Computer Noise Overview

I. Concepts

 

White Noise)

 

Random Noise with the same energy at all frequencies is called white noise. The power spectral density of white noise is evenly distributed throughout the frequency domain. White noise is a random signal or random process with the power spectrum density constant. In other words, the power of this signal in each frequency band is the same, because the white light is composed of the monochrome light of various frequencies (colors, therefore, this flat power spectrum is called "white", and this signal is also called white noise. Relatively, other noise signals that do not have this property are called colored noise.

 

The white noise is used in the frequency field.

 

The ideal white noise has infinite bandwidth, so its energy is infinite, which is impossible in the real world. In fact, we often regard flat signals with limited bandwidth as white noise, which makes it easier for us to perform mathematical analysis. However, white noise is more convenient in mathematical processing, so it is a powerful tool for system analysis. Generally, as long as the spectrum width of a noise process is much greater than the bandwidth of the system to which it is applied, and the spectrum density of the noise process can be considered as a constant, it can be processed as white noise. For example, the thermal and elastic noise have a uniform power spectral density in a wide frequency range, which can be considered as white noise.

 

The probability density function of white noise (sample statistical histogram) is uniformly distributed, so it is also called uniform white noise or random noise.

 

Gaussian Noise) 

 

It is a type of noise in which its probability density function is subject to Gaussian distribution (I .e. normal distribution. The so-called Gauss of Gaussian noise is for the probability density distribution of the amplitude.

 

Gaussian white noise

 

If a noise is subject to Gaussian distribution, and its power spectral density is uniformly distributed, it is called Gaussian white noise. Hot noise and grain noise are Gaussian white noise.

 

In the so-called Gaussian white noise, Gaussian refers to the probability distribution as a normal function, while the white noise refers to its second-order moment not correlated. The first moment is a constant, which refers to the time correlation of successive signals. This is to examine two different aspects of a signal.

 

Random Sequence

 

The concept of random mathematics refers to the uncertain sequence of random events in the physical world. It is a random generation, unpredictable, and non-repeating sequence. A signal composed of a random sequence is called a random noise, that is, a uniform white noise.

 

Pseudo-random sequence

 

If a sequence, on the one hand, can be pre-determined and can be repeatedly produced and replicated, on the other hand, it has the random characteristics of a random sequence (that is, statistical characteristics ), this is called a pseudo-random sequence.

 

A pseudo-random sequence has some random characteristics, but its sequence is definite. They are usually produced by digital computers, but they have the random characteristics of a random sequence. Because it also has a random feature, it is impossible to judge whether it is a real or pseudo-random sequence from the characteristics of an existing sequence. It can only be determined based on the sequence generation method. The pseudo-random series has good randomness and correlation functions close to white noise (Delta functions), and has pre-certainty and repeatability. These features make the pseudo-random sequence widely used, especially in CDMA systems, as a Spread Spectrum Code has become a key issue in CDMA technology.

 

Common pseudo-random sequences include M sequence, gold sequence, and orthogonal sequence.

 

Pseudo-random sequence is the basis for Computer Noise Generation.

 

(Pseudo) random number generator (RNG)

 

An algorithm or method that generates (pseudo) random sequences (usually integer.

 

 

Ii. Noise Generation Method

 

Generate pseudo-random sequence (white noise)

 

The generation of pseudo-random sequences has the following common algorithms:
● Linear coordinality algorithm (MATLAB 4.0 and earlier versions use the <rand function>)
● Shift-Register Method
● Lagged-Fibonacci algorithm (MATLAB 5.0 and later versions use the <rand function>)
● Monte Carlo Method
● Mid-square method, 1940 s
● Mersenne Twister, 1988
● Combined generators (combine the outputs of two or more rngs, eg, using)
◆ The universal generators
Combine 2 generators: lagged Fibonacci, xn + 1 = (Xn-k) mod 1677213
◆ The kiss generator (Keep it simple, stupid)
Combine three simple generators: congruential generator, 3-shift generator, multiply-with-Carry Generator
● Blum-shub (BSS) generators, 1986
● The havege Generator

 

 

Pseudo-random sequence measurement criteria

 

● Fast, especially in simulation fast
● Well distributed (pass all statistical tests known)
● Independent independent
● Portable and reproducible can be repeatedly generated on different computers (for Verifying Simulation Results)
● Long periods (for deterministic rngs) long cycle
● Unpredictable and irreproducible (for cryptography)
● Security confidentiality (for cryptography)
● Large seed spaces (for deterministic rngs) have enough seed options

 

 

Quality Test of pseudo-random sequence

 

Goodness-of-fit test is used to measure the difference between the sample distribution and the expected distribution.
There are several common testing methods:
● Pearson's chi-square test
● The Kolmogorov-smirov (KS) test (most commonly used)
● The Anderson and darling (AD) test

 

 

Statistical Test-the collision test (collision test)
● Knuth's collection Test
● 16 statistical tests recommended by NIST for password security verification
◆ Frequency (monobit) test
◆ Frequency test within a block
◆ Runs Test
◆ Tests for the longest run of ones in a block
◆ Binary matrix rank test
◆ Discrete Fourier Transform (spectral) test
◆ Non-overlapping Template Matching Test
◆ Overlapping Template Matching Test
◆ Maurer's "universal statistical" test
◆ Lampel-Ziv compression test
◆ Linear complexity Test
◆ Serial test
◆ Approximate entropy Test
◆ Cumulative sums (CUSUM) test
◆ Random excursions Test
◆ Random excursions variant Test
● Diehard rngs examining package (most commonly used)
◆ Birthday spacings (one of the three required tests. All other tests can be passed through these three tests)
◆ Gcd (one of the three mandatory tests)
◆ Gorilla gorilla (one of the three mandatory tests)
◆ Overlapping Permutations
◆ Binary rank NN
◆ Binary rank 68
◆ Monkey tests opso, oqso, DNA
◆ Count the 1's
◆ Count the 1's specific
◆ Parking lot
◆ Minimum distance
◆ Random spheres
◆ The squeeze
◆ Overlapping sums
◆ Runs up and down
◆ The craps

 

 

Convert the pseudo-random number sequence to another Distribution

 

The output of an RNG is a random integer with a uniform distribution (for example, [0, 2 ^ 32-1. However, in practice, we often use random numbers with other distributions, such:
● Uniform in [0, 1) (even distribution)
● Normal (Normal Distribution) (Gaussian noise)
● Exponential (exponential distribution)
● Gamma (gamma distribution)
● Poisson
● Binomial (Binary distribution)

 

 

This requires conversion. Common conversion methods include:
● Density distribution function (Distributed Density Function Method)
● The acceptance-rejection method (accept-reject method)
● The Monty Python method (patchwork)
● The Ziggurat method (FAST)
● The alias method for generating discrete variates
● The straightforward table look up Method)

 

 

 

Produce Pseudo-Random Sequences with normal distribution (Gaussian white noise)

 

According to the content in the previous section, the emergence of Gaussian white noise is usually divided into two steps: first, the uniform distribution of white noise is generated, and then the Gaussian white noise is obtained through the uniform distribution of white noise.

 

There are currently several common methods:
● Box-Muller method (MATLAB 4.0 and earlier versions use the <randn function>)
● Ziggurat method (<randn function> is used in MATLAB 5.0 and later versions)

 

(To be continued ......)