GATE 2016 EC (EC3) – Question 60
A voice-grade AWGN telephone channel has bandwidth 4.0 kHz and two-sided noise spectral density $\eta/2=2.5\times10^{-5}$ W/Hz. To transmit at 52 kbps with arbitrarily small bit error rate, the minimum $E_b$ in mJ/bit is ________.
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Correct answer: 30.0 to 33.0
Explanation
Shannon capacity is $C=B\log_2(1+S/N)$. At the minimum required power, C=R=52000 bits/s and B=4000 Hz, so S/N=$2^{13}-1=8191$.
The two-sided density is η/2, hence the total in-band noise power is $N=\eta B=(5\times10^{-5})(4000)=0.2$ W. Required signal power is 1638.2 W.
Divide by bit rate: $E_b=1638.2/52000=0.0315038$ J/bit, or **31.504 mJ/bit**.