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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

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  • $\begingroup$ For the formula for $x(t)$, shouldn’t there be a sine term after $n_s$? $\endgroup$ – Dan Boschen Nov 28 '18 at 14:12
  • $\begingroup$ Nice catch. Rectified it now. $\endgroup$ – Arkonaire Nov 28 '18 at 14:40
  • $\begingroup$ that formula is actually a great example why I strongly prefer to use exponential frequency notation for complete signals and avoid use of sines and cosines. The two ways of writing the Fourier Series Expansion is another good example of the unnecessary complexity. Sorry no one has answered yet; will be able to dig into it in a few days if no one else bites before then. $\endgroup$ – Dan Boschen Nov 30 '18 at 4:04
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yes, you are right. But there is misconception that figure of merit is the ratio of output to input SNR. It is actually the ratio of output SNR of a receiver to the output SNR of a baseband system (without modulation). Here, the output SNR of a baseband system is taken as a benchmark for judging noise performance of receiver. Hence, FoM = (output SNR)/(Si/NoW)

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