Noise

Chris Angove has been interested in noise for some years, especially thermal, Nyquist or kTB noise. He is familiar with the various measurement parameters that are related to the effects of noise including noise figure, excess noise, excess noise temperature, standard noise temperature, noise bandwidth, phase noise, AM noise, Y factor and G/T ratio.

In the context of 'RF over optical fiber' communications he has also built up an understanding of Shot and signal spontaneous noise in the context of optical components (couplers, isolators, WDM filters and EDFAs). He has studied several references on this type of noise but would freely admit that there is much more to learn.

1. What is Noise?

Noise is any unwanted signal. In this context it is electrical and the type present in a practical communications system serving to degrade the probability of detection.

There are many types of noise, but lets first look at the type of noise most prevalent in communication systems which has been given several names such as Johnson, Nyquist, kTB or available thermal noise. It results from random voltage variations at a molecular level.

The bandwidth limited noise power PN is given by where: k is Boltzman's constant (1.38 x 10-23 JK-1), T is the absolute temperature (k) and BN is the noise bandwidth.

This type of noise is uniformly distributed across frequency. The noise density in power per unit bandwidth (like in dBm/Hz)is constant. So, for a fixed temperature, the total noise power is proportional to bandwidth as we can see from the above equation. It is always present at the input of any device that has an input. It can be a simple load resistor like a 50 W one which is settled at the ambient temperature or it might be deliberately heated or cooled. As long as we know its temperature accurately we can calculate the noise power.

### Equivalent Noise Bandwidth

This is a fairly straightforward concept but putting it into words is tricky.

A perfect band pass filter (BPF) has a rectangular passband response. Within this passband, all power will be passed, outside of it all power will be rejected. The noise density (typically in dBm/Hz) will be zero outside of the band and a fixed value within it. A practical BPF or any realistic device like an amplifier will have a passband response which is not actually rectangular, but more rounded, the noise density will vary with the actual frequency.

If a Nyquist noise source is applied to such a device,the total noise power passing through it is an aggregate of the noise density across the actual frequency. Its equivalent noise bandwidth is the equivalent rectangular bandwidth that would pass the same total noise. So equivalent noise bandwidth is a totally theoretically concept, totally different from, say, the - 3 db bandwidth.

Probably not explained particularly well but I will read it a few times and improved it in the fullness of time as they say.

### Noise Figure

The noise figure F of a device is the ratio of the signal to noise ratio at the input to the signal to noise ratio at the output. If Nyquist noise source is applied to the input, then where

Si and So are the input and output signal powers of the device
Ni and No are the input and output noise powers of the device
and G is the noise gain of the device

Any arbitrary noise figure measured in this way depends on the temperature of the noise source that is used to do the measurement. Measuring a low noise amplifier on an antenna tower in the UK in winter would no doubt yield a different result for the same device from that measured in a similar location in California. As long as the actual measurement temperature is known, it can be adjusted to a standard temperature for comparing to data sheets, specifications etc. A standard temperature of 290 K has been chosen or about 17°C. I suppose that is a sort of reasonably average temperature. If you live in California of course.

If Nx is the excess noise power added by a noisy device, referred to its output, or assumed to be added at its output. We could have just as easily referred it to its input. Then the noisy device with Nyquist noise applied is equivalent to a totally noise free (theoretical) device with an additional noise source connected to the output and therefore contributing to all subsequent stages. Another way of describing this output noise is: Another way of expressing the noise figure is ### Equivalent Input Noise Temperature

The excess noise part can be considered to come from a noise free amplifier of noise gain G to which is applied noise from a Nyquist noise source running at a temperature known as the equivalent noise temperature of the network Te. Then Substituting for Nx into the alternative expression for noise figure gives ### Y Factor

To perform a Y factor measurement, the noise power outputs from alternate hot and cold sources are applied to the device under test. The sources don't have to be literally hot and cold but at significantly different and accurately known temperatures, so that their Nyquist noise powers can be easily determined. The Y factor is defined as the ratio of the output noise power from the device when the input is fed from a 'hot' load to that when it is fed from a 'cold' load.

The hot and cold output noise powers are and So the Y factor definition gives Re-arranging these gives and Therefore, the noise gain is given by This is of course the gain actually measured using noise instead of the more usual CW gain.

