All electronic systems are subject to noise. In general, noise is the portion of a signal that is undesirable, which means that the perspective or context is important for determining what noise is. For example, 60 Hz AC power is desirable from the perspective of the power engineer, but it is often a source of noise for electronic components that are subject to EMI, and 60 Hz power signals can show up as ‘noise’ even though the 60 Hz is not a random signal. In a more mathematical definition, noise is any signal that is described using probability and statistics, as opposed to deterministic differential equations that are used to describe sinusoids.
Statistical signals, which will be referred to as noise for the purposes of this document, are characterized by a probability distribution. In all of probability theory, the Gaussian (or normal) distribution is the simplest to handle because the mean, median, and mode are all equal. In many derivations of formulas involving Gaussian noise, a zero mean is assumed, which makes the math much easier. Gaussian noise is also referred to as white noise because the frequency spectrum is flat, so it is analogous with white light, which contains all frequencies /wavelengths.
Thermal noise tends to be very close to Gaussian noise in terms of its frequency distribution, and it cannot be avoided because it is present in any physical system that is above absolute zero (which is all of them). Due to this, there will always be some minimum amount of electronic noise due to thermal motion, and this is referred to as the noise floor. Again, not all types of noise are the same, but thermal noise is always present and is responsible for the lowest possible energy level in a macroscopic physical system.
The Units of Spectral Noise
In signal processing, we have measures of energy and power that are somewhat different than in physics, but nonetheless meaningful. The “power” of a signal is:
P = ∫x2 dt
Formally speaking there is a limit involved, but this formal definition does not translate directly into discrete-time calculations. The relevance of this equation is that squaring the signal and integration determine the units of the power. Recall from calculus that the derivative of position, which has units of meters, results in velocity, which has units of meters per second. Integration restores the velocity back, which means there is a change of units with integration just as there is with differentiation.
Because the derivative is taken with respect to time, the initial unit of meters is converted to meters per second, which means the initial unit was divided by time. With integration the inverse happens, which means the unit gets multiplied by time. The units for frequency are generally referred to as Hertz (Hz), but this is just the reciprocal of seconds, which means that multiplication by seconds is equivalent to division by Hz.
Now we can come back to the integral equation that measures the power of a signal. If the signal is measured in Volts, squaring the signal transforms the units to Volts squared, and integrating that transforms the units further to Volts squared multiplied by seconds, which is equivalent to Volts squared per Hz. The only difference between this and the conventional units used to express the noise in terms of power is taking the square root of the final quantity. Just as with root-mean-square (RMS) calculations, the square root is not essential to the process, but it is a convention that is used to scale the result. Applying the square root to the unit of Volts squared per Hz results in Volts per square root Hz:V/√Hz
Volts → Volts2 → Volts2 * seconds = Volts2/ Hz → Volts/√Hz
The arrows represent operations, and the one equality is shown for the conversion between seconds and frequency. The first arrow is the squaring operation, the second arrow is the integral operation, and the final arrow is the square root operation.
The next figure was taken from EDM-DSA, using a 50 ohm terminator on channel 1 with vertical axis set to Log Mag and spectrum type set to sqrt(EU^2/Hz). The particular value of the noise displayed by the vertical cursor fluctuates and should not be considered as the maximum noise level, but it illustrates the use of these units in the context of a real measurement.
