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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 $\frac{N_0}{2}$. 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($SNR_{max}$), which of the following holds?

  1. $E_s = E_h$ ; $SNR_{max}=\frac{2E_s}{N_0} \\$
  2. $E_s = E_h$ ; $SNR_{max}=\frac{E_s}{2N_0} \\$
  3. $E_s > E_h$ ; $SNR_{max}>\frac{2E_s}{N_0} \\ $
  4. $E_s < E_h$ ; $SNR_{max}=\frac{2E_h}{N_0}$
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