Ferkans — Interactive Telecom Tutor

The Simplest Non-Trivial Space-Time Code

The point is that Alamouti's code is the simplest non-trivial space-time code — and it achieves something remarkable: full diversity with linear decoding. Before Alamouti's 1998 paper, everyone assumed that achieving full diversity over a MIMO channel required joint maximum-likelihood detection over a multi-symbol codeword. Alamouti showed that for $n_t = 2$ transmit antennas, one specific orthogonal codeword structure allows the receiver to decouple the problem into two independent scalar decisions, one per transmitted symbol, each enjoying the full $2 n_r$ -fold diversity of the MIMO channel.

Why was this so surprising? The rank criterion of Chapter 10 (Thm. [?ch10:thm-rank-determinant]) says full diversity requires the error matrix $\boldsymbol{\Delta} = \mathbf{X} - \hat{\mathbf{X}}$ to have full row rank $n_t = 2$ for every codeword pair. Off-the-shelf constructions — replicating the same symbol on both antennas, or sending two independent symbols through a V-BLAST-style split — both fail this test. Alamouti's key insight was that by sending the matrix $\mathbf{X}_A = \begin{pmatrix} s_1 & -s_2^* \\ s_2 & s_1^* \end{pmatrix}$ over two successive channel uses, the columns of $\mathbf{X}_A$ are orthogonal in $\mathbb{C}^2$ for any choice of symbols $(s_1, s_2)$ . The error matrix $\boldsymbol{\Delta}$ inherits this orthogonality, has rank 2 automatically, and — via a two-line trick — lets the receiver split the detection into two matched-filter scalar decisions. Rate 1, full diversity, linear decoding. A 3 dB loss relative to the receive-diversity benchmark is the only price.

This section derives the Alamouti scheme from scratch, proves its full diversity via the orthogonality trick, and situates it next to the $1 \times n_r$ MRC reference. The orthogonality structure — dispersion matrices that satisfy a Hurwitz-Radon-Eckmann algebra — is the seed of the general OSTBC construction of §2.

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Definition:
Space-Time Codeword Matrix

A space-time codeword matrix over the MIMO channel with $n_t$ transmit antennas is an element $\mathbf{X} \in \mathbb{C}^{n_t \times T}$ , where $T$ is the block length in channel uses. Entry $X_{i,t}$ is the complex baseband symbol transmitted on antenna $i$ at time slot $t$ . The received block over a quasi-static MIMO channel is $\mathbf{Y} \;=\; \mathbf{H}\,\mathbf{X} \;+\; \mathbf{w}, \qquad \mathbf{Y} \in \mathbb{C}^{n_r \times T},$ with $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ constant over the block and $\mathbf{w}$ having i.i.d. $\mathcal{CN}(0, \sigma^2)$ entries.

A space-time block code (STBC) $\mathcal{C} \subset \mathbb{C}^{n_t \times T}$ is a finite codebook of such matrices. The rate of the code is $R_{\mathrm{STBC}} \;=\; \frac{\log_2 |\mathcal{C}|}{T} \text{ bits per channel use}.$ When the codebook carries $K$ complex information symbols drawn from a constellation $\mathcal{X}$ of size $M$ , the rate in symbols per channel use is $R = K/T$ ; the rate in bits per channel use is $R_{\mathrm{STBC}} = (K/T) \log_2 M$ .

The normalisation convention is that the total energy per channel use is $E_s$ independent of $n_t$ , so that each column of $\mathbf{X}$ has expected squared Frobenius norm $E_s$ . In particular, for the Alamouti code with QPSK input at average energy $E_s$ , each of the two transmit antennas radiates $E_s/2$ per channel use — the origin of the 3 dB power-splitting loss versus a single-antenna benchmark.

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Definition:
Alamouti Codeword Matrix

The Alamouti code is the STBC for $n_t = 2, T = 2, K = 2$ defined by the codeword matrix $\mathbf{X}_A(s_1, s_2) \;=\; \begin{pmatrix} s_1 & -s_2^* \\ s_2 & s_1^* \end{pmatrix} \;\in\; \mathbb{C}^{2\times 2},$ where $(s_1, s_2) \in \mathcal{X}^2$ are the two information symbols drawn from a unit-energy constellation $\mathcal{X}$ and the average transmit energy per channel use is $E_s = \mathbb{E}[|s_1|^2 + |s_2|^2]$ . Row $i$ of $\mathbf{X}_A$ is transmitted on antenna $i$ ; column $t$ is the simultaneous vector of antenna outputs at time slot $t$ .

Equivalently, over two channel uses the transmit sequence is:

Time slot	Antenna 1	Antenna 2
$t = 1$	$s_1$	$s_2$
$t = 2$	$-s_2^*$	$s_1^*$

The rate is $K/T = 2/2 = 1$ complex symbol per channel use.

The codeword has a remarkable structural property: for every $(s_1, s_2)$ , $\mathbf{X}_A \mathbf{X}_A^H \;=\; (|s_1|^2 + |s_2|^2)\,\mathbf{I}_2.$ The rows (equivalently, columns) are mutually orthogonal in $\mathbb{C}^2$ for any input. This is the single algebraic fact that makes every operational property of the Alamouti code — diversity, decoding, capacity — fall out.

