demand for information exchange is a characteristic of modern civilisation. The
transfer of information from the source to its destination has to be done in such
a way that the quality of the received information should be as close as possible
to the quality of the transmitted information.
information to be transmitted can be machine generated (e.g., images, computer
data) or human generated (e.g., speech). Regardless of its source, the information
must be translated into a set of signals optimized for the channel over which
we want to send it. The first step is to eliminate the redundant part in order
to maximize the information transmission rate. This is achieved by the source
encoder block in Figure 1-1. In order to ensure the secrecy of the transmitted
information, an encryption scheme must be used. The data must also be protected
against perturbations introduced by the communication channel which could lead
to misinterpretation of the transmitted message at the receiving end. This protection
can be achieved through error control strategies: forward error correction (FEC),
i.e., using error correcting codes that are able to correct errors at the receiving
end, or automatic repeat request (ARQ) systems.
The modulator block generates a signal suitable for the transmission channel.
In the traditional approach, the demodulator block from Figure 1-1 makes a "hard"
decision for the received symbol and passes it to the error control decoder block.
This is equivalent, in the case of a two level modulation scheme, to decide which
of two logical values, say -1 and +1, was transmitted. No information is passed
on about how reliable the hard decision is. For example, when a +1 is output by
the demodulator, it is impossible to say if it was received as a 0.2 or a 0.99
or a 1.56 value at the input to the demodulator block. Therefore, the information
concerning the confidence into the demodulated output is lost in the case of a
"hard" decision demodulator.
The capacity of a channel, which was first introduced 50 years
ago by Claude Shannon, is the theoretical maximum data rate that can be supported
by the channel with vanishing error probability. In this discussion, we restrict
our attention to the additive white Gaussian noise (AWGN) channel.Here, x is modulated
symbol modelled by arandom process with zero mean and variance Es (Es is the energy
per symbol). For the specific case of antipodal signalling 2 , x = + Es 1/2 .
z is sample from an additive white Gaussian noise process with zero mean and variance
N0/2. The capacity of the AWGN channel is given by :
per channel use. Signalling at rates close to capacity is achieved in practice
by error correction coding. An error correction code maps data sequences of k
bits to code words of n symbols. Because n>k, the code word contains structured
redundancy. The code rate, r = k/n is a measure of the spectral efficiency of
the code. In order to achieve reliable communications, the code rate cannot exceed
the channel capacity ( r < C). The minimum theoretical signal-to-noise ratio
3 Eb/N0 required to achieve arbitrarily reliable communications 4 can be found
by rearranging equation (2) This is the minimum Eb/N0 required for any arbitrary
distribution for the input x.
Equation (3) can be satisfied with equality
only if the input is Gaussian distributed. If the input is antipodal instead of
Gaussian, slightly higher Eb/N0 is required. Plots of the minimum Eb/N0 required
to achieve channel capacity as a function of code rate r are given in Figure 1
for both Gaussian distributed inputs and antipodal inputs.
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