Know Your Eb/N0 - Part 3

This is the last part of our three-part series that explains the meaning of E_b/N_0 in digital communication context.

Calculating E_b/N_0

Having understood how Gaussian noise comes to affect transmitted symbols, it remains to define an appropriate average signal energy, and hence a signal-to-noise ratio (SNR). A commonly used definition of SNR is E_b/N_0, where E_b is the average received energy per transmitted information bit, and N_0 is twice the two-sided power spectral density of the AWGN.

In digital transmission, the transmitter usually takes in information bits, encoding them (for the purposes of error detection and correction) with an encoder to produce a proportionate number of channel bits. The constant of proportionality, i.e., the ratio of the number of information bits to channel bits, is referred to as the encoder rate, and is denoted R. (For example, if every information bit on average produces two channel bits, the encoder rate is 1/2.)

The channel bits are then mapped (perhaps after interleaving or some other operation) to a signal constellation, producing a proportionate number of transmitted symbols. The constant of proportionality for a constellation of size |\Omega| is \log_2|\Omega| channel bits per symbol.

The overall transmission efficiency (or rate) is therefore given by

information bits per symbol.

 
(One can check that the units are correct: we have information bits per channel bit multiplied by channel bits per symbol, resulting in information bits per symbol.)

If the modem is capable of transmitting at a symbol rate of B symbols per second^1, then the overall rate of information bit transmission is

information bits per symbol.

 
Suppose that the transmitter selects from among the points of an N-dimensional constellation \Omega with equal probability. The average symbol energy in N dimensions is given by

E_s(\Omega)=\frac{1}{\Omega}\sum_{x \in \Omega} ||x||^2

 
Assuming energy is measured in Joules, E_s(\Omega) can be thought of as having units of Joules per symbol.

It can be shown that the average symbol energy of a symmetric regular M-PAM signal constellation with minimum squared Euclidean distance d^2_{min} min is given by

E_s=\frac{d^2_{min}(M^2 - 1)}{12}

 
Similarly, a square M^2-QAM constellation (which is really just two M-PAM systems in quadrature) has average symbol energy

E_s=\frac{d^2_{min}(M^2 - 1)}{6}

 
Now, if the transmission rate is R\log_2|\Omega| bits per symbol, then the average information bit energy is given by

E_b=\frac{E_s}{R\log_2|\Omega|}

 
measured in Joules per information bit. (Again, one can verify that the units are correct: we have Joules per symbol divided by information bits per symbol, resulting in Joules per information bit.)
Now, knowing that the noise variance per signal space dimension is \sigma^2 = N_0/2, we may calculate E_b/N_0.

In summary,

\frac{E_b}{N_0}=\frac{E_s}{2R\sigma^2\log_2|\Omega|}

where
- \Omega is an N-dimensional signal constellation,
- E_s is the average energy (in N-dimensions) of the points of \Omega,
- \sigma^2 is the Gaussian noise variance per signal dimension,
- R is the code rate.

 

Some Examples

1. A system uses a convolutional code of rate R, combined with 2-PAM constellation \Omega = {−1, 1}. The noise variance per dimension is 2. What is E_b/N_0?
Solution: The noise variance per dimension is \sigma^2 = N_0/2; hence N_0 = 2\sigma^2. Each symbol has unit energy, i.e., E_s = 1, and each symbol conveys \log_2 2 = 1 bit, so the energy per information bit is given by E_b = 1/R. It follows that

\frac{E_b}{N_0}=\frac{1}{2R\sigma^2}

 

2. The modulation of the previous example is modified to the 4-QAM constellation

\Omega = {(1, 1), (1,-1), (-1, 1), (-1,-1)}.

 
The noise variance per dimension is 2. What is E_b/N_0?
Solution: As above, N_0 = 2\sigma^2. Each symbol has energy E_s = 2. We have

E_b=\frac{E_s}{R\log|\Omega|}=\frac{2}{2R}=\frac{1}{R}.

 
It follows that

\frac{E_b}{N_0}=\frac{1}{2R\sigma^2}.

 

3. Suppose the constellation of the previous example is modified to \Omega = {(1, 0), (-1, 0), (0, 1), (0,-1)}. Now what is E_b/N_0?
Solution: Now E_s = 1, so E_b = 1/(2R), and hence

\frac{E_b}{N_0}=\frac{1}{4R\sigma^2}.

 

4. The output of a rate 5/6 binary encoder is mapped to a 64-QAM constellation with d^2_{min} = 4. The noise variance per dimension is \sigma^2. What is E_b/N_0?
Solution: For M^2-QAM (here M = 8), E_s = d^2_{min}(M^2 - 1)/6 = 4(63)/6 = 42. We have

\frac{E_b}{N_0}=\frac{E_s}{2R\sigma^2\log_2|\Omega|}=\frac{42}{2(5/6)6\sigma^2}=\frac{4.2}{\sigma^2}

 

^1 For example, an ideal QAM modem operating in a bandwidth W can send up to W symbols per second without intersymbol interference.

References:
[1] Bernard Sklar, Digital Communications: Fundamentals and Applications, Prentice Hall, 2 edition, January 2001.
[2] John Proakis, and Masoud Salehi, Digital Communications, McGraw-Hill, 5th edition, November 2007.
[3] John R. Barry, Edward A. Lee, and David G. Messerschmitt, Digital Communication, Springer, 3rd edition, September 2003.

About the author

Frank KschischangFrank R. Kschischang received the B.A.Sc. degree (with honors) from the University of British Columbia, Vancouver, BC, Canada, in 1985 and the M.A.Sc. and Ph.D. degrees from the University of Toronto, Toronto, ON, Canada, in 1988 and 1991, respectively, all in electrical engineering. He is a Professor of electrical and computer engineering at the University of Toronto, where he has been a faculty member since 1991. During 1997–98, he was a visiting scientist at MIT, Cambridge, MA and in 2005 he was a visiting professor at the ETH, Zurich.
His research interests are focused primarily on the area of channel coding techniques, applied to wireline, wireless and optical communication systems and networks. In 1999, he was a recipient of the Ontario Premier’s Excellence Research Award and in 2001 (renewed in 2008) he was awarded the Tier I Canada Research Chair in Communication Algorithms at the University of Toronto. In 2010, he was awarded the Killam Research Fellowship by the Canada Council for the Arts. Jointly with Ralf Koetter, he received the 2010 Communications Society and Information Theory Society Joint Paper Award. He is a Fellow of IEEE and of the Engineering Institute of Canada. During 1997–2000, he served as an Associate Editor for Coding Theory for the IEEE TRANSACTIONS ON INFORMATION THEORY. He also served as technical program co-chair for the 2004 IEEE International Symposium on Information Theory (ISIT), Chicago, and as general co-chair for ISIT 2008, Toronto. He served as the 2010 President of the IEEE Information Theory Society. He is currently serving as Editor-in-Chief of the IEEE Transactions on Information Theory.

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