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NOISE

Module by: Nguyen Huu Phuong

NOISE

All the unwelcome signals of random nature superimposing on our signal carrying information are named together as noise. In the electronic devices and circuits noise is generated due to the random motion of electrons (nonuniform speed, collisions…). This is thermal noise. The active electronic devices, besides thermal noise, also generate shot noise. Some phensmena, such as thunder and closure of electric switches, cause impulsive noise (high amplitudes but in burts). The sun generates both thermal and impulsive noises. Noise can be classified as internal (or system) noise, and external noise or interference.
There is a special noise, rather an interference, that we should not forget, that is the 50Hz/60Hz radiation from electric powerline. This noise induces onto our bodies, and electric circuits by way of electromagnetic wave. The power supply for a circuit is another source of 50Hz/60Hz interference.
As for frequency characteristic, one distinguishes white noise, pink noise… The most convenient to model, also the most frequently mentioned is white noise which has the power spectral density S(f) unchanged with frequency f. Figure 1 shows S(f) has a fixed value of NoNo size 12{T rSub { size 8{s} } } {}/2. When the white noise passes through a filter, the outpul noise will be no longer white due to the frequency characteristic of the filter.
Figure 1: The power spectral density of white noise
Preceding is the dependence of noise on frequency. There remains another aspect of noise even more important, i.e. the probability of occurence at difference noise amplitudes. For this the amplitude of noise is a random variable, denoted x. This random variable is judged by its probability density function (PDF), or just probality density f(x), and probability distribution function, or cumulative distribution function (CDF) F(x). These two functions are illustrated through the two popular distributions: Uniform and Guassian.
For the uniform distribution we have
f ( x ) = 1, − 1 2 ≤ x ≤ 1 2 0, otherwise f ( x ) = 1, − 1 2 ≤ x ≤ 1 2 0, otherwise alignl { stack { size 12{f \( x \) =1, matrix { {} # {} # {} } { { - 1} over {2} } <= x <= { {1} over {2} } } {} # size 12{ matrix { {} # {} # {} # 0, matrix { {} # {} # {} } {} } ital "otherwise"} {} } } {} (1)
F ( x ) = ∫ − ∞ x f x dx = ∫ − 1 / 2 x f x dx F ( x ) = ∫ − ∞ x f x dx = ∫ − 1 / 2 x f x dx size 12{F \( x \) = Int rSub { size 8{ - infinity } } rSup { size 8{x} } {f left (x right ) bold italic"dx"= Int rSub { size 8{ - {1} slash {2} } } rSup { size 8{x} } {f left (x right ) bold italic"dx"} } } {} (2)
These are shown in Figure 2.
If the limits are a and b instead of -1/2 and 1/2, then
f x = 1 b − a a ≤ x ≤ b f x = 1 b − a a ≤ x ≤ b size 12{f left (x right )= { {1} over {b - a} } matrix { {} # {} # {} } a <= x} {} (3)
Figure 2: Uniform distribution
The mean and the variance of the distribution are respectively
μ=a+b2μ=a+b2 size 12{μ= { {a+b} over {2} } } {}(4)
σ2=b−a212σ2=b−a212 size 12{σ rSup { size 8{2} } = { { left (b - a right ) rSup { size 8{2} } } over {"12"} } } {}(5)
In reality, many random variables has gaussian distribution or near to it, white noise is an example. Its PDF and CDF are respectively
fx=12πσe−x2/2σ2fx=12πσe−x2/2σ2 size 12{f left (x right )= { {1} over { sqrt {2π} σ} } e rSup { size 8{ - x rSup { size 6{2} } ital "/2"σ rSup { size 6{2} } } } } {}(6)
Fx=∫−∞xfxdxFx=∫−∞xfxdx size 12{F left (x right )= Int rSub { size 8{ - infinity } } rSup { size 8{x} } {f left (x right ) bold italic"dx"} } {}(7)
Figure 3: Gaussian distribution having zero mean and variance σ2σ2 size 12{σ rSup { size 8{2} } } {}
Where σ2σ2 size 12{σ rSup { size 8{2} } } {} is the variance ( σσ size 12{σ} {} is the standard deviation). The distribution has a shape of a ringing bell (Figure 3). Peak probability (at x = 0) is
fP=12πσe−0/2σ2=12πσfP=12πσe−0/2σ2=12πσ size 12{f rSub { size 8{P} } = { {1} over { sqrt {2π} σ} } e rSup { size 8{ - 0/2σ rSup { size 6{2} } } } = { {1} over { sqrt {2π} σ} } } {}(8)
and at distance x=±σx=±σ size 12{x= +- σ} {}, the problalslity is
fσ=12πσe−σ2/2σ2=1e.12πσ≈0,60612πσfσ=12πσe−σ2/2σ2=1e.12πσ≈0,60612πσ size 12{f left (σ right )= { {1} over { sqrt {2π} σ} } e rSup { size 8{ - σ rSup { size 6{2} } /2σ rSup { size 6{2} } } } = { {1} over { sqrt {e} } } "." { {1} over { sqrt {2π} σ} } approx 0,"606" { {1} over { sqrt {2π} σ} } } {}(9)
The smaller σσ size 12{σ} {} (hence σ2σ2 size 12{σ rSup { size 8{2} } } {}) the narrower the bell, i.e the more centralized the distribution.
When a DC voltager is corrupted by adding noise, the probability distribution is the gaussian distribution of the noise but the mean is shifted to the new mean μ Figure 4. The mean μ can be positive or negative. The probability distribution becomes
fx=12πσe−x−μ2/2σ2fx=12πσe−x−μ2/2σ2 size 12{f left (x right )= { {1} over { sqrt { bold italic"2π"} σ} } e rSup { size 8{ - left (x - μ right ) rSup { size 6{2} } bold italic"/2σ" rSup { size 6{2} } } } } {}(10)
Figure 4: Gaussian distribution having positive mean μ

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