Correct Answer - Option 3 :

\(\sqrt {SN{R_1}}\)
__Shannon-Hartley Theorem__- It tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise.

\(C = B{\log _2}\left( {1 + \frac{S}{N}} \right)\)

Where C is the channel capacity in bits per second

B is the bandwidth of the channel in hertz

S is the average received signal power over the bandwidth

N is the average noise

S/N is the signal-to-noise ratio (SNR)

For transmitting data without error R ≤ C where R = information rate

Assume information Rate = R

Form Shanon hartley theorem

R_{max} = C

\(R = B{\log _2}\;\left( {1 + SNR} \right)\)

When channel Band width (B) = B_{1}

Signal to noise ratio = SNR_{1}

\({R_1} = {B_1}{\log _2}\left( {1 + SNR} \right)\)

If SNR ≫ 1

\({R_1} = {B_1}{\log _2}SN{R_1}\) ---(1)

When channel B ⋅ ω = 2B_{1}

Let signal to noise Ration = SNR_{2}

Information Rate = R_{2}

\({R_2} = 2{B_1}{\log _2}SN{R_2}\) ---(2)

Given- R_{2} = R_{1}

\(2{B_1}{\log _2}SN{R_2} = {B_1}{\log _2}SN{R_1}\)

\(SN{R_2} = {\left( {SN{R_1}} \right)^{\frac{1}{2}}}\)