The question arises from time to time, is a receiver test with different resistors connected it its input terminals meaningful?

## Scope

This discussion applies to linear receivers. A receiver using a diode AM detector, with or without BFO injection is NOT a linear receiver for this purpose, nor is an FM receiver. A good traditional superheterodyne SSB Communications Receiver is a linear receiver for the purpose of this discussion, but for example any techniques designed to reduce / cancel noise will render it non-linear

## Noise in resistors

Thermal agitation within a resistor gives rise to broadband noise (Johnson-Nyquist noise, thermal noise), the noise power that can be captured from a resistor in a given bandwidth is given by \(P=k_0 T B\) where:

- k0 is Boltzman’s constant;
- T is the absolute temperature; and
- B is the bandwidth.

## Receiver sensitivity

If:

- a receiver is designed for a 50+j0Ω source; and
- has noise figure specifications or specifications that imply a noise figure; and
- and is tested to specification

the measurement is done with a 50+j0Ω source that contributes thermal noise (50Ω @ 290K) and the equivalent internal noise contribution of the receiver can be calculated.

There are many ways to specify ‘sensitivity’, the poorest are bandwidth dependent, and others are bandwidth independent.

Bandwidth independent measures allow like for like comparison, whereas bandwidth independent ones do not but have appeal to buyers with less knowledge. ARRL tried to solve the latter problems with its Minimum Discernable Signal metric which looks like a sensitivity measurement (ie µV for 0dB S/N), but whilst it pretends to be bandwidth independent, it relies on a rubbery bandwidth specification (see ARRL Test Procedures Manual (Rev L) – Noise Figure calculation).

## Receiver sensitivity specification example

Let’s say that a certain receiver has sensitivity specified as 50Ω input, 0.1µV for 10dB S/N. The specifications probably do not give the Equivalent Noise Bandwidth, let us assume it is 2000Hz for this example (appropriate to an SSB telephony receiver but the actual bandwidth is critically important, it can be measured).

The S part of the sensitivity specification above is a sine wave, and its amplitude is 0.1µV RMS at the receiver terminals (ie loaded by the nominal receiver input impedance). We can calculate the power simply as \(P_s=\frac{E^2}{R}=2 \text{e-16 W}\).

The N part is a bit more complicated. Noise voltage can be calculated as \(V_n=V_s*10^(\frac{S/N}{20})=0.0316 \text{ µV}\). Noise power will simply be \(P_n=P_s*10^(\frac{S/N}{10})=2 \text{e-17 W}\).

The standard test procedure uses a Standard Signal Generator with source impedance 50Ω and it assumes that it supplies approximately 290K thermal noise.

Now there are two components to that noise power, part due to the source (the attenuator in a signal generator produces thermal noise in its output resistance) and receiver internal noise which can be expressed as a power at the input terminals.

The power in the 50Ω source in 2000Hz bandwidth at 290K is 8e-18 W, it is an appreciable part of the total equivalent noise power of 2e-17 W calculated above, making the internal part 1.2e-17 W.

Nothing about the specification implies the internal implementation. Nor does the specification define behavior with source resistance other than 50Ω.

Note that whilst the sensitivity test assumes that the test source supplies 290K noise, in normal use, the noise supplied to the input connector depends on external noise, antenna and feed system losses etc, and it may be less than or greater than 290K, it may range from a few K to millions of millions of K.

An antenna has two noise sources:

- thermal noise at 290K (or whatever the temperature of the components) generated in loss resistances (eg conductors, integrated feed lines and waveguides etc); and
- received external noise (ambient noise).

## Discussion

Now it might seem that simply connecting a variable resistance to the input terminals would allow supplying a variable amount of noise power, and finding the noise power at which the receiver output power doubled could be used to find sensitivity.

There are a number of problems, including:

- the receiver performance is specified with a 50Ω source, and that requirement is not met by the variable resistor (the internal noise may be affected by the source impedance); and
- the noise produced by the variable resistor is loaded by the receiver, and it turns out to be maximum at 50Ω, not nearly enough for testing most Communications Receivers.

Sensitivity determined by a standard sensitivity test does not imply the internal noise of a receiver for any other input source impedance than 50Ω.

## A question to ponder

So, if the standard sensitivity test does imply the receiver internal noise for a 50Ω source, any 50Ω source, and we know the noise power from a room temperature 50Ω resistor connected to its input, can we find the noise power contributed by a 50Ω antenna using a receiver of known sensitivity?