Shannon Capacity and Limit( Theorems
and an Application)
Report Done by:
Rana Zuhair
Lutfi Abdul Kadhum
Abbas Ghanim
Under Supervision of:
Dr.Ali
Control and System Engineering Dept.
Postgraduate Studies (2022-2023)/Computers Branch.
14th Sep 2022
Shannon Capacity and Limit Theorems
Definition
No discussion on coding should be concluded without at least a mention of
the Shannon capacity theorem. The Shannon capacity theorem defines the
maximum amount of information, or data capacity, which can be sent over
any channel or medium (wireless, coax, twister pair, fiber etc.).
C= B* Log2(1+S/N)
Where:
C is the channel capacity in bits per second (or maximum rate of data)
B is the bandwidth in Hz available for data transmission
S is the received signal power
N is the total channel noise power across bandwidth B
What this says is that higher the signal-to-noise (SNR) ratio and more
the channel bandwidth, the higher the possible data rate. This equation sets
the theoretical upper limit on data rate, which of course is not fully achieved
in practice.
It does not make any limitation on how low the achievable error rate will be.
That is dependent on the coding method used.
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.
Here is a graph showing the relationship between C/B and S/N (in dB):
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Examples
Here are two examples of the use of Shannon's Theorem.
Modem
For a typical telephone line with a signal-to-noise ratio of 30dB and an audio
bandwidth of 3kHz, we get a maximum data rate of:
C= 3000 * log2(1001)1
which is a little less than 30 kbps.
Satellite TV Channel
For a satellite TV channel with a signal-to noise ratio of 20 dB and a video
bandwidth of 10MHz, we get a maximum data rate of:
C= 10000000* log2(101)
which is about 66 Mbps.
1
Review Logarithm Base Change where (Logb a=log10 a/log10 b)
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Increase the throughput of wireless Channel Methodologies
According to Shannon Formula shown above, there are many technologies
to expand the channels’ capacity and then increase the system throughput,
and the designers of new generation of mobile network (like 4G and 5G)
were not far from these methodologies as roadmap for developing such
networks.
From these technologies: OFDMA, Single Carrier FDMA, MC-CDMA and
Multiple input and multiple output.
Multiple-Input and Multiple-Output Systems
Multiple-input and multiple-output (MIMO) is an extremely important
development for wireless communications systems. The transmitter and
receiver have multiple antennas, as shown below MIMO is an enhanced
wireless communication technique that has been used
in many commercial systems, such as Wi-Fi (802.11n and beyond) and LongTerm Evolution (LTE). Because multipath propagation causes Rayleigh
fading, which will affect a signal transmitted and this will have an impact on
the SNR.
MIMO offers diversity, and this principle provides many versions of the
signal transmitted to the receiver. If each transmitted signal is affected in a
different way by multipath, then the probability of the signals being affected
in the same way is reduced. Therefore, diversity allows a reduction in bad
fading. For example, if the signal is transmitted from four transmitters, each
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signal is affected by multipath in a different way and therefore the receiver
is able to reproduce the original signal.
There are different types of diversity that could be employed. The first type
is ‘time diversity’, where the signal may be transmitted at different times,
while the second type is ‘frequency diversity’, where the signal transmitted
may be transmitted at different frequencies, which is achieved by
transmitting on different sub-channels for the case of OFDM. The third type
of diversity is ‘space diversity’, which is the basis of MIMO, in which the
signal is transmitted in different directions and actually takes advantage of
multipath.
MIMO systems have more than one antenna at the transmitter and receiver
side, and this allows different paths for the signals, so that each signal path
is affected in a different way. One of the main advantages of MIMO systems
is that the system can increase the channel capacity significantly while still
obeying Shannon’s law. By increasing the number of antennas on both sides,
it is possible to increase the information throughput of the wireless channel
linearly.
This makes MIMO technology essential in many communications systems, to
improve their capacity.
Shannon’s law states that there is a limit to the rate at which data can be
transmitted in a channel in the presence of noise. MIMO offers a method of
increasing data rates beyond what is achievable in a regular wireless
channel; therefore, exploring this law is important.
Shannon’s law defines the maximum rate of which error-free data can be
transmitted over a certain bandwidth with the presence of various noise
signals. From this formula it can be seen that the limit of the data rate that
can be transmitted depends on the bandwidth available and the SNR of the
received signal. From these limits, it can be seen that a decision should be
taken with regards to the manner of transmission.
The modulation scheme used is a major factor determining the capacity;
using higher-level modulation schemes, increases the capacity but this
would require a better SNR. Thus, a balance is required between the data
rate and the acceptable error rate, SNR and transmitter power. Even though
this option of changing the modulation scheme is possible, it is not easy to
achieve and could cost more in practice. Therefore, other methods should
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be utilized to increase the capacity of the channel, which brings us to MIMO
technology that offers a capacity increase. To allow the enhanced
throughput capability, MIMO employs a set of multiple antennas.
MIMO wireless systems use a matrix mathematical approach to achieve the
advantage of the additional throughput. Data streams t1, t2, … tn are
transmitted from antennas 1, 2,…n.
There are various propagating paths that can be used, and each path has its
own unique channel characteristics. To enable the receiver to discriminate
between the different signal paths it is necessary to use a variable to
represent the different channel characteristics. This variable can be
represented by hij, which is the channel characteristic from transmitter
antenna i to receiver antenna j. To explain further, assume we have three
antennas in the transmitter and receiver side, and then we are able to set up
a matrix as follows:
r1= h11t1 + h21t2+h31t3
r2=h12t1+h22t2+h32t3
r3=h13t1+h23t2+h33t3
where r1 is the signal received at antenna 1, r2 is the signal received at antenna 2,
and so forth.
This can be represented in a matrix form as:
R=H⋅T
To restore the transmitted information at the receiving terminal, it is important to
include many steps of digital signal processing. The first step is that the system
decoder has to predict the specific channel characteristic hij and create the
channel transfer matrix. Once the channel estimations are completed, the matrix
[H] could be produced and that can be used to reconstruct the transmitted
information streams by multiplying the received vector by the inverse of the
estimated matrix:
T = H−1 ⋅ R
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This represents the basis of MIMO wireless systems, but in practice the situation
is more complicated as the signal propagation is not simple. In addition, each
variable in the equation consists of a continuous information stream, which
means that it is not constant for a long time.
References
Michael Parker ,Digital signal processing everything you need to know to get
started/ p 145
Sasan Adibi, Fourth-generation Wireless Networks_ Applications and Innovations
/p 155
Ian Robertson, Microwave and Millimetre-Wave Design for Wireless
Communications/page 486.
L L Peterson and B S Davie, Computer Networks: a systems approach (Morgan
Kaufmann), 1996. ISBN: 1-55860-368-9 (Paperback ISBN: 1-55860-404-9 ) pp 9495.
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