Know Your Eb/N0 - Part 2

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

Signal Constellations

Often, particularly in the specification of signal sets used in modems, we rely on the notion of an
orthonormal basis. A set \Phi=\{\phi_1(t), \phi_2(t), \cdots, \phi_N(t)\} is an orthonormal basis if

\langle\phi_i(t),\phi_j(t)\rangle=\delta_{ij}:=\begin{cases} 1               & \text{if} \: i=j; \\0   & \text{otherwise.}\end{cases}

 
The linear span L(\Phi) of \Phi consists of the space of all possible linear combinations of the elements
of \Phi; we call this the signal space spanned by \Phi. Given any signal x(t) in the signal space L(\Phi), we can express x(t) uniquely as a linear combination of the elements of \Phi, i.e.,

 x(t) = x_1\phi_1(t) + x_2\phi_2(t) + \cdots + x_N\phi_n(t),

 
where x = (x_1, x_2,\cdots,x_N) forms a vector of signal space coordinates for the signal x(t). Since
there are N basis vectors, and every signal is defined uniquely by N coordinate values, the space L(\Phi) is said to be N-dimensional.
It is often convenient to manipulate the coordinates of x(t), rather than work with x(t) directly.
For example, if x(t) has coordinate vector x = (x_1,\cdots,x_N), then

\begin{eqnarray*}
\|x(t)\|^2 & = & \langle x(t), x(t) \rangle \\
& = & \langle \sum_{i=1}^N x_i \phi_i(t), \sum_{j=1}^N x_j \phi_j(t)\rangle \\
& = & \sum_{i=1}^N \sum_{j=1}^N x_i x_j^* \langle \phi_i \phi_j\rangle \\
& = & \sum_{i=1}^N x_i2 \\
& = & \|x\|^2
\end{eqnarray*}

 
where we have used the fact that \langle\phi_i,\phi_j\rangle = \delta_{ij}. Likewise, the squared Euclidean distance between x(t) and y(t) is equal to d^2(x, y), where x and y are the coordinate vectors corresponding to x(t) and y(t), respectively.
A finite subset of an N-dimensional signal space is often referred to as an N-dimensional signal constellation. Modems typically use 1- or 2-dimensional signal constellations, though higher dimensional signal constellations are not unheard of, and indeed, most coding schemes can be viewed as defining a higher-dimensional constellation. Two-dimensional constellations are usually referred to as having two “subchannels:” the in-phase or “I” channel and the quadrature or “Q” channel. These are often implemented with two orthonormal basis functions

\phi_1(t) = p(t) \text{cos} \:2\pi f_ct \quad \text{and} \quad \phi_2(t) = p(t) \text{sin} 2\pi f_ct

 
where f_c is a carrier frequency, and p(t) is an appropriate pulse shaping function chosen to make \phi_1(t) and \phi_2(t) orthogonal (or nearly so).

Various signal constellations are shown in Fig. 2, where each black dot represents an element of the signal constellation with the corresponding signal space coordinates.
PAM and QAM
Figure 2: Various pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM) signal constellations: (a) 2-PAM (or BPSK), (b) 4-PAM, (c) 4-QAM (or QPSK) (d) 16-QAM.

The Receiver

Suppose we have an orthonormal basis \Phi = \{\phi_1,\cdots,\phi_N\} for an N-dimensional signal space. A
sufficient statistic for the detection in AWGN of signals drawn from this signal space is generated by the “front end” shown in Fig. 3.
    Suppose that a signal x(t) with coordinate vector x = (x_1,\cdots,x_N) is transmitted, and r(t) = x(t) + n(t) is received, where n(t) is a sample function from a zero-mean white Gaussian noise process of two-sided power spectral density N_0/2. The receiver front end produces the vector
(r_1, r_2,\cdots, r_N), where

\begin{eqnarray*}
r_i& = & \langle r(t), \phi_i(t) \rangle \\
& = & \langle x(t)+n(t), \phi_i(t) \rangle \\
& = & \langle x(t),\phi_i(t)\rangle + \langle n(t),\phi_i(t)\rangle \\
& = & x_i + n_i
\end{eqnarray*}

 
receiver front end
Figure 3: A receiver “front end” for detection in AWGN of signals drawn from a signal space with orthonormal basis \{\phi_1,\cdots,\phi_N\}.
Here n_i is a zero-mean Gaussian random variable with variance \sigma^2 = N_0/2. For i = j, observe that n_i is independent of n_j.
    In other words, when vector x is transmitted, vector r = x+n is received, where n = (n_1,\cdots,n_N) is a vector of independent zero-mean Gaussian random variables, each having variance \sigma^2 = N_0/2. Thus we obtain the following important normalization.

Assuming the receiver of Fig. 3 is used, the Gaussian noise variance per signal space dimension is \sigma^2 = N_0/2.

In N dimensions, the total expected noise energy per transmitted symbol is, therefore, N N_0/2.

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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