# Signals and Systems/Noise

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Noise is an unfortunate phenomenon that is the greatest single enemy of an electrical engineer. Without noise, digital communication rates would increase almost to infinity.

## White Noise (Gaussian Noise)

White Noise, or Gaussian Noise is called white because it affects all the frequency components of a signal equally. This noise can be modeled as a Gaussian noise process. Gaussian processes are stochastic processes for which the random variables are jointly Gaussian. We don't talk about Frequency Domain analysis till a later chapter, but it is important to know this terminology now.

## Colored Noise

Colored noise is different from white noise in that it affects different frequency components differently. For example, Pink Noise is random noise with an equal amount of power in each frequency octave band.

## White Noise and Autocorrelation

White Noise is completely random, so it would make intuitive sense to think that White Noise has zero autocorrelation. As the noise signal is time shifted, there is no correlation between the values. In fact, there is no correlation at all until the point where t = 0, and the noise signal perfectly overlaps itself. At this point, the correlation spikes upward. In other words, the autocorrelation of noise is an Impulse Function centered at the point t = 0.

${\displaystyle {\mathcal {C}}[n(t),n(t)]=\delta (t)}$

Where n(t) is the noise signal.

## Noise Power

Noise signals have a certain amount of energy associated with them. The more energy and transmitted power that a noise signal has, the more interference the noise can cause in a transmitted data signal. We will talk more about the power associated with noise in later chapters.

## Thermal Noise

Thermal noise is a fact of life for electronics. As components heat up, the resistance of resistors change, and even the capacitance and inductance of energy storage elements can be affected. This change amounts to noise in the circuit output. In this chapter, we will study the effects of thermal noise.

The thermal noise or white noise or Johnson noise is the random noise which is generated in a resistor or the resistive component of a complex impedance due to rapid and random motion of the molecules, atoms and electrons. According to the kinetic theory of thermodynamics, the temperature of a particle denotes its internal kinetic energy. This means that the temperature of a body expresses the rms value of the velocity of motion of the particles in a body. As per this kinetic theory, the kinetic energy of these particles becomes approximately zero (i.e. zero velocity) at absolute zero. Therefore, the noise power produced in a resistor is proportional to its absolute temperature. Also the noise power is proportional to the bandwidth over which the noise is measured. Therefore the expression for maximum noise power output of a resistor may be given as:

${\displaystyle Pn=k\cdot T\cdot B}$

where

k is Boltzmann's constant
T is the absolute temperature, in Kelvin degrees
B is the bandwidth of interest, in Hertz.

Disclaimer: The above section contains significant errors. The noise power equation is wrong. The physical reasons given for Johnson noise are wrong. Do not use this source of information.