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.

  1. Noiseless Channel : Nyquist Bit Rate –
    For a noiseless channel, the Nyquist bit rate formula defines the theoretical maximum bit rate

    BitRate = 2 * Bandwidth * log2(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 * log2(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 * log2(L)
    log2(L) = 6.625
    L = 2^6.625 = 98.7 levels

  2. 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 * log2(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 * log10(S/N)

    so for example a signal-to-noise ratio of 1000 is commonly expressed as:

    10 * log10(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 * log2(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*log10(SNR)
    SNR = 10^(SNR(dB)/10)
    SNR = 10^3.6 = 3981

    Hence, C = 2 * 10^6 * log2(3982) = 24 MHz

Reference:
Data Communications and Networking – Book



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