Noise

Noise
Noise is classified as an electric disturbance that interferes or distorts a signal of information passing through a communication system. There are several classes of noise that are cause by different phenomena in nature (ex. thermal noise (aka Nyquist noise) is caused by the thermal agitation of electrons flowing through a conductor).

Power-Law Noise
Many definitions of noise assume a signal with components at all frequencies, some which are more apparent than others. Most noise falls under the Power-Law noise. The Power-law noise states that the power spectral density (PSD) is proportional to 1/f^β:

Eq. 1) $$ S(f) \propto \frac {1} {f^\beta} $$

White Noise
White Noise has a β = 0, which means that it has a uniformly distributed PSD. Since white noise theoretically has an infinite bandwidth, it can cause interference in all communication systems. The power spectral density for white noise is:

Eq. 2) $$S(\omega) = \frac {N_0} {2}$$

N0 is usually referenced as the input stage of the receiver, where k is the Boltzman constant (1.38*10^-23), and Te is the Equivalent noise temperature of the receiver.

The amount of white noise passed through a system can be defined by the amount of bandwidth the system will need to allocate to transfer a signal. For both a Low-Pass filter and a High-Pass filter, the noise-equivalent bandwidth is defined as: Eq. 3)$$BW_N = \frac {\int_0^\infty |H(f)|^2 df} {H^2(0)} $$

Pink Noise
Pink noise has a β = 1, which means that the frequency components of its PSD decrease as the frequency increases. The rate at which the noise decreases is 10dB/decade, which means that the interference caused by pink noise is negligible at higher friequencies. A type of pink noise is Flicker noise. Flicker noise is a type of electronic noise that is always related to a direct current, therefore, Flicker noise is predominant over lower frequencies. Flicker noise is commonly found in carbon composition resistors.

Brownian Noise
Brownian Noise (also know as Red noise or simply Brown noise) is a physical phenomenon caused by the random movement of particles suspended in a fluid. Brownian noise has a β = 2, which means that the frequency components of its PSD decrease rapidly as the frequency increases. The rate at which the PSD decreases is -20dB/decade, which means that the interference caused by Brownian noise is negligible at higher frequencies. The Power Spectrum is given by:

Eq. 4)$$ S(\omega) = \frac {S_0^2} {\omega^2}$$

Brownian noise is found in coherent optical communications, where the lasers used exhibit instabilities which can be modeled as Brownian motion.

Thermal Noise
Thermal noise is electrical noise arising from the random motion of electrons in a conductor. The PSD of thermal noise is: Eq. 5) $$ S_n(f) = 2kTR$$ where k = Boltzmann Constant, T= temperature, in kelvins, and R is the resistance of the noisy resistor.

The thermal noise over a band $$\Delta f$$ is: Eq. 6) $$\overline{n^2(t)} = (2kTR)(B_W) = (2kTR)(2\Delta f) = 4kTR\Delta f$$ The thermal noise rms voltage at the output of a system is: Eq. 7) $$S_{\text Vo}(\omega) |H(\omega)|^2 S_n(\omega)$$ Eq. 8) $$\sigma _{\text Vout}^2 = \frac {1} {2\pi} \int_{-\infty}^{\infty} S_{\text Vout}(\omega) d\omega$$

Eq. 9) $$\sigma _{\text Vout} = \sqrt{\frac {1} {2\pi} \int_{-\infty}^{\infty} S_{\text Vout}(\omega) d\omega}$$

References:

 * Bak, Per. Tang, Chao. Wiesenfeld, Kurt. "Self-Organized Critically: An explanation of 1/f noise" In Physical Review Letters, Volume 59, Number 4, July 1987.
 * 1) Mazo, J.E.. Shamai Shlomo. "Theory of FM clicks with Brownian Motion Phase Noise." In IEEE Transactions on Communications, Vol. 38, NO. 7, July 1990.
 * 2) Keshner Marvin S. "1/f Noise" In the Proceedings of the IEEE, Vol. 70, NO. 3, March 1982.
 * 3) Lathi, B.P. (1998). Modern Digital and Analog Communication Systems (Third Edition), Oxford University Press, pp. 414