# Computer Network | Maximum data rate (channel capacity) for noiseless and noisy channels

Data rate governs the speed of data transmission. A very important consideration in data communication is how fast we can send data, in bits per second, over a channel. Data rate depends upon 3 factors:

- The bandwidth available
- Number of levels in digital signal
- The quality of the channel – level of noise

Two theoretical formulas were developed to calculate the data rate: one by Nyquist for a noiseless channel, another by Shannon for a noisy channel.

**Noiseless Channel : Nyquist Bit Rate –**

For a noiseless channel, the Nyquist bit rate formula defines the theoretical maximum bit rateBitRate = 2 * Bandwidth * log

_{2}(L)In the above equation, bandwidth is the bandwidth of the channel, L is the number of signal levels used to represent data, and BitRate is the bit rate in bits per second.

Bandwidth is a fixed quantity, so it cannot be changed. Hence, the data rate is directly proportional to the number of signal levels.

**Note –**Increasing the levels of a signal may reduce the reliability of the system.**Examples:**

**Input1 :**Consider a noiseless channel with a bandwidth of 3000 Hz transmitting a signal with two signal levels. What can be the maximum bit rate?

**Output1 :**BitRate = 2 * 3000 * log_{2}(2) = 6000bps**Input2 :**We need to send 265 kbps over a noiseless channel with a bandwidth of 20 kHz. How many signal levels do we need?

**Output2 :**265000 = 2 * 20000 * log_{2}(L)

log_{2}(L) = 6.625

L = 2^{6.625}= 98.7 levels**Noisy Channel : Shannon Capacity –**

In reality, we cannot have a noiseless channel; the channel is always noisy. Shannon capacity is

used, to determine the theoretical highest data rate for a noisy channel:Capacity = bandwidth * log

_{2}(1 + SNR)In the above equation, bandwidth is the bandwidth of the channel, SNR is the signal-to-noise ratio, and capacity is the capacity of the channel in bits per second.

Bandwidth is a fixed quantity, so it cannot be changed. Hence, the channel capacity is directly proportional to the power of the signal, as SNR = (Power of signal) / (power of noise).

The signal-to-noise ratio (S/N) is usually expressed in decibels (dB) given by the formula:10 * log

_{10}(S/N)so for example a signal-to-noise ratio of 1000 is commonly expressed as:

10 * log

_{10}(1000) = 30 dB.**Examples:****Input1 :**A telephone line normally has a bandwidth of 3000 Hz (300 to 3300 Hz) assigned for data communication. The SNR is usually 3162. What will be the capacity for this channel?

**Output1 :**C = 3000 * log_{2}(1 + SNR) = 3000 * 11.62 = 34860 bps**Input2 :**The SNR is often given in decibels. Assume that SNR(dB) is 36 and the channel bandwidth is 2 MHz. Calculate the theoretical channel capacity.

**Output2 :**SNR(dB) = 10 * log_{10}(SNR)

SNR = 10^{(SNR(dB)/10)}

SNR = 10^{3.6}= 3981Hence, C = 2 * 10

^{6}* log_{2}(3982) = 24 MHz

**Reference:**

Data Communications and Networking – Book

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