Questions tagged [gaussian]

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17
votes
3answers
47k views

Gaussian Blur - Standard Deviation, Radius and Kernel Size

I've implemented a gaussian blur fragment shader in GLSL. I understand the main concepts behind all of it: convolution, separation of x and y using linearity, multiple passes to increase radius... I ...
16
votes
5answers
23k views

How to Generate Band Limited Gaussian White Noise?

In some papers, I read that the additive noise is band limited Gaussian white. How can I simulate this type of noise use Matlab?
16
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3answers
6k views

How do I add AWGN to an I and Q representation of a signal?

I have a wireless communication system that I am simulating in Matlab. I am performing some watermarking through slightly adjusting the phase of the transmitted signal. My simulation takes the ...
14
votes
2answers
5k views

Which IIR filters approximate a Gaussian filter?

So it recently dawned on me that Bessel filters, despite being listed along with the other common types, are really an oddball that belongs in a different "class", and I'm trying to learn more about ...
11
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2answers
7k views

What is the relationship between the sigma in the Laplacian of Gaussian and the two sigmas in the Difference of Gaussians?

I understand that a Laplacian-of-Gaussian filter can be approximated by a Difference-of-Gaussians filter, and that the ratio of the two sigmas for the latter should be 1:1.6 for the best approximation....
10
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5answers
4k views

Why is random noise assumed to be normally distributed? [duplicate]

From residual in the linear regression to noise in signal processing are assumed to be normally distributed? By considering them as normally distributed we are kind of telling the pattern in the noise ...
10
votes
5answers
8k views

Low-pass filter parameters for image downsampling

I need to downscale an image in a factor of $s_x$ horizontally and $s_y$ vertically ($s_x$, $s_y$ < $1$). I want to use a finite $n\times m$ low-pass filter before downsampling. How should I ...
9
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2answers
2k views

Why is Gaussian noise called so?

Can you please explain: why is a specific type of noise called "Gaussian noise"? Why is it relevant to call it Gaussian? Please, explain in layman's terms.
8
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3answers
622 views

Capacity of AWGN channel

I am confused understanding basic concepts of communication over AWGN channels. I know the capacity of a discrete time AWGN channel is: $$C=\frac{1}{2}\log_2\left(1+\frac{S}{N}\right)$$ and it is ...
8
votes
2answers
7k views

What is a $BT$ (Bandwidth-Time) product with reference to modulation?

I would like to know what does Bandwidth-Time product mean. I understand that Bandwidth ($B$) = 1/Symbol Time ($T$), hence $BT = 1$. But how can it vary? What is its significance? e.g. When we say ...
8
votes
1answer
1k views

Is the discrete Gaussian kernel an eigenfunction of the DFT?

So the Gaussian function is an eigenfunction of the Fourier transform because it transforms into itself, right? But this isn't true for the sampled Gaussian in the DFT because the tails of the ...
7
votes
2answers
248 views

What is the difference between a simple gaussian filter and gaussian filter multiplied by its sum of it elements?

1 2 1 1 2 1 2 4 2 (A) (1/16) * 2 4 2 (B) 1 2 1 1 2 1 Both matrices are the same ...
7
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1answer
2k views

Why does the separable filter reduce the cost of computing the operator?

A separable filter in image processing can be written as product of two more simple filters. Typically a 2-dimensional convolution operation is separated into 2 onedimensional filters. This reduces ...
7
votes
1answer
1k views

Is there a mathematical method to determine if noise is Gaussian?

Is there a mathematical method to determine if a signal's noise is Gaussian? The only way I know so far is to analyze the histogram and layover a Gaussian distribution to visually determine if the ...
7
votes
1answer
2k views

Choice of Gaussian kernel parameters when lowpass filtering before image resampling?

I need to decimate a signal by a factor of q. More specifically my signal is a 3D "image": $\ I(x_i,y_j,z_k)$, which I need to downsample by a factor of two in the z direction. I want to do lowpass ...
5
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2answers
2k views

Why should an image be blurred using a Gaussian Kernel before downsampling?

