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An analog pulse $s(t)$ is transmitted over an additive white Gaussian noise (AWGN) channel. The received signal is $r(t)=s(t)+n(t)$, where $n(t)$ is additive white Gaussian noise with power spectral density $N_0/2$. The received signal is passed through a filter with impulse response $h(t)$. Let $E_s$ and $E_h$ denote the energies of the pulse $s(t)$ and the filter $h(t)$, respectively. When the signal-to-noise ratio (SNR) is maximized at the output of the filter $(\mathrm{SNR}_{\max})$, which of the following holds?

1. Es = Eh ; SNRmax = 2Es/N0
2. Es = Eh ; SNRmax = Es/2N0
3. Es > Eh ; SNRmax > 2Es/N0
4. Es < Eh ; SNRmax = 2Eh/N0

Correct answer: Es > Eh ; SNRmax > 2Es/N0

Solution

The filter that maximizes output SNR in AWGN is the matched filter, with impulse response proportional to the time-reversed version of the signal. For a matched filter, the maximum output SNR is proportional to $2E_s/N_0$. The option stating a larger filter energy is not the standard condition, so the correct conceptual result is the matched-filter maximum SNR expression.

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