Ferkans — Interactive Telecom Tutor

From Diversity to Coding in Space and Time

In Chapter 10 we saw that the Alamouti scheme extracts full transmit diversity from a $2 \times N_r$ system with a remarkably simple linear decoder. This section develops the general theory of space-time codes (STCs): how to design codeword matrices that jointly exploit spatial and temporal dimensions to maximise diversity and coding gain over MIMO fading channels.

Definition:
Space-Time Code

A space-time code (STC) for an $N_t$ -antenna transmitter over $T$ symbol periods is a finite set $\mathcal{C}$ of $N_t \times T$ complex matrices:

$\mathcal{C} = \{\mathbf{C}_1, \mathbf{C}_2, \ldots, \mathbf{C}_L\}$

During each codeword transmission, the $i$ -th row of $\mathbf{C}_\ell$ is sent from antenna $i$ , and the $t$ -th column is sent during symbol period $t$ . The received signal at the $N_r$ receive antennas is:

$\mathbf{Y} = \mathbf{H}\mathbf{C}_\ell + \mathbf{N}$

where $\mathbf{Y}$ is $N_r \times T$ , $\mathbf{H}$ is $N_r \times N_t$ , and $\mathbf{N}$ is the $N_r \times T$ noise matrix with i.i.d. $\mathcal{CN}(0, \sigma^2)$ entries.

The code rate is $R = \frac{\log_2 L}{T}$ bits per channel use. The Alamouti code uses $N_t = 2$ , $T = 2$ , and transmits 2 symbols in 2 time slots, achieving rate $R = 1$ symbol per channel use.

,

Definition:
Pairwise Error Probability (PEP)

The pairwise error probability (PEP) is the probability that the ML decoder selects codeword $\mathbf{C}_j$ when $\mathbf{C}_i$ was transmitted. For a Rayleigh fading channel with $N_r$ receive antennas, the PEP averaged over the channel is upper-bounded at high SNR by:

$P(\mathbf{C}_i \to \mathbf{C}_j) \leq \left(\prod_{k=1}^{r} \lambda_k\right)^{-N_r} \left(\frac{\text{SNR}}{4}\right)^{-rN_r}$

where $\lambda_1, \ldots, \lambda_r$ are the nonzero eigenvalues of $\Delta\mathbf{C}\,\Delta\mathbf{C}^H$ with difference matrix $\Delta\mathbf{C} = \mathbf{C}_i - \mathbf{C}_j$ , and $r = \text{rank}(\Delta\mathbf{C})$ .

Theorem: Rank and Determinant Criteria for Space-Time Code Design

To minimise the pairwise error probability at high SNR for a MIMO system with $N_r$ receive antennas:

Rank criterion (diversity gain): For every pair of distinct codewords $\mathbf{C}_i \neq \mathbf{C}_j$ in $\mathcal{C}$ , the difference matrix $\Delta\mathbf{C} = \mathbf{C}_i - \mathbf{C}_j$ should have rank $r$ as large as possible. The diversity order is $d = r \cdot N_r$ . Maximum diversity requires $r = \min(N_t, T)$ for all codeword pairs.

Determinant criterion (coding gain): Subject to full rank, the minimum determinant

$\delta_{\min} = \min_{\mathbf{C}_i \neq \mathbf{C}_j} \det(\Delta\mathbf{C}\,\Delta\mathbf{C}^H) = \min_{\mathbf{C}_i \neq \mathbf{C}_j} \prod_{k=1}^{r} \lambda_k$

should be maximised. This quantity determines the coding gain of the space-time code.

The rank criterion ensures that the error probability decays as $\text{SNR}^{-rN_r}$ : more nonzero eigenvalues mean steeper decay (higher diversity). The determinant criterion controls the vertical shift of the error curve: a larger product of eigenvalues means better coding gain. Think of rank as the slope and determinant as the intercept of the PEP curve on a log-log scale.

