Gaussian Noise

Gaussian noise, also known as Gaussian white noise, is a kind of statistical noise characterised by the probability density function of the normal distribution.It became a form of white noise that is a random signal containing frequencies of equal power. It is also used today in artificial intelligence, statistics, computer systems, computer vision, natural language processing and computational biology.

These are some of the types of random noise which has a Gaussian distribution and is also referred to as Gaussian noise. It is defined by the measure of central tendency and dispersion. The mean commonly refers to the middle position of the distribution while the variance shows the extent of the distribution. It will also be observed that the actual values of the items in a distribution of Gaussian form are symmetric about the mean value and most of the items are concentrated in the immediate vicinity of this value.

Synthetic Gaussian Noise

The noise that is synthesized or produced using mathematical equations or formulae. It’s common when doing simulations or testing on how well a certain system or algorithm is going to hold up.

Gaussian Noise for Signal Processing

This is noise that is intentionally added to signals to mimic real signal in real systems, which has variations and interferences. It is popular in digital signal processing where it is used for testing the efficiency of noise elimination functions or the stability of the processing schemes.

Gaussian Noise for Machine Learning

This is the noise that is introduced in the data to make machine learning algorithms perform better. It is frequently employed in data augmentation to enhance the range of variation of the data and as a mend for overfitting.

Convolutional Neural Network (CNN) Model for Gaussian Noise Detection

This is another categorization of Gaussian noise that is characterized by deep convolutional neural networks to be able to detect the Gaussian noise and its density in an image.

The foundation of the working principle of Gaussian noise is fanatical and is defined by the characteristics of the Gaussian probability density function. Gaussian noise is added to a system or a signal as an error or disturbance in a given communication system. This noise follows the Gaussian distribution, and it implies that most of the time, the amplitude of the disturbance is small most of the time, since most frequency’s values are closely located to the mean, but in some instances, the amplitude of the disturbance can be big. This is due the fact that many harmonic progressions contain the coefficients of Gaussian distribution which are infinitely extended to left and right than many of the distributions.

Gaussian noise has two main components commonly referred to as the mean and the variance of the noise.

• Mean : Average of Gaussian noise gives the idea about the long term Gaussian noise. It pointed to the fact on average the noise distribution will lie around this value. A mean of zero simply means that in the average sense the noise on the upside and downside is equal.
• Variance : The measure of spread in Gaussian noise is indicated by the term variance and denotes how off the values of the noise are to or from the statistical mean. It measures the amount of growth or spread of the noise distribution. A higher variance means that the representation of noise values will be spread out from the mean while a low variance will mean that the representation is close to the mean.

Gaussian noise is developed by using random numbers that are Gauss distributed or often referred to as normally distributed. To create Gaussian noise different techniques can be applied, such as Box-Muller transform and Marsaglia polar.

• Box-Muller Transform : The Box-Muller transform is widely known as one of the powerful techniques to generate the Gaussian random samples. It entail using two series of random numbers that are equally distributed in the range 0 to 1 and then converting the two into two independent Gaussian variables. In the transformation process trigonometric functions are used to transform the uniform random numbers into Gaussian samples. Although this proposal of the Box-Muller transform is easy to perform, it is costly in terms of computational complexity because of the trigonometric functions.
• Marsaglia Polar Method : The Marsaglia polar method is another way of generating from N(0,1) distribution other than using the Box-Muller transform, which requires trigonometric functions. From these random numbers, it also produces two new independent Gaussian random variables also. The method scales the random numbers from a different interval; for example [-1, 1] to generate the Gaussian samples. Besides, the Marsaglia polar method is computationally cheaper if compared to the Box-Muller transform.

These methods accompanied by the other methods are commonly used to produce Gaussian noise in fields like signal processing, image processing and simulation. Randomness is added to the signals or data by adding Gaussian noise in many practical applications. It is also beneficial in the realisation of signal processing techniques in different applications.

• Gaussian Distribution: It is a probability distribution for a continuous variable that can take on any real value. The probability density function of the general form is also equal on both sides and the graph of the function in the form of the bell curve.
• Variance : It is a measure of the variation in the difference between the two values of the numbers in a particular data set. In other words, variance quantifies the degree of dispersion of each value in the given set from the mean and, therefore, from each member of the set.
• Mean : A typical example of an arithmetic mean of a set of figures obtained after the establishment of the sum total of the figures and division of the total with the number of figures.