#### Practical Measurements Using Y Factor

Many types of noise figure measuring equipment use the Y factor or 'hot' and 'cold' load principle. Sometimes the noise sources are literally 50 W ones but at precisely known temperatures. For example the cold one may be at the temperature of liquid nitrogen and the other at room temperature. Alternatively a specifically designed noise source can be used, such as a noise diode. One of these can generate a level of noise equivalent to quite a high noise temperature.

A noise figure meter (NFM) typically uses the Y factor principle. It is essentially a noise receiver, used for measuring noise power levels accurately and a means of controlling a noise diode. Usually it incorporates some form of internal processor. How does it perform a measurement?

The instrument must initially be calibrated for the second stage contribution. The second stage is actually formed from all of the components between the output of the device under test and the detector part of the NFM.

From the equations above, the noise temperature of the second page T2 is given by and its noise gain G2 is given by: When the noise source is connected to the input of the DUT and its output is connected to the NFM,the second stage comprises the DUT itself plus the NFM. Thus for this combination, the overall noise temperature T12 and the overall gain G12 are given by:  The equivalent noise figure for the 2 cascaded stages is given by the following equation, also known as Fris's formula: The gain for the 2 cascaded stages is Thus  Substitution gives and This is how the NFM calculates the noise temperature of the DUT.

### Excess Noise Ratio

#### Excess Noise Ratio

The excess noise ratio (ENR) expressed in dB is: There is a mistake in the diagram here, it should of course be 10*log base 10 not 1-log base 10. The hyphen (minus) sign and zero are next to each other on my keyboard and my finger slipped. Here the ENR is expressed in dB above 290 K and T is the equivalent noise temperature of the source in Kelvin. The 'cold' temperature of 290 K is actually the standard noise temperature. Typically, a graph of ENR against frequency is supplied with noise sources, for example with the HP 346B which is the noise source commonly used with the Hp 8972A noise figure meter.

### Noise Figure of a Matched Attenuator

The following equations can apply to a matched attenuator as well as an amplifier.   The excess noise Nx is the equivalent noise that would occur at the output, if the actual amplifier was replaced by a noise-free one. It could be referred to the input but I haven't done that.

A matched attenuator with a linear loss ratio of L has a linear gain ratio of An input noise of Ni will result in an output noise of Ni/L. therefore the excess noise is given by Substituting gives   Therefore the loss of the matched attenuator in dB is equal to its noise figure in dB.

### Phase Noise

Phase noise, also called phase jitter or short-term instability, results from the random variation in the phase of a noise voltage. This is apparent in oscillators and vectorially adds to the fundamental signal voltage generated by the oscillator. The resulting power against frequency spectrum of a typical oscillator, subject to phase noise is shown below. The phase noise spectrum is symmetrical and is therefore often represented by in single sideband (SSB) form comprising a graph of phase noise power density, usually in dBc/Hz, against frequency offset from the fundamental.

### Phase Noise in Oscillators

There is an excellent reference on this by Dr. Rohde Designing Low Phase Noise Oscillators. Most oscillators require the phase noise density to be as small as possible but its value in terms of the offset from the fundamental is important. If the oscillator is destined to be used in a phase locked loop (PLL), the action of the PLL is to actually reduce the overall phase noise of the PLL closer to the carrier within certain limits whilst that further from the carrier is not affected. At greater frequency offsets the phase noise of the VCO itself dominates the overall phase noise of the PLL. Here adequate phase noise performance is down to the design of the oscillator itself.

The parameters which determine the phase noise are:

• The Flicker (or 1/f) corner frequency.
• The loaded Q of the resonator.
• The noise contribution due to the oscillator's noise figure.

The Flicker or 1/f noise corner frequency increases as the bias current of the active device is increased. For example, in a bipolar transistor oscillator, a typical collector current (IC) of 0.25 mA gives a 1/f frequency of 1 kHz, whereas for IC = 5 mA, 1/f = 9.3 kHz. So it is best to keep IC as small as possible.

Recent VCO developments have been towards smaller size and therefore physically smaller resonators. Unfortunately this reduces the Q factor and increases the phase noise contribution but normally provides a larger frequency tuning range.

Nyquist noise of the type described in Section TBA is always present in the internal oscillator components and therefore adds to the phase noise. Its level relative to the wanted fundamental signal may be significant.