# 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.

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.

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*}

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.