1.- The SNR is a parameter attributed to signals, not systems
Although often one may read 'this system has this or that SNR', in general such expression is referred to a system with a known input output signals.
2.- SNR is a power ratio : SIGNAL / NOISE
There's no such thing as 'negative' (linear) power values of signals.
I mention 'linear' because when using
dBmW then obviously there may be positive or/and negative
dBW dBmW values.
Some people, I've worked with, shorten
dBm and then started mixing signal amplitude values with power values, not a good idea.
3.- DC is not noise
AWGN or interferring signals are ususally unknown, apart from statistical moments or direct measurements that are usually samples.
For noise, given the random nature, that is what goes to the denominator of SNR.
In analog cellular
CIR measurements were/are common Carrier to Interference,
obviating that the signal power contents is small compared to the carrier power, because the Interferrer has a similar value to the signal carrier, both carrier and Interferrer >> signal, without carrier.
What signal information is being conveyed by a DC level anyway?
Constellations for modulation/demodulation do not include DC.
Something related but out of context here is the (modulation constellation) Error Vector Measurement, EVM and there may be constellation points that on purpose or by noise/interference are biased, but it's all referred to the constellation coordinates origin, that doesn't include DC.
You may want to consider that a transistor bias is getting into the signal path or a power supply leaking DC, or a slightly rippled DC.
4.- Added negative mean implies increasing power
the DC sign doesn't matter when calculating power.
s be zero mean signal. Then
s+S0 is the DC-ed signal.
P() power operator cancells DC sign.