Question

In the communication system, if for a given rate of information transmission requires channel bandwidth, B₁ and signal-to-noise ratio SNR₁. If the channel bandwidth is doubled for same rate of information then a new signal-to-noise ratio will be

1. SNR₁
2. 2SNR₁
3. \(\sqrt {SN{R_1}}\)
4. \(\frac{{SN{R_1}}}{2}\)

Answer 1

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) = B₁

Signal to noise ratio = SNR₁

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

If SNR ≫ 1

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

When channel B ⋅ ω = 2B₁

Let signal to noise Ration = SNR₂

Information Rate = R₂

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

Given- R₂ = R₁

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

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