Proof

Conditioned PEP

Conditioned on a fixed channel realisation $\mathbf{H}$ , the ML decoder compares $\|\mathbf{Y} - \mathbf{H}\mathbf{C}_i\|_F^2$ to $\|\mathbf{Y} - \mathbf{H}\mathbf{C}_j\|_F^2$ . The conditional PEP is:

$P(\mathbf{C}_i \to \mathbf{C}_j \mid \mathbf{H}) = Q\!\left(\sqrt{\frac{\|\mathbf{H}\Delta\mathbf{C}\|_F^2}{2\sigma^2}}\right)$

Using the Chernoff bound $Q(x) \leq \frac{1}{2}e^{-x^2/2}$ :

$P(\mathbf{C}_i \to \mathbf{C}_j \mid \mathbf{H}) \leq \frac{1}{2}\exp\!\left(-\frac{\|\mathbf{H}\Delta\mathbf{C}\|_F^2}{4\sigma^2}\right)$

Averaging over Rayleigh fading

Let $\mathbf{A} = \Delta\mathbf{C}\,\Delta\mathbf{C}^H$ with eigendecomposition $\mathbf{A} = \mathbf{U}\boldsymbol{\Lambda}\mathbf{U}^H$ . Since $\mathbf{H}$ has i.i.d. $\mathcal{CN}(0,1)$ entries, $\tilde{\mathbf{H}} = \mathbf{H}\mathbf{U}$ has the same distribution. Then $\|\mathbf{H}\Delta\mathbf{C}\|_F^2 = \sum_{k=1}^{r}\lambda_k \|\tilde{\mathbf{h}}_k\|^2$ where $\tilde{\mathbf{h}}_k$ are independent $\mathcal{CN}(\mathbf{0}, \mathbf{I}_{N_r})$ vectors.

The MGF of $\|\tilde{\mathbf{h}}_k\|^2 \sim \text{Gamma}(N_r, 1)$ gives:

$\mathbb{E}\!\left[e^{-s\|\tilde{\mathbf{h}}_k\|^2}\right] = \frac{1}{(1+s)^{N_r}}$

Averaging the Chernoff bound with $s = \lambda_k/(4\sigma^2)$ :

$P(\mathbf{C}_i \to \mathbf{C}_j) \leq \frac{1}{2} \prod_{k=1}^{r} \frac{1}{\left(1 + \frac{\lambda_k \cdot \text{SNR}}{4N_t}\right)^{N_r}}$

High-SNR approximation

At high SNR, $1 + \frac{\lambda_k \cdot \text{SNR}}{4N_t} \approx \frac{\lambda_k \cdot \text{SNR}}{4N_t}$ , yielding:

$P(\mathbf{C}_i \to \mathbf{C}_j) \leq \frac{1}{2} \left(\prod_{k=1}^{r}\lambda_k\right)^{-N_r} \left(\frac{\text{SNR}}{4N_t}\right)^{-rN_r}$

The exponent $rN_r$ is the diversity order (rank criterion), and $\left(\prod_{k=1}^{r}\lambda_k\right)^{1/r} = (\det \mathbf{A})^{1/r}$ is the coding gain (determinant criterion). $\blacksquare$

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Definition:
Orthogonal Space-Time Block Code (OSTBC)

An orthogonal STBC (OSTBC) is a space-time block code whose codeword matrix $\mathbf{C}$ satisfies the orthogonality condition:

$\mathbf{C}\mathbf{C}^H = c \left(\sum_{k=1}^{K}|x_k|^2\right) \mathbf{I}_{N_t}$

where $x_1, \ldots, x_K$ are the $K$ information symbols encoded in $\mathbf{C}$ , and $c$ is a positive constant.

This orthogonality enables single-symbol ML decoding: the joint ML detection of $K$ symbols decouples into $K$ independent scalar detections, each equivalent to MRC over $N_t N_r$ diversity branches.

The Alamouti code (Chapter 10) is the only complex OSTBC that achieves rate $R = 1$ for $N_t = 2$ . For $N_t > 2$ with complex constellations, the maximum rate of an OSTBC is strictly less than 1. This is the fundamental rate limitation of orthogonal designs.

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Definition:
Rate Limitation of Complex OSTBCs

For complex signal constellations and $N_t$ transmit antennas, the maximum rate (in symbols per channel use) achievable by an OSTBC is:

$R_{\max} = \begin{cases} 1 & N_t = 2 \text{ (Alamouti)} \\ 3/4 & N_t = 3, 4 \\ \text{decreasing} & N_t > 4 \end{cases}$

For real constellations (e.g., BPSK, PAM), rate-1 OSTBCs exist for any $N_t$ as the Hurwitz-Radon construction provides sufficient orthogonal designs.

