Please pay attention to this discussion because you will find it nowhere else. I have an idea why you will find it nowhere else but I will let you find it as we move forward. We will break the question down to its components to make its solution digestible.

A CW receiver with the AGC off has an equivalent input noise power density of -174 dBm/Hz. What would be the level of an unmodulated carrier input to this receiver that would yield an audio output SNR of 0 dB in a 400 Hz noise bandwidth?

With the AGC (automatic gain control) we are dealing with a pure radio with no fluff. CW operation tells us it is a constant signal in.

The -174 dBm here is the operative key. This is telling us that this is the impossible “perfect” receiver that introduces no noise of its own. The question now becomes very simple, in essence, completely removing the radio from the question. It asks how much environmental noise 400 Hz introduces no matter where you are on planet earth. Even temperature becomes irrelevant since it is by default telling us to use the standard 290^{o}K.

Given no input, the output is at -174 dBm. That -174 dBm is an absolute measure, but now we switch to a relative ratio and thus we are looking at dB to define the environment instead of dBm.

We are going to supply an input, but before we hook up anything to the input, we are told that the operating environment will be at a bandwidth of 400 Hz. We therefore need to re-calculate the noise figure.

P_{400Hz} = 1.38e-23 J/^{o}K * 290^{o}K * 400 Hz = 1.60e-18 J.Hz

dBm = 10 log_{10} (1.60e-18/1e-3) = -148 dBm

For an operational bandwidth of 400 Hz, the noise floor will be -148 dBm, and it doesn’t matter where on planet Earth the station will be. We have not yet introduced a signal in for the station.

We are now asked to specify the required level of signal in such that the signal-to-noise (SNR) ratio will be 0 dB. Why is it that all of a sudden we are given dB to define the problem?

What does this mean? It means that the signal in will be equal to the noise in. Thus, the two will be indistinguishable from the other. And thus, the correct answer is -148 dBm. At a 400 Hz bandwidth, this is the noise floor or figure. Any signal in must be greater than this value in order to be detected and amplified.

## dBm or dB?

But this brings to the floor a perplexing question. Again, hold on to your hat because you will see this discussed nowhere else.

The question E4C06 asks for a “relative” level in producing a SNR of 0 dB. All of our input for the question was in dBm (an absolute measure using a reference of 1 mW) but it requests a result of 0 dB for a signal-to-noise ratio. This is okay because although our known absolute noise level in is given in dBm, we are countering it with an equal dBm and thus in so doing define a ratio. Because the two are equal, their ratio is unity. A log base 10 (log_{10}) value for unity is zero.