GATE 2016 EC – Question 60
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 ($\text{SNR}_{max}$), which of the following holds?
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Correct answer: (A) $E_s = E_h$; $\text{SNR}_{max} = \frac{2E_s}{N_0}$
Explanation
The SNR is maximized by the matched filter, whose impulse response is a scaled and time-reversed copy of the pulse, so the filter and the pulse have the same energy, $E_s = E_h$ (with the usual normalisation). The maximum SNR is $\frac{2E_s}{N_0}$.