input reffered noise
Input reffered noise density is just mathematical model which enables comparision of TIA with different gain and bandwididth.
Notice that you can not:
-calculate input noise power directly by integrating input noise density because you have no information on bandwidth , and integral would not convergate
-you can not measure it directly, because it is not measurable (does not exist - just mathematical model !!!)
The key point for your measuerement should be rms noise at the output.
1. From this value you can determine input noise rms (and thus sensitivity) by dividing it by the TIA gain Rf.
2. You can also determine input reffered noise in,rms^2 by dividing output noise power at the ourput vn,out,rms^2, by Rf^2 and TIA Bandwidth. Someone insetad of whole bandith takes just part where ampliification is lowest (for example before dominant pole = 50k), which gives smaller (but inaccurate) input reffered noise. Be sure that in such papers with unusually small input reffered noise, without specified freq range in which noise was measured, or noise rms, are usually inacurrate.
Then when someones say this "the input referred noise current density was below 20pA/√Hz within the amplifer frequency band", are they telling the operating band freqeuncy of the TIA they design. Like my case my operating frequency is from 50kHz to 875Mhz (low cutoff frequecy - Tia BW) or something else. Please clarify me at this part as well.
Your TIA frequency band starts from 0 if you do not have ac coupling. In this case noise is also transferred to the output and should be considered. Upper limit can be TIAbw,and you should specify it.
rms noise
Ideally you should measure output rms noise in the whole frequency range from 0 to infinity. In reality uper limit is defined by your measurement equipement.
How much is enough? I would take as starting point some value greater than TIAbw, ENB, or some higher value say 1.25G ..2G.
After some frequency, because of noise shaping input reffered noise will not affect on output. This is because noise is low passed shaped from freq TIAbw with slope -40dB/dec (notice that at TIAbw is not enough).
How much? It should be defined defined by ENB (1.1bw for Butterworth response).
For your case you can prove simulating your circuit noise by Cadence, taking different values for your rmsNoise upper limit.
If same rmsnoise was calculated for intervals 0.1..1G, 0.1 .. 2G , 0.1 .. 10G and 0.1 .. 100G, which differs significantly from 0.1 .. 675k, 1G should be enough.
To sumarize:
-The most valuable information for your receiver is rms noise voltage. Rms noise is determined for the range from 0 to frequency that is larger for some factor of TIA bw.
-Input reffered noise concept is suitable for different receivers comparision, because it has noise information normalized by gain, and TIA bw. For complete comparision you should know also the bandwidth in which input reffered noise was calculated, and this bw should be equal to TIA bw.
-If someone insetad of whole TIA bw, takes part where noise is more or less attenuated, than it can have smaller/higher values of input reffered noise than actual.