I recently read that before downsampling an image, it should be blurred using a Gaussian Kernel. This way, the downsampled image is better than just picking a single pixel out of a NxN block or ...
5
votes
1answer
1k views

Common Use Cases for 2D Non Separable Convolution Filters

In the image processing world, I've noticed that a lot of the popular convolution filters are separable. Here's a quick list of common separable filters: Sobel Gaussian blur Box filter (all ones, for ...
5
votes
2answers
4k views

Standard Deviation in Gaussian Blur

I have a function that performs gaussian blur on image for some specific $\sigma$ (the standard deviation). It first computes kernel of size $\lceil 3\sigma \rceil$ and then performs convolution with ...
5
votes
1answer
110 views

Is there a transformation filter to decode light signal through Glass Bricks?

A friend is doing renovations and getting his entire ground-floor street-side wall replaced with glass bricks. I told him that with all those bricks giving similar distorted views of the same room, ...
5
votes
1answer
1k views

Convolution of Two Kernels [OpenCV]

I've been completely stuck on a portion of my assignment for a few days now. After plenty of searching around, I have been unsuccessful in discovering information that leads me to the correct solution....
5
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0answers
162 views

Distribution of the output of AWGN channel

$$\mathbf{Y}=\mathbf{X}+\mathbf{W}$$ $\mathbf{X}$ is input, an $n$-dimension equiprobable vector in a finite subset of complex space $\mathcal{A}^n \subset \mathbb{C}^n$ with norm $\sqrt{\rho n}$ (...
4
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1answer
948 views

How to generate random samples of Gaussian distribution directly in the frequency domain?

One can easily draw (pseudo-)random samples from a normal (Gaussian) distribution by using, say, NumPy: ...
4
votes
1answer
207 views

Zero padding effect on a FFT of gaussian noise

I have a gaussian noise $\nu(t)$ with variance $\sigma^2$. After a FFT I get $X(\omega)$. If now I do the IFFT on the $X(\omega)$ can I say that the result is still a gaussian noise of variance $\...
4
votes
1answer
2k views

Difference between Gaussian and moving average filters for peak detection and doppler shift detection?

Consider having a signal in the time domain, and you want to smooth the signal. Moving average and Gaussian filters that are used. How do you choose which is used for what? What are the conditions ...
4
votes
2answers
4k views

Gaussian Filter Close to Image Border

When applying Gaussian filters close to the borders of an image, algorithms usually zero-pad or replicate/mirror/wrap the borders. This is not good enough for my case, so I wonder if there is ...
4
votes
1answer
288 views

Shannon capacity with distribution different from Gaussian

If I understand correctly, in AWGN channel, for a given SNR, there is a code for $M$ input codeword $\mathbf{X}_{i, 1 \leq i \leq M}$ that $$\lim_{n \to \infty} R=\lim_{n \to \infty}\frac{\log(M)}{n} ...
4
votes
1answer
682 views

Relation between frequency spectrum and PDF of a random variable

I have a random variable that is being generated according to some probability distribution function (e.g. a Gaussian PDF). When looking at the frequency spectrum of the generated data does the ...
3
votes
3answers
2k views

Gaussian random generator

I have quite a straight-forward question. What I aim for is the generation of a certain set of random numbers with a normal distribution (mu = 0, ...
3
votes
1answer
7k views

What is use of a gaussian filter over histogram?

I am working in the field of digital image processing and currently i am reading this paper and I have two questions about it.... After applying a gaussian filter to a histogram, will the pixel ...
3
votes
2answers
3k views

When should the sum of all elements of a gaussian kernel be zero?

I found an approximation of a 5x5 2D convolution kernel like this : Here, the sum of the elements is zero and this one was used for Laplacian of Gaussian! Another one here : This one has all ...
3
votes
1answer
811 views

Determine Parameters of gaussian distribution to fit a certain Image-Filter

I'd like to know what the parameters for the two-dimensional gaussian distributions are, which create this (I think) quite common image-processing filter: $$ h_\sigma(x,y)= \left[ \matrix { 1&2&...
3
votes
2answers
192 views

STFT calculation with Gaussian Window

$$ f(t)=\exp(jat^2) \,\,\, and \,\,\, g(t)\,\,is\,\, a\,\, Gaussian\,\, Window:$$ $$ g(t)= \left (πσ^2\right)^{\frac{-1}{4}}\exp\left (\frac{-t^2}{2σ^2} \right ) , \,\,\,\,\,\,\left \|g(t) \right \|...
3
votes
1answer
73 views

