Last Updated on October 4, 2023 by John Berry

Radio amateurs are interested in the signal to noise ratio of received signals. In professional digital systems the interest is in E_{b}/N_{0}, the energy per bit over noise in a 1Hz bandwidth. Radio amateurs often use noise in the the 2.5kHz SSB channel as reference. To include electrical noise (from the various system components) we need to be able to calculate the threshold degradation.

The following considers the electrical noise in a receiver system. The system comprises an array of four antennas, combiner, low noise amplifier, feeder, and receiver in the rig.

There are two steps needed.

## The antenna

First, calculate the noise figure of the antenna in its environment and have this available for use in a subsequent cascade calculation.

To calculate this, we need to revert to noise temperatures and look up the antenna noise temperature on the manufacturer’s website.

The formulas concerned are:

t_{s} is the antenna noise temperature, f_{s }is the corresponding noise factor.

F_{s} is the noise figure, f_{s} is the corresponding noise factor from above*.*

A handy converter is available at https://www.microwaves101.com/calculators/1339-noise-conversion-calculator.

In my case, the noise temperature of my 11-element 2m Yagi is about 257°K (from manufacturer data). This gives a noise figure of about 2.8dB.

Note that the manufacturer has included the antenna gain in the antenna noise temperature measurement. The gain as a result of focussing of the beam (typically called forward gain) is included elsewhere in the path budget.

There are four antennas in the array, and hence in principle, the four noise temperatures must add. However, each antenna will be directional. It’s likely that the array will have a significantly narrower beamwidth than a single antenna of the same type. With a narrower beamwidth, the array will capture less noise. Hence the noise temperature of the array will be lower than that for the single antenna.

To complicate things further, the optimised four-stack array may have significantly fewer responses off the main beam when compared to a single antenna. This is another argument for improved noise performance from the array.

So, it’s complicated. It might be that the noise is increased, and it might be that it is reduced. Modelling is needed to conclude this definitively.

To consider the antenna array in the noise calculation, I have assumed that a four-stack has the same noise temperature as a single antenna.

## System noise figure

Second, calculate the cascaded noise figure for all receiver elements to enable the threshold degradation from system noise to be deduced.

This is done using the classic cascade formula below:

Where f_{s} is the system noise factor and the various f_{n} and g_{n} are the noise factors and gains of the system stages. The system noise figure F_{s} in dB is then:

A calculator and good discussion on this is given by Everything RF at: https://www.everythingrf.com/rf-calculators/cascaded-noise-figure-gain-calculator. Note however that the authors there have not obeyed the convention that scalar variables are in lower case notation while decibel values are in upper case.

I assume a five-stage system comprising receiver, feeder and low noise amplifier, combiner and antenna array as shown below.

In my case, the LNA has a noise figure of 0.6dB and a gain of 20dB. I’ve assumed a manufacturer’s insertion loss figure for the antenna combiner of 0.5dB. I estimate the main feeder (and all connectors and pass-through losses) at 3dB, with the corresponding gain at -3dB. I don’t have an accurate noise figure for the rig receiver, so I’ve estimated that at 3dB and assumed the gain of the first receiver stage at 20dB.

The system noise figure from above is 3.5dB.

## System sensitivity

Given that the rig receiver sensitivity (for 0dB signal to noise for an SSB signal in a 2.5kHz channel) is -159dBW for a noise figure of 3dB, and there’s system noise figure from the cascade (including that receiver) of 3.5dB, the system sensitivity is -158.5dB. That’s just half a dB worse than using the receiver alone. But of course, the big advantage is that we win 20dB gain in the signal path, accounted for elsewhere. The receiver sensitivity is from the manufacturer’s equipment specification.

Just as a side exercise, it’s well worth using the cascade calculator to see where the system noise sensitivities are. Try reducing the LNA noise figure – the effect is big, justifying its use. Try increasing the feeder loss (simulating poor quality feeder) – its effect is small, so paying good money for good feeder is a waste (considering the RX side in isolation). And see why using a low noise antenna (with narrow beamwidth and suppressed side lobes) is key – its noise adds dB for dB. The formula above shows why it is the primary contributor of noise.

The threshold degradation is therefore 0.5dB, and the system sensitivity of -158.5 is used elsewhere in calculating the path budget.

For reference, if the antenna is in an urban environment, there is may be a significant increase in noise. Noise into an omni-directional antenna is some 16dB worse than rural giving a system sensitivity of about 141dBW. This is worst case since antennas are generally directional. Hence much of the noise is attenuated by antenna directivity. There will nonetheless be some degradation and many radio amateurs report high noise and poor system sensitivities. This environmental noise has not been considered above.