Chris Angove, Independent Professional Engineer

Chris Angove is a highly experienced and MSc qualified chartered electronics engineer specialising in electrical and electronics engineering. He manages and owns Faraday Consultancy Limited (FCL).

Electrical Noise

Of all my jobs supporting clients I estimate that, in well over 50% of them, there have been significant issues involving electrical noise in one or more of its various forms. Many of these were difficult to solve, but one thing is clear. Attention to detail, using sound RF techniques and calling on previous experience with similar problems has brought solutions.

Electrical noise, both natural and man-made, is unwanted and affects all types of electronic equipment because it is either inherent in, or gets coupled to, the components and circuits that have been designed to detect and process the wanted signals. In many cases it is difficult or impossible to filter out. Some examples of natural and man-made noise are described below [1].

Additive White Gaussian Noise

Here we will study additive white Gaussian noise (AWGN), a form of naturally generated electrical noise and arguably the most important. It is also known as Johnson, Nyquist, thermal or simply kTB noise, after perhaps it's most well-known equation. Most of the information included here is from references by Pozar [1] and Sklar [2].

Why is Additive White Gaussian Noise so Important in Communications Engineering?

The Derivation of AWGN from Black Body Radiation

Gaussian Distribution of the Random Noise Voltages

Applying the Auto-correlation Function to AWGN Noise Samples

Why is Additive White Gaussian Noise so Important in Communications Engineering?

Noise comprises unwanted signals which are present in a communications system and are added to the wanted signals which we require to detect. There are many different types of noise but the subject of this discussion is additive white Gaussian noise (AWGN).

AWGN originates from the resistive or dissipative components which form parts of all electronic circuits and is present at all temperatures of the source above absolute zero. Sources include not only pure resistors but the resistive or lossy parts of imperfect reactive components, inductors and capacitors. The mean power level of AWGN is directly proportional to the product of just two variables: the absolute temperature of the AWGN source, T in kelvin (K) and the noise bandwidth of the communications channels which it affects, B in hertz (Hz). Therefore, the only measures we can take to minimise AWGN for the noise critical components are by reducing the bandwidth, reducing the temperature or reducing both of these parameters simultaneously.

The laws of physics tell us that the frequency range of AWGN covers to well in excess of the frequency range of radio equipment that exists with the current technology, to over 100 GHz. AWGN voltage and phase components are random, so it cannot be filtered out and it is therefore always present with the signal. The best signal to noise ratio (SNR) performance with AWGN is achieved by minimising the AWGN power and maximising signal power simultaneously. The influence of AWGN is at its greatest in the small signal or 'front end' stages of a receiver. The AWGN noise performance of communications equipment is closely related to its noise figure.

The Derivation AWGN from Black Body Radiation

The Gaussian nature of AWGN means that its RMS voltage has a Gaussian distribution with a mean value of zero and its mean power is finite for a given bandwidth. The RMS value of the noise voltage, Vn, is given by Planck's black body radiation law, (2-1) [1].

V n = 4hfBR e hf kT 1
Equation 2-1. The RMS AWGN voltage, derived from Planck's black body radiation law.

In (2-1):

Using Matlab® 2020b, (2-1) is plotted in Figure 2-1 for some arbitrary values of resistance (50 Ω), temperature (290 K) and noise bandwidth (1 GHz) and using a rather hypothetical frequency range 10 MHz to 100000 GHz.

Figure 2-1. The relationship between AWGN RMS voltage and frequency for the parameters shown. This is constant at about 29 dBµV to well past 100 GHz (1011 Hz) and the limit of communications equipment functionality using conductive transmission lines with the current technology.

This shows the AWGN RMS noise voltage to be constant at about 29 dBµV to well beyond 1011 Hz or 100 GHz. In fact, it does not start to fall significantly to well past 1000 GHz. In the RF and microwave frequency range, hf<<kT and (2-1) approximates to (2-2), also known as the Raleigh-Jeans formula.[3].