Diversity-Multiplexing Tradeoff

Animated visualization of the optimal DMT curve

d^*(r) = (N_t - r)(N_r - r)

for a

2 \times 2

MIMO channel. The curve shows how space-time codes (full diversity,

r = 0

) and spatial multiplexing (full rate,

d = 0

) are endpoints of a continuous tradeoff, with the Alamouti code achieving

(r, d) = (1, 1)

.

The piecewise-linear DMT curve for a

2 \times 2

channel. The achievable region lies on or below the curve.

Space-Time Codeword Transmission

Watch how the Alamouti

2 \times 2

codeword matrix maps symbols to antennas and time slots, achieving full diversity with orthogonal structure that enables single-symbol ML decoding.

The Alamouti codeword matrix: rows correspond to antennas, columns to time slots. Orthogonality ensures

\mathbf{C}\mathbf{C}^H \propto \mathbf{I}

.

Pairwise Error Probability vs. SNR

Visualise how the PEP bound decays with SNR for different ranks (diversity orders) and determinants (coding gains) of the codeword difference matrix.

Parameters

N_r

(receive antennas)2

Rank

r

of

\Delta\mathbf{C}

2

\det(\Delta\mathbf{C}\Delta\mathbf{C}^H)

[dB]0

Example: Verifying the Rank/Determinant Criteria for the Alamouti Code

The Alamouti codeword matrix for symbols $x_1, x_2$ is:

$\mathbf{C} = \begin{bmatrix} x_1 & -x_2^* \\ x_2 & x_1^* \end{bmatrix}$

Show that the Alamouti code achieves full rank ( $r = 2$ ) for all distinct codeword pairs and compute the coding gain.

Solution

Codeword difference matrix

Let $\mathbf{C}_i$ encode $(x_1, x_2)$ and $\mathbf{C}_j$ encode $(y_1, y_2)$ . The difference is:

$\Delta\mathbf{C} = \begin{bmatrix} x_1 - y_1 & -(x_2 - y_2)^* \\ x_2 - y_2 & (x_1 - y_1)^* \end{bmatrix} = \begin{bmatrix} d_1 & -d_2^* \\ d_2 & d_1^* \end{bmatrix}$

where $d_k = x_k - y_k$ .

Computing the rank

$\Delta\mathbf{C}\,\Delta\mathbf{C}^H = \begin{bmatrix} |d_1|^2 + |d_2|^2 & 0 \\ 0 & |d_1|^2 + |d_2|^2 \end{bmatrix} = (|d_1|^2 + |d_2|^2)\,\mathbf{I}_2$ $Since$ \mathbf{C}_i \neq \mathbf{C}_j $implies$ (d_1, d_2) \neq (0, 0) $, we have$ |d_1|^2 + |d_2|^2 > 0 $, so$ \text{rank}(\Delta\mathbf{C}) = 2 = N_t $. The Alamouti code achieves **full diversity**$ d = 2N_r$.

Coding gain

$\det(\Delta\mathbf{C}\,\Delta\mathbf{C}^H) = (|d_1|^2 + |d_2|^2)^2KATEXPLACEHOLDER0END\delta_{\min} = \min_{(d_1,d_2) \neq (0,0)} (|d_1|^2 + |d_2|^2)^2 = d_{\min}^4$ $where$ d_{\min} $is the minimum Euclidean distance of the underlying scalar constellation. The coding gain is$ \delta_{\min}^{1/N_t} = d_{\min}^2 $.$ \blacksquare$

Quick Check

A space-time code for $N_t = 3$ transmit antennas has a codeword pair whose difference matrix $\Delta\mathbf{C}$ has rank 2. With $N_r = 2$ receive antennas, what is the diversity order contributed by this codeword pair?

2

3

4

6

Correction:

4

The diversity order is $r \times N_r = 2 \times 2 = 4$ . Full diversity would be $N_t \times N_r = 6$ , which requires $\text{rank}(\Delta\mathbf{C}) = 3$ for all codeword pairs.

Common Mistake: Confusing Code Rate with Diversity Order

Mistake:

Assuming that a higher-rate space-time code always provides the same diversity order. For example, believing that a rate-1 STC for $N_t = 4$ antennas achieves full diversity.

