12 How to determine figure of merit for DSB-SC using coherent detector?

Figure of merit, \(\gamma \) is defined as \(\frac {SNR_{o}}{SNR_{c}}\) where \(SNR_{o}\) is the signal-to-noise ratio on output from modulator, and \(SNR_{c}\) is signal-to-noise ratio for the channel, assuming channel has AWGN added. The following diagram shows the calculations. I used a coherent demodulator.

Figure 3: figure of merit for DSB-SC

Question: Verify the above.