What happens if the SINR of a signal is less than 1 in a (wireless) network? Is that practical at all to have a SINR < 1? Using the Shannon capacity formula, it seems that we must be able to have SINR < 1, but we get a rate < 1 bits/sec/Hz.
Yes, it is practical.
For example, if you have an ASK signal with modulation depth 60%:
Using a low-pass filter
you can reduce the output signal bandwidth in order to be nice to the neighbors:
The recipient gets a noisy signal
and also uses a lowpass filter to reconstruct it:
This signal is sufficiently similar to the original signal that you can decide between 0 and 1 bits, but you need a rather narrow filter here in order to remove the noise -- in my case, 1% of the sampling rate (that is the
In order to be able to reconstruct that signal, I had to use rather long symbols (100 samples). That is, with 100 Hz sampling rate, which would allow me to express frequencies up to 50 Hz, I can only transfer 1 bit per second.
There are plenty of examples where SINR (or SNR) is less than 1 and still quite usable. The signal on an antenna (that you want to receive) maybe only 1 microvolt but all the other signals you don't want (from 50Hz upwards) totalize a few millivolts - radios manage to overcome this problem by filtering.
Even after filtering at a particular frequency for a wanted frequency, the result may still be SNR<1 but, if the "system" is a direct sequence spread spectrum (DSSS) transmission that doesn't matter because the sum of all the different frequencies transmitted at means that noise cancels.
Not only is it practical, it is commonly done. See, for instance, http://www.gpssource.com/faqs/15 which discusses the SNR of GPS signals (typically about -26 db, or ~ 1/400.