Correction:

There is a fundamental rate-diversity trade-off for space-time codes. For complex OSTBCs, achieving full diversity $N_t N_r$ limits the rate to $R \leq 3/4$ for $N_t > 2$ . Higher-rate codes (e.g., spatial multiplexing) sacrifice diversity order for throughput. The diversity-multiplexing trade-off (DMT) formalises this: at multiplexing gain $r_{\text{mux}}$ , the maximum diversity order is $d^*(r_{\text{mux}}) = (N_t - r_{\text{mux}})(N_r - r_{\text{mux}})$ .

Historical Note: The Birth of Space-Time Coding

1998

Space-time coding emerged from two parallel efforts in the late 1990s. Vahid Tarokh, Nambi Seshadri, and Robert Calderbank published the rank and determinant criteria in 1998, establishing the theoretical foundation for STC design. Independently, Siavash Alamouti at AT&T proposed his elegant $2 \times 1$ transmit diversity scheme in 1998, which became one of the most cited papers in wireless communications. Tarokh, Jafarkhani, and Calderbank subsequently generalised Alamouti's construction to orthogonal STBCs for arbitrary numbers of antennas, connecting the design problem to the classical theory of Hurwitz-Radon families of matrices.

,

Why This Matters: Space-Time Codes and Coded Modulation

The rank and determinant criteria developed here connect directly to the theory of coded modulation over MIMO channels. The CM book extends these ideas to bit-interleaved coded modulation (BICM) with space-time coding, quasi-orthogonal designs that trade some orthogonality for higher rates, and the algebraic number theory behind cyclic division algebra codes (including the DMT-optimal constructions by Elia/Kumar/Caire). Readers interested in the deep algebraic structure of space-time code design should consult the CM book's chapters on lattice codes and DMT-optimal constructions.

🎓CommIT Contribution(2006)

Explicit DMT-Optimal Space-Time Codes

P. Elia, K. R. Kumar, S. A. Pawar, P. V. Kumar, H.-F. Lu, G. Caire — IEEE Transactions on Information Theory

The diversity-multiplexing tradeoff (DMT), established by Zheng and Tse (2003), defines the optimal curve $d^*(r)$ relating diversity order $d$ to multiplexing gain $r$ for a MIMO channel. A key open question was whether explicit, constructable codes could achieve every point on this curve.

Elia, Kumar, Pawar, Kumar, and Lu (with connections to Caire's group) provided the first family of explicit algebraic space-time codes that achieve the optimal DMT for any $N_t \times N_r$ system. These codes use cyclic division algebra constructions and can be encoded/decoded with polynomial complexity.

dmtspace-time-codesalgebraic-codes

🎓CommIT Contribution(2003)

LAST Codes — Lattice Space-Time Codes Achieving the Optimal DMT

H. El Gamal, G. Caire, M. O. Damen — IEEE Transactions on Information Theory

El Gamal, Caire, and Damen showed that lattice-based space-time codes (LAST codes) paired with lattice decoding achieve the full DMT of MIMO channels. This result is significant because:

It provides a constructive proof that the DMT is achievable, not just an information-theoretic bound.
Lattice decoding (via sphere decoding) has polynomial average complexity, making the codes practically relevant.
The algebraic structure of lattices enables systematic code design, connecting MIMO coding to number theory.

This work was a foundational CommIT contribution that bridged information-theoretic DMT analysis with practical code construction.

dmtlattice-codesmimo-coding

Space-Time Code (STC)

A coding scheme that jointly encodes data across multiple transmit antennas and multiple time slots, represented by a codeword matrix $\mathbf{C} \in \mathbb{C}^{N_t \times T}$ .

Orthogonal Space-Time Block Code (OSTBC)

A space-time block code satisfying $\mathbf{C}\mathbf{C}^H \propto \mathbf{I}$ , enabling single-symbol ML decoding with full diversity.

Rank Criterion

The design rule that the codeword difference matrix $\Delta\mathbf{C} = \mathbf{C}_i - \mathbf{C}_j$ should have maximum rank for all distinct codeword pairs, ensuring the highest achievable diversity order.

Determinant Criterion

The design rule that $\det(\Delta\mathbf{C}\,\Delta\mathbf{C}^H)$ should be maximised over all codeword pairs, determining the coding gain of the space-time code.

Space-Time Codes