# Noise

Wednesday, November 28th, 2018 - Analog Electronics

These unwanted signals are called either noise or interference, depending on their origin. The term noise comes from the acoustic analogy of unwanted sounds. What is wanted and what is not is, of course, a matter of definition. One man’s signal is another man’s noise. My neighbour’s music (signal) may very well be my noise. In the same way, a radio astronomer’s signal from a distant galaxy may well be the very annoying noise in the receiver of a satellite communication system. Noise is one of the terms often used in practice without sufficient precision. The authors consider it important to distinguish between interference and noise because of the very different measures required to reduce them to acceptable levels.

Interference, as its name imphes, is an unwanted signal originating in a man made source. In theory ehminating the source can always eliminate this totally. In practice, this hardly ever possible! One can not just switch off the nearby TV transmitter because it interferes with one’s measuring equipment. On the other hand, passengers are prohibited to use electronic equipment on board aircraft. Interference between different electronic systems is an increasingly important design consideration. The specifications to be met by various types of equipment are laid down in legislation relating to Electro-Magnetic Compatibility (EMC). Interference may also take place between different parts of the same system or equipment. For example, the power supply may emit signals which interfere with the input stage of an amplifier. Interference is controlled by the good design of circuits, the layout of the printed circuit boards, the method of earthing and screening, etc.

Noise is an unwanted signal generated by natural mechanisms. Clearly, these can not be eliminated, even in theory. However, a good understanding of the various sources of noise can lead to the minimization of the effects of the noise and the optimization of system performance by good circuit design.

## Types Of Noise

### Thermal (Johnson) noise

Thermal noise is generated by the random motion of free electrons in a conductor resulting from thermal agitation. The magnitude of the motion is proportional to the temperature of the conductor. The random motion of the electrons constitutes a random current in the conductor and thus a random noise voltage appears across its terminals. The magnitude of the thermal noise is measured in terms of its average power Pave which is a function of the temperature of the conductor T and the width of the frequency band B included in the measurement.

Pave =4kTB

where k is Boltzmann’s constant, k = 1.3 x 10-23 J/K, Tis the absolute temperature and B is the bandwidth over which the noise is measured.

The rms noise voltage Vn across the terminals of a conductor of resistance R can be found from the average power Pave$V_{n}^{2}/R$ to be

$V_{n}=2\sqrt{kTBR}$

Thermal noise is often described as Gaussian white noise. The term white refers to the distribution of power over the frequency spectrum. This is assumed to be uniform. Just as white Hght contains all the colours in the spectrum to an equal extent, the spectrum of white noise contains all frequencies to an equal extent.

Note that this is theoretically impossible since it would imply that the noise signal contains an infinite amount of energy. There is a frequency limit where the noise power is no longer equally distributed. However, this limit is very much higher than any frequency used in electronics. So, the assumption is safe to use in the case of all electronic systems.

The term Gaussian refers to the distribution of the voltage magnitudes of the noise signal. Imagine that a great many measurements were made of the magnitudes of the noise voltage across a resistor over a substantial period. The range of voltages can be divided into segments and the number of readings in each segment can be plotted as shown in the picture below. This type of plot is called a histogram. If the number of samples is increased and the width of the segments is reduced the envelope of the curve approaches that of a Gaussian or normal distribution. This is shown in the picture below. The magnitude of the Gaussian frequency function (of occurrence of voltage as a function of voltage) f(v) is given by

$f(v)=\frac{e^{-\frac{1}{2}}\left&space;[&space;\frac{v}{V_{rms}}&space;\right&space;]^{2}}{V_{rms}\sqrt{2\pi&space;}}$

where Vrms is the rms value of the noise waveform. The continuous function f(v) is a probability density function which has the dimension 1/V. The area under a given segment of the curve, such as the shaded area in the picture below, is a measure of the probability that the

histogram of thermal noise voltage samples

An illustration of the Gaussian or normal distribution

voltage has a value within the base of the area (here between V1 and V2). It can be shown that 68% of all the voltage samples in Gaussian noise are smaller than Vrms (between +Vrms and -Vrms), 95% are less than 2 Vrms and 99.5% are less than 3 Vrms.

Note that the maximum average noise power Pavemax is delivered from a noisy resistor R to a load when the load also has a resistance R as stated by the maximum power transfer theorem.

$P_{avemax}=\frac{[V_{n}/2]^{2}}{R}=\frac{V_{n}^{2}}{4R}=\frac{4kTBR}{4R}=kTB$

### Shot Noise

The flow of electrons in a conductor is not a smooth regular process. There are statistical fluctuations in the number of electrons arriving at the terminals of a conductor owing to their discrete nature. Although the average number of arriving electrons, and therefore the average current, remains constant. The randomness of the number of arriving electrons introduces a random fluctuation, noise, in the current. This noise is a function of the average current Idc The noise current is expressed in terms of its rms value Isnrms as.

$I_{snrms}=\sqrt{2qBI_{dc}}$

where q is the charge of an electron (1.6 x 10-19 coulomb) and B is the bandwidth over which the noise is measured in Hz. Shot noise, as thermal noise, is white and has a Gaussian distribution of magnitude.

### Galactic Noise

Electromagnetic emissions from distant stars etc. are picked up by the aerials of satellite and other communication systems. These signals are not wanted by the users of these systems and therefore they are termed noise. Different amounts of noise are received from different parts of the sky.

### Flicker, Excess Or 1/f Noise

More noise is measured in resistors and semiconductors than would be expected by calculations of the thermal noise and the shot noise. This noise is called the excess noise. The mechanisms producing this noise are complex and only partly understood (see Fish, 1993). It is known to depend on the construction of devices. Therefore, it can be minimized by the appropriate choice of components. Flicker noise has a higher magnitude at low frequencies. It is sometimes called pink noise, in comparison to the uniformly distributed white noise. The power density varies inversely with frequency, hence its other name 1/f noise.