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Shannon Capacity Explained

The document summarizes Shannon's channel capacity theorem, which establishes fundamental limits on communication. The key points are: 1) Shannon's theorem defines the maximum data rate (channel capacity C) that can be transmitted over a channel with a given bandwidth B in the presence of noise. 2) The capacity is determined by the signal-to-noise ratio (SNR) and is given by C = B log2(1+S/N). 3) There exists a minimum energy per bit (Eb/N0) required for error-free communication, known as the Shannon limit, below which communication is impossible regardless of bandwidth.

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368 views15 pages

Shannon Capacity Explained

The document summarizes Shannon's channel capacity theorem, which establishes fundamental limits on communication. The key points are: 1) Shannon's theorem defines the maximum data rate (channel capacity C) that can be transmitted over a channel with a given bandwidth B in the presence of noise. 2) The capacity is determined by the signal-to-noise ratio (SNR) and is given by C = B log2(1+S/N). 3) There exists a minimum energy per bit (Eb/N0) required for error-free communication, known as the Shannon limit, below which communication is impossible regardless of bandwidth.

Uploaded by

vijeta diwan
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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1

SHANNON CHANNEL CAPACITY THEOREMl


Shannon Channel model
Noiseless Channels and Nyquist Theorem

For a noiseless channel, Nyquist theorem gives the relationship


between the channel bandwidth and maximum data rate that can
be transmitted over this channel.
Nyquist Theorem

C  2B log 22 m
C m
C: channel capacity (bps)
B: RF bandwidth
m: number of finite states in a symbol of transmitted signal

Example: A noiseless channel with 3kHz bandwidth can only transmit


a maximum of 6Kbps if the symbols are binary symbols.

4
5
6
Shannon limit …

• Shannon theorem puts a limit on transmission


data rate, not on error probability:

– Theoretically possible to transmit information at any


rate Rb , where Rb  C with an arbitrary small error
probability by using a sufficiently complicated coding
scheme.

– For an information rate Rb > C , it is not possible to find


a code that can achieve an arbitrary small error
probability.

7
Shannon limit …

Unattainable
region
C/W [bits/s/Hz]

Practical region

8
SNR [dB]
9
Shannon Limit

10
11
Shannon limit …

 S
C  W log 2 1  
 N C  Eb C 
 log 2 1  
S  EbC

W  N0 W 
 N  N 0W
C
As W   or  0, we get :
W
Shannon limit
Eb 1
  0.693  1.6 [dB]
N0 log 2 e

– There exists a limiting value of Eb / N 0 below which there can be no error-


free communication at any information rate.

– By increasing the bandwidth alone, the capacity cannot be increased to


any desired value. 12
Shannon limit …

W/C [Hz/bits/s] Practical region

Unattainable
region

-1.6 [dB] Eb / N 0 [dB] 13


Bandwidth efficiency plane
R>C
Unattainable region M=256
M=64
R=C
M=16
M=8
R/W [bits/s/Hz]

M=4
Bandwidth limited
M=2

M=4 M=2 R<C


M=8 Practical region
M=16

Shannon limit MPSK


Power limited
MQAM PB  105
MFSK

Eb / N 0 [dB] 14
Shannon capacity formula
• Shannon capacity formula – establishes Minimum energy per bit normalized to noise
fundamental limits on communication power density that is required for a given
spectrum utilization
• In the case of AWGN satellite channel

 S
C  B  log 2 1   Eb  Eb  2g  1
 N  min   
N0  N0  g
C – capacity of the channel in bits/sec
B – bandwidth of the channel in Hz
S/N – signal to noise ratio (linear) Note: g is the fundamental
measure of spectrum utilization.
Define g = R/B - bandwidth utilization in bps/Hz, Ultimate goal of every wireless
where R is the information rate in bps. communication system is to
provide largest g for a given set
C  E R of constraints.
g  log 2 1  b 
B  N0 B 
 E 
g  log 2 1  b  g 
Page 15  N0 

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