Higher-order moment of output of LTI system

Assume a very simple LTI system. Assume $x$ is white Gaussian i.i.d. with variance $\sigma^2$. The output variance is straightforward to obtain. For example, for a continuous-time system: $$\mbox{...
3
votes
1answer
14k views

Fit data to Gaussian distribution

I want some data to fit the corresponding Gaussian distribution. The data is meant to be Gaussian already, but for some filtering reasons, they will not perfectly match the prescribed and expected ...
3
votes
1answer
549 views

Discrepancy between Gaussian FFT and its Fourier Transform

I am trying to do the FFT of a Gaussian signal and comparing it to the theoretical Fourier transform. For infinitely small time step $dt$ and infinitely long signal length $T$, the 2 should become ...
3
votes
1answer
118 views

Does delta-sigma ADC also reduce Gaussian noise on input signal to ADC or just quantization noise?

The motive for posing this questions arises from a difference of analysis between a colleague an myself. Our general environment is in the construction of an analog front-end which takes in signals ...
3
votes
1answer
135 views

Is $A\cos(\omega t+\theta)$ a Gaussian random process?

$Z(t) = A\cos(\omega t+\theta)$ where $A$~$N(0,\sigma ^2) $ and $\theta $~$(0,2\pi)$ are independent. I'm trying to figure out if $Z(t)$ is a Gaussian random process and whether it is strict sense ...
3
votes
1answer
608 views

Poles of an analog Gaussian filter

I am trying to calculate the poles of an analog Gaussian filter. Its characteristic function, $e^{-\log(2)x^2}$, can be expanded into MacLaurin series: $$2^{-x^2} = e^{-\log(2)x^2} = \lim_{N \to \...
3
votes
2answers
202 views

Equivalent of Fourier transform for Gaussians?

I have a long signal (million of samples) containing a lot of Gaussian peaks, whose standard deviation is random and about $5$ to $50$ samples wide. Sometimes, these peaks overlap, but not often. The ...
3
votes
2answers
134 views

If noise is your signal, what is your noise?

Consider the following contrived situation. Imagine a Gaussian white noise process $x[t]$, with bandwidth $Δf$, with PSD equal to some quantity $A$ which you would like to measure. So the way to ...
3
votes
1answer
108 views

Resources on Non-Gaussian Channels

I understand why most of the classic textbooks only consider Additive White Gaussian Noise channels in digital data transmission. Unfortunately in my case noise is not Gaussian and it might even be ...
3
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0answers
137 views

Compensate for standard deviation loss

I am not sure if this question will be a little off-topic on this forum, that I will give it a try anyway, since it implies signal process arguments. By using ...
2
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5answers
907 views

Why does the Kalman filter remove only Gaussian noise?

What and where in the derivation of the Kalman filter is the assumption of Gaussian noise? Why and how does this assumption help?
2
votes
2answers
266 views

Situations where complexity is too big to exceed linearity and gaussianity

I'm studing about Kalman Filter and Particle Filter in multiple target tracking in computer vision (tracking pedestrians). Reading sientific papers I'm colliding with a lot of sentences like: The ...
2
votes
3answers
1k views

On coloured Gaussian noise

It is known that the PSD of additive white Gaussian noise (AWGN) is constant and equal to its variance. What about coloured Gaussian noise (CGN)? For example, given the following PSD of CGN $$S(f) ...
2
votes
3answers
4k views

Which sigma to use for edge detection

I know that in many edge detection algorithm the first step is choosing a scale to work on. We actually operate on the chosen scale by applying a gaussian blur on the original image. I wanted to ...
2
votes
2answers
306 views

PSD of complex white gaussian noise

It may be a really simple question, but I'm not sure about this one: Given a complex white Gaussian noise process with iid real and imaginary parts and a double sided power spectral density of $N_0/2$...
2
votes
2answers
134 views

Conditions for symmetric and unimodal windows in both time and frequency domains

After a lecture on harmonic analysis and time/frequency methods, I reconsidered the Gaussian kernel, defined in continuous time. It is unimodal and symmetric, and its continuous Fourier transform is ...
2
votes
1answer
397 views

Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary process in time (in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary ...
2
votes
1answer
5k views

Determine the optimum receiver and the corresponding $P_{eM}$ for an AWGN channel

I have a source that emits $M$ equiprobable messages, which are assigned signals $s_1, \dots,s_M,$ that are equidistant by $a$. That is, if we plot the $s_k$ signals in a horizontal axis they are dots ...