Exact definition of soft bits in digital receiver
Soft bits or more generally speaking, soft metrics, are usually taken at the output of the demodulator (e.g. the output of a matched filter sampled at $T_b$). A soft bit can be therefore a real value, but it is usually quantified (which is required if a DSP comes after).

A hard bit is obtained by performing a hard decision on a soft metric. So for instance, if $x[kT_b]$ is the output of the matched filter when a BPSK signal is applied, then a hard-bit can be obtained by simply comparing $x[kTb]&gt;0$, which is $0$ or $1$.

This figure from wikipedia is self-explanatory.



As you see, the output of the matched filter is actually a continuous signal. Some part of your receiver (which is not included in the figure) must perform also timing estimation in order to determine at which instant the output of the matched filter must be sampled. Those samples are the red points in the figure, and might be considered as soft bits. So the higher the magnitude of those samples, the more information they provide about what the original transmitted symbol was. That is the advantage of having soft-bits: they don't only tell you what symbol has been probably sent, they also give an idea about that probability.