19 votes
Accepted

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

An image "should not be blurred using a Gaussian Kernel" in general. This however can be a safe bet for a lot of basic image processing needs, and a smoothing is almost mandatory when you ...
Laurent Duval's user avatar
14 votes

Why is Gaussian noise called so?

Noise is random, but like most random phenomena, it follows a certain pattern. Different patterns are given different names. Consider rolling a die. This is clearly random. Roll the die 1000 times, ...
MBaz's user avatar
  • 15.3k
14 votes

Capacity of AWGN channel

Assuming a channel whose input at each time is a continuous random variable $X$ and its output is $Y=X+Z$, where $Z\sim\mathcal{N}(0,N)$ and $Z$ is independent of $X$, then $$C_{\text{CI-AWGN}}=\frac{...
msm's user avatar
  • 4,285
12 votes

Why is random noise assumed to be normally distributed?

Starting at an even more basic level than the other (much smarter) answers, I'd like to pick up on this part of the question: This seems contradictory to me as on one side it is random then on the ...
gidds's user avatar
  • 221
10 votes
Accepted

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

Assume you have a $N\times M$ sized image. If you know take what is classically used, a square filter kernel, of let's say size $L\times L$, you'd need to convolve that with the picture – which gives ...
Marcus Müller's user avatar
9 votes
Accepted

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

The $BT$ product is the bandwidth-symbol time product where $B$ is the $-3\textrm{ dB}$(half-power) bandwidth of the pulse/filter and $T$ is the symbol duration. For different applications you will ...
Gilles's user avatar
  • 3,386
9 votes
Accepted

Correct way to add AWGN to a signal

Short answer 10*log(bw/fs) to take into account the oversampling operation because the awgn() function specifies the signal-to-...
AlexTP's user avatar
  • 6,595
9 votes

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

According to (digital) sampling theorem, signals should be properly bandlimited, before they are (down) sampled. A practical digital filter approximately limits the bandwidth of the signal and makes ...
Fat32's user avatar
  • 28.2k
8 votes

Why is random noise assumed to be normally distributed?

normal distribution (i like to call it "gaussian") remains normal after addition of normally distributed numbers. so if gaussian goes into an LTI filter, a gaussian distribution comes out. but because ...
robert bristow-johnson's user avatar
8 votes

PSD of complex white gaussian noise

With reference to $N_o$ this usually is the symbol for the power spectral density (PSD) of thermal noise, where $N_o = kT$, where k is Boltzmann's Constant and T is the temperature in Kelvin. With ...
Dan Boschen's user avatar
  • 51.4k
8 votes
Accepted

How to approximate gaussian kernel for image blur

The continuous Gaussian, whatever its dimension (1D, 2D), is a very important function in signal and image processing. As most data is discrete, and filtering can be costly, it has been and still is, ...
Laurent Duval's user avatar
7 votes

Gaussian Blur - Standard Deviation, Radius and Kernel Size

It turns out that the rows of Pascal's Triangle approximate a Gaussian quite nicely and have the practical advantage of having integer values whose sum is a power of 2 (we can store these values ...
wcochran's user avatar
  • 243
7 votes

Capacity of AWGN channel

The capacity formula $$C = 0.5 \log (1+\frac{S}{N}) \tag{1}$$ is for discrete time channel. Assuming you have a sequence of data $\left\lbrace a_n \right\rbrace$ to send out, you need an orthonormal ...
AlexTP's user avatar
  • 6,595
7 votes
Accepted

The Distribution of Filtered Gaussian White Noise

If you filter a Gaussian random process with an LTI system, the output will also be Gaussian. You can make intuitive sense of this by considering that a linear combination (which is what filtering ...
Matt L.'s user avatar
  • 90k
7 votes

Why is random noise assumed to be normally distributed?

