Analog vs. digital communications in PCM and noise
This is quite a broad topic.  Let's narrow the scope to toll quality telephone audio.  Further assume both analog and digital channels are bandlimited to the same passband.  I assume you don't want to figure out the difference between a pair of wires and a PCM channel (i.e. a pair of wires has a very slow decay in frequency response where as a digital channel is hard band limited).

Ok a for sort of apples to apples comparison of a roughly 4Khz voice band channel, you can look for sampling artifacts in the channel.  The sampling frequency for telephone audio is 8Khz.  The true passband for telephone quality audio is 200Hz-3600Hz (still roughly speaking here).

Look for quantization noise.  Quantization noise is not random and can be made nearly periodic by choosing the right test signal.  The PCM encoding for standard CODEC ditigal telephone channels uses a type of compression that follows a log scale (u-law or A-law).  The analog reconstruction filter does not completely eliminate the quantization noise.  Because of the compression scheme, the quantization steps increase with increasing signal amplitude.  If you create a test signal that is a ramp running from -Vp to +Vp (where Vp is the maximum signal amplitude the channel can handle), you will see discontinuities that are closely spaced near 0V and spaced farther appart near +Vp and -Vp.  Use a frequency of 1KHz so the signal frequency is harmonically related to the sampling frequency.  This will place the quantization errors nearly at the same location for each period of the ramp signal.

You can also use a spectrum analyzer and look for spurs in the frequency content of a test signal.  In this case, use a 1Khz sine wave.  Once again the quantization errors will remain at nearly the same position on each period of the sinewave.  The quantization errors will appear as distortion of the sine wave and show up as harmonics in the signal spectrum.  These harmonics will be greatest when the sampling frequency is harmonically related to the signal frequency so 1KHz and 2KHz are good choices.  As the test tone drifts away from intger divisions of the sampling rate, the quantization noise spreads and begins to look like a spectral contribution from random noise.  In telephone systems, a 1KHz test tone is generally not used for this reason.  The normal test tone frequency is 1004Hz.

Note that modern CODECS with high compression rates such as G.729 have a difficult time passing periodic signals.  These CODECs are designed in such a way as to interpret all signals as human voice or noise, so the reproduced periodic signal will be highly distored or modulated.