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Definition:
Diversity Order of an STBC

The diversity order of an STBC $\mathcal{C}$ on the i.i.d. Rayleigh MIMO channel is $d(\mathcal{C}) \;=\; -\lim_{\text{SNR}\to\infty} \frac{\log P_e(\text{SNR})}{\log \text{SNR}},$ i.e., the exponent at which the codeword error probability decays with SNR. By the rank criterion of Ch. 10 (Thm. [?ch10:thm-rank-determinant]), $d(\mathcal{C}) \;=\; n_r \cdot \min_{\mathbf{X}\ne\hat{\mathbf{X}}} \mathrm{rank}(\mathbf{X} - \hat{\mathbf{X}}).$ An STBC is said to achieve full diversity if the minimum rank is $n_t$ , so that $d(\mathcal{C}) = n_t n_r$ — the maximum possible for the i.i.d. Rayleigh MIMO channel.

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Alamouti Scheme: $\mathbf{X}_A$ Encoding and Orthogonal Decoding

Animated walk-through of the Alamouti encoder and decoder. The video builds the

2\times 2

codeword matrix

\mathbf{X}_A

column by column, shows the two channel uses with the channel coefficients

(h_1, h_2)

multiplying each column, and then traces how the matched filter

\mathbf{H}^{H}

exploits the orthogonality

\mathbf{X}_A\mathbf{X}_A^H \propto \mathbf{I}

to decouple the received

2\times 2

block into two independent scalar channels for

s_1

and

s_2

.

Alamouti encoding and orthogonal decoding flow. Rate 1 symbol per channel use, diversity

2 n_r

, linear matched-filter ML decoder.

Theorem: Alamouti Achieves Full Diversity $2 n_r$ with Linear Decoding

Consider the Alamouti code $\mathbf{X}_A$ of Definition DAlamouti Codeword Matrix over a quasi-static i.i.d. Rayleigh MIMO channel with $n_t = 2, n_r \ge 1$ . Then:

(a) The ML decoder for $(s_1, s_2)$ reduces to two decoupled scalar decisions: for each $k \in \{1, 2\}$ the receiver forms a sufficient statistic $\tilde y_k \;=\; \left(\sum_{i,j} |h_{i,j}|^2\right) s_k \;+\; \tilde w_k,$ where $\tilde w_k$ is $\mathcal{CN}$ noise independent across $k$ .

(b) The effective per-symbol SNR is $\text{SNR}_{\mathrm{eff}} \;=\; \frac{\text{SNR}}{2}\sum_{i=1}^{n_r}\sum_{j=1}^{2} |h_{i,j}|^2,$ i.e., $2 n_r$ channel magnitudes sum in the effective SNR, at a half-of-total-power penalty.

(c) Consequently, the Alamouti code achieves diversity $d = 2 n_r$ — the full MIMO diversity order for $n_t = 2$ .

Alamouti's orthogonality means the two transmitted symbols live in orthogonal subspaces of the received signal space. Projecting onto those subspaces is exactly what a matched filter does, and the projections are noise-plus-scalar just as in the $1 \times n_r$ MRC receiver — with $2 n_r$ instead of $n_r$ diversity branches, at the cost of splitting the total transmit energy across two antennas. This is diversity-for-free in the sense that no bandwidth is spent (rate 1), and complexity-for-free in the sense that decoding is scalar.

Show Hint

Write the received matrix $\mathbf{Y} = \mathbf{H}\mathbf{X}_A + \mathbf{w}$ explicitly column by column for $n_r = 1$ .

Conjugate the second column and stack the two channel uses into a $2 \times 1$ vector: the effective channel becomes a matrix with orthogonal columns.

Multiply by the Hermitian transpose of the effective channel to decouple $s_1$ and $s_2$ .

Verify that the post-matched-filter noise is still white Gaussian (because unitary transformations preserve whiteness).

Proof

System model and column-wise expansion

Consider first $n_r = 1$ ; the $n_r > 1$ case is an identical argument in parallel over the $n_r$ receive antennas. Denote the channel row as $\mathbf{h} = (h_1, h_2)$ . Over the two channel uses of one Alamouti block, the received samples are: $y_1 \;=\; h_1 s_1 + h_2 s_2 + w_1,$ $y_2 \;=\; -h_1 s_2^* + h_2 s_1^* + w_2.$ Conjugating the second equation gives $y_2^* = -h_1^* s_2 + h_2^* s_1 + w_2^*$ . Stacking $(y_1, y_2^*)^T$ as a column vector: $\underbrace{\begin{pmatrix} y_1 \\ y_2^* \end{pmatrix}}_{=: \tilde{\mathbf{y}}} \;=\; \underbrace{\begin{pmatrix} h_1 & h_2 \\ h_2^* & -h_1^* \end{pmatrix}}_{=: \tilde{\mathbf{H}}} \begin{pmatrix} s_1 \\ s_2 \end{pmatrix} + \underbrace{\begin{pmatrix} w_1 \\ w_2^* \end{pmatrix}}_{=: \tilde{\mathbf{w}}}.$