A real-life use of Gaussian noise is described in image processing, where the type of noise called Gaussian is used to represent noise in captured photographs. Another example can be seen in the application of the field of audio processing where Gaussian noise is used to represent the background noises or static .

As with any other type of noise, Gaussian noise is characterised by its probability distribution, and it is different from, for example, uniform noise. While Gaussian noise is normally distributed, the uniform noise is uniformly distributed; this is, each value has the same probability of occurrence.

1. Gaussian noise is relatively amicable as it is very easy to work with mathematically.
2. It is a good model for many different actual random processes.
3. Most general types of noise are modelled as additive noise where the signal is contaminated by Gaussian noise. This makes it possible to add the noise to the original signal in order to obtain the noisy signal and this is very convenient for analysing and processing the noisy signal.
4. Gaussian models are easy to deal with and they meet several conditions: The first and second order moments are sufficient to define the distribution; when viewed from any orientation of the coordinate axes the bell-shaped curve appears to be the same.
5. Gaussian noise is used as additive white noise to generate additive white Gaussian noise, making it a crucial component in the analysis and design of communication systems.

1. Gaussian noise is not suitable for many of the actual signals that we use in practice. Most of the signals have non-Gaussian nature.
2. Gaussian noise is also called noise with unlimited variability since it has no upper and lower bounds; its relative values can range from negative infinity to positive infinity. For a huge variety of real signals, it is impossible because they are always bounded, or to be more precise, they always have minimum and maximum values.
3. Nevertheless, Gaussian noise is also undesired since it can complicate the signal processing and often hides the signal of interest.
4. Mainly, Gaussian noise may become an issue in such systems as a different number of outliers can be encountered due to the fact that the tails of the Gaussian distribution are infinite.
5. It may not accurately model the noise characteristics of certain real-world signals. Many signals have non-Gaussian distributions, and the assumption of Gaussian noise may not accurately represent the true nature of the noise in these signals.

1. Signal Processing: In signal processing, Gaussian noise is used to model the noise which is existing in the signals.
2. Image Processing: In the image processing, Gaussian noise is incorporated to show the presence of noise when photographs are taken.
3. Audio Processing: Gaussian noise as its name suggests is used to formulate background noise or static.
4. Telecommunications: Particularly, Gaussian noise models are applied in assessing and designing communication systems.
5. Artificial Intelligence: Different type of filters can be used in artificial intelligence for instance in Gaussian processes for machine learning.
6. Statistics: In statistical models the error term is depicted by the use of Gaussian noise models.
7. Data Science: Added randomness in models is incorporated in data science using what is known as the Gaussian noise models.
8. Astronomy: Gaussian processes has been applied in using probabilistic models for astronomical time series .
9. Molecular Biology: Based on the current description and classification, Gaussian processes are used as predictors of molecular properties.
10. Speech Recognition: There are GMMs which are based on Gaussian distribution and have been used for extracting features from spoken data for use in the speech recognition systems .

The basic of Gaussian noise lies in the Gaussian distribution which is a powerful tool in statistic, signal processing and machine learning and other fields. Although it is highly simple, it holds a very strong position when it comes to explaining the element of risk in real-life environments.

What is the difference between Gaussian noise and white noise.

This type of noise is a stochastic noise since the value or the signal level of the noise is generated from a Gaussian distribution. On the other hand white noise is that class of noise whose signal levels are equivalent to the intensity of frequency.

Why does probability and statistics utilise Gaussian noise?

Gaussian noise is also common in probability and statistics due to the Central Limit Theorem, which explains that no matter the inputs’ distribution, the many sums of them will asymptotically look like Gaussian. This means that adding Gaussian noise is beneficial for many complex systems and real-world processes, which is why this approach is frequently used.

Can you explain, in general, the process of removing Gaussian noise from the signal?

It is important to state that Gaussian noise can be removed or its effect in a signal can be minimised by employing different signal processing techniques. These comprise the methods of filtering like the use of the low-pass filter and the more complex methods like the adaptive noise cancellation and wavelet transform methods.