For the FOM calculation of an AM receiver, the input and output signal powers are respectively, $$P_{in} = \frac{A_c^2(1 + k_a^2P_m)}{2}$$ $$P_{out} = k_a^2A_c^2P_m$$ These are obtained from the input and output expressions for the receiver as given by, $$x(t) = A_c(1 + k_am(t))\cos(2\pi f_ct) + n_c(t)\cos(2\pi f_ct) - n_s(t)\sin(2\pi f_ct)$$ $$y(t) = k_aA_cm(t) + n_c(t)$$ where $n_c(t)$ and $n_s(t)$ are the in phase and quadrature components of the complex noise envelope, and the output is obtained via envelope detection. Now, while calculating output noise power, we multiply the PSD of $n_c(t)$ with the bandwidth. So, $$P_{noise,out} = 2N_0W$$ because $n_c(t)$ has a PSD of $N_0$ spread out in a frequency range of $-W$ to $W$. This gives us the SNR of output as, $$SNR_{out} = \frac{k_a^2A_c^2P_m}{2N_0W}$$ Now, in the literature, as in Wikipedia for instance, (link below) the input SNR is, $$SNR_{in} = \frac{A_c^2(1 + k_a^2P_m)}{2N_0W}$$ which corresponds to $$P_{noise,in} = N_0W$$ Now, it is my understanding that the input noise is a bandpass AWGN noise with a double sided PSD of $\frac{N_0}{2}$ spread out in a frequency range of $f_c - W$ to $f_c + W$ and $-f_c - W$ to $-f_c + W$, which then should correspond to a noise power of $2N_0W$. But it seems that the negative frequencies have not been accounted for while calculating input noise power. Where am I going wrong here?
Link to wikipedia page: https://en.wikipedia.org/wiki/Signal-to-noise_ratio
