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We had defined spectral efficiency as
spectral efficiency in a system is limited by the signal
would be favourable to maximize the spectral efficiency in order to utilize the available
bandwidth efficiently and transmit a maximum amount of data in that bandwidth. All the advanced
modulation formats, QPSK, 16QAM and so on, are the attempts to
the efficiency of the system.
The question arises, can the spectral efficiency be increased as much as desired, or is there a
limit on the spectral efficiency? It turns out that
spectral efficiency is limited by the SNR in the system, as given by
theorem establishes the theoretical
white Gaussian noise (AWGN) channel.
Limit for Spectral Efficiency
We had defined spectral efficiency as ௧௦௧௧ௗ ௧ ௧
௨௧௭ௗ ௗ௪ௗ௧
, and we also saw that qualitatively, the
stem is limited by the signal-to-noise ratio (SNR) requirements. It
would be favourable to maximize the spectral efficiency in order to utilize the available
bandwidth efficiently and transmit a maximum amount of data in that bandwidth. All the advanced
lation formats, QPSK, 16QAM and so on, are the attempts to enhance
The question arises, can the spectral efficiency be increased as much as desired, or is there a
limit on the spectral efficiency? It turns out that there does exist a theoretical limit; the maximum
spectral efficiency is limited by the SNR in the system, as given by Shannon’s theorem
theoretical upper limit for reliable information transfer over a
channel. The noise in an AWGN channel has a normal (Gaussian)
Page 1
, and we also saw that qualitatively, the
noise ratio (SNR) requirements. It
would be favourable to maximize the spectral efficiency in order to utilize the available
bandwidth efficiently and transmit maximum amount of data in that bandwidth. All the advanced
enhance the spectral
The question arises, can the spectral efficiency be increased as much as desired, or is there a
there does exist a theoretical limit; the maximum
Shannon’s theorem. The
the upper limit for reliable information transfer over an additive
The noise in an AWGN channel has a normal (Gaussian)
distribution is additive to the signal, and its power spectral density is constant with respect to
frequency (white spectrum). In case of such a channel, the channel capacity C, the theoretical
maximum bit rate that can be transmitted in a one-sided bandwidth B is given as
C = B × logଶ(1 +
S
N
)
Where ௌ
ே
represents the SNR (as a ratio, not in dB). Please note that the logarithm is to the base
of 2, not to the base 10 or e. The information rate R must always satisfy R ≤ Blogଶ(1 + ௌ
ே
).
Thus the fundamental limit on the maximum achievable spectral efficiency of a linear channel is
given as
R
B
= logଶ ൬1 +
S
N
൰
logଶ x =
logଵ x
logଵ 2
=
logଵ x
0.3
As the noise in the system increases, the SNR decreases, and as a result, the maximum possible
spectral efficiency reduces. In order to achieve high spectral efficiencies, we need to guarantee
very low noise levels in the system, thereby ensuring high SNR. This can be done by identifying
the various noise sources in the system (transmitter, channel, receiver) and find the aggregated
noise to obtain the SNR at the receiver, and thus evaluate the maximum spectral efficiency. Thus
it is important to know the noise characteristics of the sources, the fiber channel, the receivers,
the amplifiers because all these noise sources have an influence on the channel capacity as
predicted by Shannon’s theorem.
As an example, let us consider a channel with SNR = 1, which means that the signal power and
the noise power are equal. Shannon’s theorem itself does not impose any limitation on the
SNR, the system has to be designed in such a manner that it can distinguish signal from noise.
For example, in the case of GPS communication, the signal is buried in noise, so the SNR is even
less than 1. Now consider the system with SNR = 1 and available bandwidth 1 MHz. The
available bandwidth also depends on the transmitter, channel, and receiver. For example, the
frequency range of operation of a coaxial cable is limited to a certain band due to the attenuation.
For such a system, the maximum information rate is given by R = 1 × 10 × logଶ(1 + 1) =
1 Mbps. Thus, it is necessary to study the bandwidth characteristics as well as the noise
characteristics of the transmitters, channel, and receivers, since both of these have an impact on the maximum permissible data rate in a system.
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