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Interference General Case

A binary data...

Question

A binary data is transmitted through an ideal AWGN channel with infinte bandwidth. The two-sided power spectral density of the noise is $N_{0} / 2$ . If the average energy is transmitted per bit is $E_{b}$ then the minimum value of $(E_{b} / N_{0})$ required, for error free transmission, will be_______dB.
-1.6

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Solution

The correct option is A -1.6
From the Shannon's channel capacity theorem,
$c = B l o g_{2} (1 + \frac{S}{N});$ $S = E_{b} R_{b}$ and $N = N_{0} B$
When the channel is ideal with infinite BW,
$C = \frac{S}{N_{0}} l o g_{2} (e)$
For error free transmission,
$R_{b} \leq C$
$R_{b} \leq \frac{S}{N_{0}} l o g_{2} (e) = \frac{E_{b} R_{b}}{N_{0}} l o g_{2} (e)$
$\frac{E_{b}}{N_{0}} = l o g_{e} (2)$
$(\frac{E_{b}}{N_{0}})_{m i n} = l o g_{e} (2) = 0.693$
In decilogs,
$[(\frac{E_{b}}{N_{0}})_{m i n}]_{d B} = 10 l o g_{10} (0.693) d B \approx - 1.60 d B$

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