Calculating the power of AWGN should be equal to infinity as PSD is constant and its integration is infinity over all frequencies. Hence for any signal with finite power mixed with AWGN, SNR should zero. But this doesn't seems right??

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    $\begingroup$ Before your measuring device burns out from the infinitely powerful noise being applied to its input. it might help if you first apply an ideal bandpass filter that passes the signal with unit gain and suppresses all the out-of-band noise. See also this answer of mine to a related question. $\endgroup$ Feb 5, 2014 at 19:43
  • $\begingroup$ Normally, there is an explicit or implicit finite bandwidth associated with an SNR measurement. This might be the bandwidth of your signal, for example. $\endgroup$
    – John
    Feb 5, 2014 at 21:33