*This is the second part of our three-part series that explains the meaning of 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 is an orthonormal basis if

The linear span of consists of the space of all possible linear combinations of the elements

of ; we call this the signal space spanned by . Given any signal in the signal space , we can express uniquely as a linear combination of the elements of , i.e.,

where forms a vector of signal space coordinates for the signal . Since

there are basis vectors, and every signal is defined uniquely by coordinate values, the space is said to be -dimensional.

It is often convenient to manipulate the coordinates of , rather than work with directly.

For example, if has coordinate vector , then

\|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 . Likewise, the squared Euclidean distance between and is equal to , where and are the coordinate vectors corresponding to and , respectively.

A finite subset of an -dimensional signal space is often referred to as an -dimensional signal . 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

where is a carrier frequency, and is an appropriate pulse shaping function chosen to make and 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.

### The Receiver

Suppose we have an orthonormal basis for an -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 with coordinate vector is transmitted, and is received, where is a sample function from a zero-mean white Gaussian noise process of two-sided power spectral density . The receiver front end produces the vector

, where

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 .

Here is a zero-mean Gaussian random variable with variance . For , observe that is independent of .

In other words, when vector is transmitted, vector is received, where is a vector of independent zero-mean Gaussian random variables, each having variance . Thus we obtain the following important normalization.

In dimensions, the total expected noise energy per transmitted symbol is, therefore, .

**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.

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