a $r \...
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Bounding Detection and Estimation by SNR in Gaussian Channel
Assume the following problem: A deterministic signal $X$ whose magnitude is known to satisfy $0 \leq \Vert X \Vert_2 < \Delta$ for some known constant $\Delta$ is transmitted through a Gaussian ...
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