How to find SNR of received DSB:AM signal with white noise after demodulation & LPF'ed?

Question

This is a homework problem so please do not give a full solution. I intend to keep this at discussion-level.

Problem Statement

I am given the Power Spectral Density (PSD) of a DSB-Amplitude modulated signal and am told that it is corrupted with additive noise with PSD = N/2 "within the passband of the signal". That signal (with noise) is demodulated and passed through a Low-Pass Filter (LPF) whose frequency spectrum is unity between -W and W. I am tasked to find the SNR at the output of the LPF.

Attempt

SNR is defined to be the ratio of (1) the power of the message component of the demodulated received signal to (2) the power of the noise component of the same demodulated received signal.

We just need to take the ratio: (1)/(2).

Assuming the answer to question 2 (below) is yes. The power of the white noise component between -W and W is $2W*N/2 = WN$.

Questions

1. I'm given the (plot of the) PSD of the modulated signal. Is this (1)?
2. How does LPF affect the PSD of a signal? (Suppose we're given the plot of both the LPF (unity gain) passband and the PSD of the signal. Do we just point-wise multiply the PSD by the LPF function?).

Answer 1

Since you don't want a full answer, I'll just provide a couple of hints. Your calculation for the noise power is correct. Regarding your questions:

1. The power of a signal is the integral of its PSD.

2. You're looking for the Wiener-Khinchin theorem, and especially its application to LTI systems.