Constellations

Dimensionality of the Signal Space

The dimensionality \( N \) of the signal space equals \( M \) (the number of input waveforms) if and only if all the waveforms are linearly independent. If one or more waveforms are linear combinations of others, the dimensionality is reduced.

Signal Space Representation (Constellation)

After constructing a set of orthonormal waveforms \( \{\phi_n(t)\} \), we can express the \( M \) signals \( \{s_m(t)\} \) as linear combinations of these waveforms. Specifically:

\[ s_m(t) = \sum_{n=1}^{N} s_{mn} \phi_n(t), \quad m = 1, 2, \ldots, M \]

Each signal \( s_m(t) \) can then be represented as a vector in the \( N \)-dimensional signal space:

\[ \vec{s}_m = [s_{m1}, s_{m2}, \ldots, s_{mN}]^{\sf T} \]

This vector corresponds to a point in the \( N \)-dimensional (real or complex) signal space, where the coordinates are \( \{s_{mn}, n = 1, 2, \ldots, N\} \).

Definition: Signal Space Representation

A set of \( M \) signals \( \{s_m(t)\}_{m=1}^M \) can be represented by a set of \( M \) vectors \( \{\vec{s}_m\}_{m=1}^M \) in an \( N \)-dimensional space, where \( N \leq M \).

This set of vectors is called the signal space representation, or the constellation, of the signal set \( \{s_m(t)\}_{m=1}^M \).

Real vs. Complex Signal Space

• If the original signals \( s_m(t) \) are real, their vector representations lie in \( \mathbb{R}^N \).

• If the signals are complex, their vector representations lie in \( \mathbb{C}^N \).

This representation allows signals to be visualized as points or vectors in the signal space, making it easier to analyze properties such as orthogonality, distance, and error performance.

Bandpass and Lowpass Orthonormal Basis

Bandpass Signal Representation

Consider a set of bandpass signal waveforms represented as:

\[ s_m(t) = \Re\{s_{m,lp}(t)e^{j2\pi f_0 t}\}, \quad m = 1, 2, \ldots, M \]

where:

• \( s_{m,lp}(t) \) denotes the lowpass equivalent signals.

• \( f_0 \) is the center frequency of the bandpass signal.

Recall that if two lowpass equivalent signals are orthogonal, their corresponding bandpass signals are also orthogonal.

Lowpass to Bandpass Basis Transformation

If \( \{\phi_{nl}(t), n = 1, \ldots, N\} \) is an orthonormal basis for the set of lowpass signals \( \{s_{m,lp}(t)\} \), we can transform this into a set of functions for the corresponding bandpass signals:

\[ \phi_n(t) = \sqrt{2} \Re\{\phi_{nl}(t)e^{j2\pi f_0 t}\}, \quad n = 1, 2, \ldots, N \]

The factor \( \sqrt{2} \) ensures that each \( \phi_n(t) \) has unit energy.

Limitations of the Bandpass Basis

However, this transformed set \( \{\phi_n(t)\} \) is not necessarily an orthonormal basis for expanding the entire set of signals \( \{s_m(t), m = 1, \ldots, M\} \). Specifically:

• There is no guarantee that \( \{\phi_n(t)\} \) forms a complete basis for the expansion of \( \{s_m(t)\} \).

• A complete basis must span the entire subspace of bandpass signals \( \{s_m(t)\} \).

The goal is to determine how to construct an orthonormal basis for representing bandpass signals using the orthonormal basis for their lowpass equivalents. This involves ensuring that the transformed basis \( \{\phi_n(t)\} \) spans the same signal space as \( \{s_m(t)\} \), thereby achieving a complete and orthonormal representation for the bandpass signals.