V n = 4kTBR
Equation 2-2. The Rayleigh-Jeans approximation and practical equation for the RMS voltage of AWGN across the RF and microwave frequency range

This equation is sufficiently accurate to use for all communications equipment using the current technology. For very low temperatures and/or very high frequencies Equation 2-1 should be used.

One of the reasons why an AWGN source such as this is known as a 'white noise' source is by analogy of the flat mean power against frequency response in Figure 2-1 with a similar one for white light in the visible optical spectrum comprising the 'colors of the rainbow' with wavelengths from approximately 380 nm to 750 nm. Another reason results from the auto correlation function applied to AWGN noise samples described in Section TBA.

We can show that the mean noise power (Pn) is not a function of resistance by creating a Thevenin equivalent circuit for the AWGN source with a noise-free source resistance R connected to a normal load of resistance R via a perfect bandpass filter (BPF). This will transfer the maximum possible power from the source to the load. This is shown in Figure 2-2. The BPF is necessary to define the applicable bandwidth of AWGN (B in (2-2).

Figure 2-2. A schematic showing a Thevenin constant voltage source for the AWGN source connected for maximum power transfer to a load resistance R.

The mean AWGN noise power in the load resistor (Pn) is therefore given by (2-3).

P n =kTB
Equation 2-3. The mean AWGN noise power as a function of bandwidth (B) and absolute temperature (T)

The bandpass filter in Figure 2-2 has a hypothetical 'brick wall' frequency response: 100% transmission (no insertion loss) in the transmission band f1 to f2 and zero transmission outside of this band.

Gaussian Distribution of the Random Noise Voltages

AWGN is caused by large numbers of independent charges in random motion and the Gaussian property originates from the Central Limit Theorem (4) (5). The voltage generated from each charge may be represented as a voltage vector whose amplitude and phase are also in random motion. The noise voltages will interact with each other according to their time and phase relationships, the latter of which are uniformly distributed between 0 and 2π radians. The resultant noise voltages will have a Gaussian distribution with a mean value of zero.

(3-1) represents the probability density function (PDF) for a Gaussian distribution of an independent random variable x with a mean (μ) and standard deviation (σ). A special case of the PDF is the normalised form of (3-1) with a mean of zero (μ =0) and a standard deviation of one (σ =1). This is shown in (3-2).

p(x)= 1 σ 2π exp 1 2 xμ σ 2
Equation 3-1. The probability density function for a Gaussian (Normal) distribution of an independent variable 'x' of mean value μ=0 and standard deviation σ.

In order to consider any AWGN power, the normalized Gaussian distribution function is a useful starting point. For AWGN, the mean is zero (μ =0) and the standard deviation is 1 (σ =1) for the normalized condition, so (3-1) becomes (3-2).

p(x)= 1 2π exp x 2 2
(3-2) The normalized Gaussian PDF for a mean value of zero (μ=0) and standard deviation of one (σ =1).

(3-2) is plotted in Figure 3-1.

Figure 3-1.The Gaussian (normalised) PDF for the distribution of AWGN noise voltage amplitudes with a mean value μ =0 and standard deviation σ =1. In this normalised case only, the variance is also equal to one (σ2 =1).

As the variance is the square of the standard deviation, these values will both be numerically equal to one (σ2 =σ =1 but just for the normalised case). For a practical, un-normalised example which we will look at shortly, we will also need to account for the units necessary for the variance and the standard deviation. The normalized condition means that the area under the curve is equal to one because the voltage range is from -∞ to +∞, so every voltage must be included within this range.

If we apply this to AWGN, 'x' becomes the spread of independent noise voltages originating from the AWGN charges and also has a mean of zero (μ =0). Furthermore, the (band limited) AWGN mean noise power is equal to the variance of the distribution.