I'll try to clear one possible source of confusion. If picking each sample value from a single distribution feels "not random enough", then let's try to make things "more random" by adding another ...
Olli Niemitalo's user avatar
7 votes
Accepted

The Effect of the Finite Radius of Gaussian Kernel

If 3𝜎 is enough for ignoring, why surely select ⎡6𝜎⎤ × ⎡6𝜎⎤ for size of kernel, and not ⎡3𝜎⎤ × ⎡3𝜎⎤ or ⎡4𝜎⎤ × ⎡4𝜎⎤? It is 3𝜎 on either side of the middle, so the total width of the kernel is ...
Cris Luengo's user avatar
  • 2,519
6 votes

Is the discrete Gaussian kernel an eigenfunction of the DFT?

This answer complements @CedronDawg's answer which introduced this family of eigenvectors. More specifically, this answer presents three algorithms and a hybrid algorithm for generating for a given ...
Olli Niemitalo's user avatar
6 votes

Is the discrete Gaussian kernel an eigenfunction of the DFT?

I have made a tremendous amount of progress on this issue in the last few weeks. The Zeroing Sine Family of Window Functions I have discovered a previously unrecognized class of window functions. ...
Cedron Dawg's user avatar
  • 7,560
6 votes

How Is Laplacian of Gaussian (LoG) Implemented as Four 1D Convolutions?

There are two ways to compute the Laplace of Gaussian operator: As Royi suggests, by computing $f * \nabla^2 * g$,where we take the operator $\nabla^2$ as a convolution kernel created using the ...
Cris Luengo's user avatar
  • 2,519
6 votes
Accepted

Gaussian filter: The best parameters for an implementation

The effect is in sidelobe level and is evaluated with consideration to number of samples per symbol used, waveform quality and the spectral mask. Increasing the length of the filter in total number of ...
Dan Boschen's user avatar
  • 51.4k
6 votes
Accepted

Choosing sigma values for Gaussian blurring on an anisotropic image

Answer: neither. The first approach is nonsense. Convince yourself of that by considering padding of arbitrary size. That'd affect the kernel when it should not because you considered the absolute ...
Christoph Rackwitz's user avatar
5 votes
Accepted

generating white gaussian noise in matlab using two different functions

A white noise sequence is one for which each (random) element is uncorrelated from every other element: $$ E[y[n]y[m]] = \left \{ \begin{array}{ll} 0 & \mbox{for } n\not=m\\ \sigma_y^2 & \mbox{...
Peter K.'s user avatar
  • 25.7k
5 votes
Accepted

Poles of an analog Gaussian filter

First of all, computing the poles of an (ideal) Gaussian filter is an impossible task, because its transfer function is not a rational function, and there are simply no poles. This is in contrast to ...
Matt L.'s user avatar
  • 90k
5 votes
Accepted

Value of power spectral density $N_0$ or effect of scaling bandwidth to SNR

The noise power continues to be $N_0/2$, independent of bandwidth. The reason is that the noise variance at the output of a filter with frequency response $H(f)$ is $$\sigma_n^2=\int_{-\infty}^\infty \...
MBaz's user avatar
  • 15.3k
5 votes

Why Does the Kalman Filter Remove Only Gaussian Noise?

First of all let us assure that a Kalman filter (estimator) does not only remove Gaussian noise, but can remove (with certain success) any other type of noise as long as it's designed accordingly. ...
Fat32's user avatar
  • 28.2k
5 votes
Accepted

On coloured Gaussian noise

Colored Gaussian noise is by definition a wide-sense-stationary (WSS) process; that is, it has constant mean (all the random variables constituting the process have the same mean) and its ...
Dilip Sarwate's user avatar
5 votes
Accepted

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

You can, but... you'll need to keep symmetry if your original time-domain signal is real-valued. If a signal $x$ is real-valued, then its DFT $X$ will exhibit complex-conjugate symmetry: $$ X[k] = X^*...
Peter K.'s user avatar
  • 25.7k

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