Relationship between Bandwidth, Data Rate and Channel Capacity

This posts describes the relationship between signal bandwidth, channel bandwidth and maximum achievable data rate. Before, going into detail, knowing the definitions of the following terms would help:

• Signal Bandwidth – the bandwidth of the transmitted signal or the range of frequencies present in the signal, as constrained by the transmitter.
• Channel Bandwidth – the range of signal bandwidths allowed by a communication channel without significant loss of energy (attenuation).
• Channel Capacity or Maximum Data rate – the maximum rate (in bps) at which data can be transmitted over a given communication link, or channel.

In general, information is conveyed by change in values of the signal in time.  Since frequency of a signal is a direct measure of the rate of change in values of the signal, the more the frequency of a signal, more is the achievable data rate or information transfer rate. This can be illustrated by taking the example of both an analog and a digital signal.

If we take analog transmission line coding techniques like Binary ASK, Binary FSK or Binary PSK, information is tranferred by altering the property of a high frequency carrier wave. If we increase the frequency of this carrier wave to a higher value, then this reduces the bit interval T (= 1/f) duration, thereby enabling us to transfer more bits per second.

Similarly, if we take digital transmission techniques like NRZ, Manchester encoding etc., these signals can be modelled as periodic signals and hence is composed of an infinite number of sinusoids, consisting of a fundamental frequency (f) and its harmonics. Here too, the bit interval (T) is equal to the reciprocal of the fundamental frequency (T =  1/f). Hence, if the fundamental frequency is increased, then this would represent a digital signal with shorter bit interval and hence this would increase the data rate.

So, whether it is analog or digital transmission, an increase in the bandwidth of the signal, implies a corresponding increase in the data rate. For e.g. if we double the signal bandwidth,  then the data rate would also double.

In practise however, we cannot keep increasing the signal bandwidth infinitely. The telecommunication link or the communication channel acts as a police and has limitations on the maximum bandwidth that it would allow. Apart from this, there are standard transmission constraints in the form of different channel noise sources that strictly limit the signal bandwidth to be used.  So the achievable data rate is influenced more by the channel’s bandwidth and noise characteristics than the signal bandwidth.

Nyquist and Shannon have given methods for calculating the channel capacity (C) of bandwidth limited communication channels.

Nyquist Criteria for maximum data rate for noiseless channels

Given a noiseless channel with bandwidth B Hz., Nyquist stated that it can be used to carry atmost  2B signal changes (symbols) per second. The converse is also true, namely for achieving a signal transmission rate of 2B symbols per second over a channel, it is enough if the channel allows signals with frequencies upto B Hz.

Another implication of the above result is the sampling theorem, which states that for a signal whose maximum bandwidth is f Hz., it is enough to sample the signals at 2f samples per second for the purpose of quantization (A/D conversion) and also for reconstruction of the signal at the receiver (D/A conversion). This is because, even if the signals are sampled at a higher rate than 2f ( and thereby including the higher harmonic components), the channel would anyway filter out those higher frequency components.

Also,  symbols could have more than two different values, as is the case in line coding schemes like QAM, QPSK etc. In such cases, each symbol value could represent more than 1 digital bit.

Nyquist’s formulae for multi-level signalling for a noiseless channel is

C = 2 * B * log M,

where C is the channel capacity in bits per second, B is the maximum bandwidth allowed by the channel, M is the number of different signalling values or symbols and log is to the base 2.

For example, assume a noiseless 3-kHz channel.

1. If binary signals are used, then M= 2 and hence maximum channel capacity or achievable data rate is C = 2 * 3000 * log 2 = 6000 bps.
2. Similarly, if QPSK is used instead of binary signalling, then M = 4. In that case, the maximum channel capacity  is C = 2 * 3000 * log 4 = 2 * 3000 * 2 = 12000bps.

Thus, theoritically, by increasing the number of signalling values or symbols, we could keep on increasing the channel capacity C indefinitely. But however, in practise, no channel is noiseless and so we cannot simply keep increasing the number of symbols indefinitely, as the receiver would not be able to distinguish between different symbols in the presence of channel noise.

It is here that Shannon’s theorem comes in handy, as he specifies a maximum theoritical limit for the channel capacity C of a noisy channel.

Shannon’s channel capacity criteria for noisy channels

Given a communication channel with bandwidth of B Hz. and a signal-to-noise ratio of S/N, where S is the signal power and N is the noise power, Shannon’s formulae for the maximum channel capacity C of such a channel is

C = B log (1 + S/N)    

(log is to base 2)

For example, for a channel with bandwidth of 3 KHz and with a S/N value of 1000, like that of a typical telephone line, the maximum channel capacity is

    C = 3000 * log (1 + 1000)  = 30000 bps (approx.)

Using the previous examples of Nyquist criteria, we saw that for a channel with bandwidth 3 KHz, we could double the data rate from 6000 bps to 12000 bps., by using QPSK instead of binary signalling as the line encoding technique. Using Shannon’s criteria for the same channel, we can conclude that irrespective of the line encoding technique used, we cannot increase the channel capacity of this channel beyond 30000bps.

In practise however, due to receiver constraints and due to external noise sources, Shannon’s theoritical limit is never achieved in practise.

Thus to summarize the relationship between bandwidth, data rate and channel capacity,

• In general, greater the signal bandwidth, the higher the information-carrying capacity
• But transmission system & receiver’s capability limit the bandwidth that can be transmitted

Hence data rate depends on

• Available bandwidth for transmission
• Channel capacity and Signal-to-Noise Ratio
• Receiver Capability

More the frequency allotted,  more the channel bandwidth, more the processing capability of the receiver, greater the information transfer rate that can be achieved.

 

 

 

What is baud rate and how is it different from bit rate?

This post briefly describes the concept of baud and bit rates.

As seen from previous posts, multi-level line encoding techniques like QAM-8, QAM-16, QAM-64 etc. use multiple values of amplitudes and phases to form multiple symbol values or signalling levels. The number of different symbol values/signalling levels in a line encoding technique decides the number of binary bits (0 and 1) that can be represented by each symbol.

For e.g.

QAM-8 uses two different Amplitudes and four phases thereby giving eight different  symbols. Since there are 8 different symbols, each symbol could be used to represent 3 binary digits (for e.g. symbol1 represents 000, symbol2 represents 001, … symbol8 represents 111), thereby giving a data rate that is 3 times that of the symbol rate.

QAM-16 uses four amplitudes and four phases, therey giving sixteen different symbols. Since there are 16 different symbols, each symbol could be used to represent 4 binary digits (for e.g. symbol1 represents 000, symbol2 represents 001, …. symbol16 represents 1111). Hence the data rate of QAM-16 is four times that of the symbol rate.

Baud rate or symbol rate is the number of symbols that can be sent on the telecommunication link per second. It is also known as the signalling speed or the number of signal value changes per second. The baud rate mainly depends on the transmission capacity and noise tolerance level of the channel/link. Additionally, the sender and the receiver needs to have the processing capability to encode and decode symbols at the baud rate.

Bit rate or data rate is the number of bits that can be transmitted on the link per second.

The relationship between bit rate and baud rate is given below:

Let “M” denote the number of signalling levels or the number of symbols. For e.g. M = 8 for QAM-8 and M=16 for QAM-16.

•   Baud Rate = No: of symbol changes per second

•   Bit rate= (Baud rate) * log (M) ,   where log is to the base 2.

For a baud rate of 100 symbols per second, the table below gives the bit rate for three different line encoding methods.

Bit Rates for different encoding methods

 Thus, it is seen that  for the same baud rate, a change in the line encoding technique results in different bit rates.