Every practical wireless and cable communication system has some form of bandwidth limit. This is required to conserve frequency spectrum and to limit interference between users and is the coefficient B in (2-3). This must be the noise bandwidth as opposed to the CW or -3 dB bandwidth, as they have different definitions.

To relate the Gaussian PDF to an un-normalized version of AWGN, we may examine the following band-limited example: a noise bandwidth (B) of 100 MHz, an absolute temperature (T) of 290 K and a load resistance (R) of 50 Ω. The mean AWGN power Pn, using (2-3) is 4×10-13 W. It is useful to calculate both the mean square AWGN voltage, VMS, (3-3) and the RMS AWGN voltage, VRMS,(3-4) . These are equivalent to the variance (σ2) and the standard deviation (σ) respectively.

V MS = P n R
(3-3) The mean square voltage of a band limited source of AWGN with mean power Pn and load resistance R. This is equivalent to the variance (σ2) of the associated Gaussian distribution.
V RMS = P n R
(3-4)The root mean square (RMS) voltage of a band limited source of AWGN with mean power Pn and load resistance R. This is equivalent to the standard deviation (σ) of the associated Gaussian distribution.

So, for our example, using (3-3) for the variance and (3-4) for the standard deviation, we have σ2 =2×10-11 V2 and σ =4.4×10-6 V. We can then de-normalise the Gaussian independent variable (x) axis using σ =4.4×10-6 V. However, this will also de-normalize the area under the PDF, which should be 1, so the vertical (probability) scale therefore requires the inverse de-normalisation. The result of this is shown in Figure 3-2.

Figure 3-2 The Gaussian PDF for the example of bandwidth limited AWGN described in Section 2. The x axis has been de-normalised to absolute values of RMS AWGN voltage.

Looking again at the classic Gaussian PDF in Figure 3-2, also known as a normal distribution or bell curve because of its resemblance to the shape of a bell, there are some interesting features. The probability of a noise voltage being between, say, x1 and x2, (p(x1≤x<x2), would be the area under the curve between x1 and x2. (3-5) is the equation for this.

p( x 1 x x 2 )= x 1 x 2 p(x)dx
(3-5)

Applying the Auto-correlation Function to AWGN Noise Samples

Further to the comments in Section 2, another way of understanding use of the term 'white' in additive white Gaussian noise is to apply the Auto correlation Function followed by a Fourier transform to convert the result from the time domain to a frequency spectrum (6).

The individual charges which generate AWGN voltages are in large numbers, random and independent and AWGN is the result of a random process. Then by applying the Auto-correlation function to a sampled AWGN voltage against time waveform, the result is a delta function of the time difference (τ) with a weighting of N0/2 where N0 is the noise power density in watts/hertz (W/Hz). (4-1).

R N (τ)=E N(t)N(t+τ) = N 0 2 δ(τ)
(4-1) The result of the Auto-correlation function applied to a AWGN waveform to give a delta function weighted by N0/2

In (4-1), the weighting of the delta function is divided by 2 to relate to the power on each side of a double sideband frequency spectrum, covering both positive and negative frequencies. For a single sided spectrum the weighting would be N0.

Errors in Quadrature Amplitude Modulation Caused by AWGN

Under preparation, please check back soon.

References

  1. Microwave Engineering - Third Edition
  2. Pozar, David M.; John Wiley & Sons Inc. (2005); pp. 487 - 493; ISBN 0-471-44878-8

  3. Digital Communications, Fundamentals and Applications, Second Edition
  4. Sklar, Bernard; Prentice Hall PTR, Upper Saddle River, New Jersey 07458 (2006); ISBN 0130847887 (2006)

  5. Pozar (op. cit.)pp. 191 - 192
  6. Sklar (op. cit.); pp. 30 - 33.
  7. You Tube® Video; Professor Iain Collings, Central Limit Theorem.
  8. You Tube® Video; Professor Iain Collings, What is White Gaussian Noise (WGN).