The effective channel has orthogonal columns

Compute the Gramian: $\tilde{\mathbf{H}}^H \tilde{\mathbf{H}} \;=\; \begin{pmatrix} h_1^* & h_2 \\ h_2^* & -h_1 \end{pmatrix} \begin{pmatrix} h_1 & h_2 \\ h_2^* & -h_1^* \end{pmatrix} \;=\; (|h_1|^2 + |h_2|^2)\,\mathbf{I}_2.$ Notice that the columns of $\tilde{\mathbf{H}}$ are orthogonal in $\mathbb{C}^2$ and each has squared norm $|h_1|^2 + |h_2|^2$ . This is where Alamouti's orthogonality $\mathbf{X}_A \mathbf{X}_A^H \propto \mathbf{I}$ shows up after "folding" the two channel uses into one virtual $2\times 2$ channel.

Matched-filter decoupling

Apply the matched filter $\tilde{\mathbf{H}}^H$ to $\tilde{\mathbf{y}}$ : $\tilde{\mathbf{H}}^H \tilde{\mathbf{y}} \;=\; (|h_1|^2 + |h_2|^2)\begin{pmatrix} s_1 \\ s_2 \end{pmatrix} + \tilde{\mathbf{H}}^H \tilde{\mathbf{w}}.$ The post-filter noise is $\mathcal{CN}(0, (|h_1|^2 + |h_2|^2)\,\sigma^2 \mathbf{I}_2)$ — its covariance is a scalar multiple of $\mathbf{I}$ because $\tilde{\mathbf{H}}^H \tilde{\mathbf{H}}$ is, and because the original noise is $\mathcal{CN}(0, \sigma^2\mathbf{I}_2)$ . So the two components are independent and identically distributed — exactly the structure of two parallel scalar channels.

Per-symbol effective SNR and diversity

The SNR on each scalar decision is $\text{SNR}_{\mathrm{eff}} \;=\; \frac{(|h_1|^2 + |h_2|^2)^2 \cdot \mathbb{E}[|s_k|^2]}{(|h_1|^2 + |h_2|^2)\,\sigma^2} \;=\; \frac{(|h_1|^2 + |h_2|^2)\,E_s/2}{\sigma^2} \;=\; \frac{\text{SNR}}{2}(|h_1|^2 + |h_2|^2),$ using $E_s = \mathbb{E}[|s_1|^2] + \mathbb{E}[|s_2|^2]$ and $\mathbb{E}[|s_k|^2] = E_s/2$ .

For general $n_r$ , the argument repeats on every receive antenna and the $n_r$ matched-filter outputs are combined coherently, giving $\text{SNR}_{\mathrm{eff}} \;=\; \frac{\text{SNR}}{2} \sum_{i=1}^{n_r}\sum_{j=1}^{2} |h_{i,j}|^2.$ This is a sum of $2 n_r$ i.i.d. exponential random variables (Rayleigh squared magnitudes), so the tail of $1/\text{SNR}_{\mathrm{eff}}$ decays as $\text{SNR}^{-2 n_r}$ . Therefore the error probability on the scalar constellation decays as $\text{SNR}^{-2 n_r}$ — diversity $d = 2 n_r$ . $\blacksquare$

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Key Takeaway

Operationally, Alamouti gives the same diversity order as a $1\times (2 n_r)$ MRC receiver, but at a $3$ dB cost because the total transmit energy is split between two antennas. The benefit is that no channel knowledge is required at the transmitter — the code is the same matrix regardless of $(h_1, h_2)$ . Alamouti is therefore the canonical open-loop transmit-diversity scheme, and for more than a decade it was the only practical way to get transmit diversity over a MIMO link in a standards- compliant way (LTE, WiMAX, Wi-Fi).

Alamouti vs. MRC: Why the 3 dB?

Compare three schemes at the same total transmit power:

SISO (1 Tx, 1 Rx): effective SNR $= \text{SNR} |h|^2$ , diversity 1.
$1 \times 2$ MRC (1 Tx, 2 Rx): effective SNR $= \text{SNR} (|h_1|^2 + |h_2|^2)$ , diversity 2.
Alamouti $2 \times 1$ (2 Tx, 1 Rx): effective SNR $= (\text{SNR}/2) (|h_1|^2 + |h_2|^2)$ , diversity 2.

The Alamouti and MRC curves have the same slope (diversity 2) but Alamouti is shifted to the right by a factor of 2, i.e. 3 dB worse in SNR. The factor-of-2 comes from splitting the transmit energy: each transmit antenna radiates only $E_s/2$ , whereas the MRC receiver collects from an antenna that radiated the full $E_s$ . The diversity benefit is real; the 3 dB is the price for not knowing the channel at the transmitter (an MRT — maximum-ratio-transmission — beamformer would close the 3 dB, but it requires CSIT).

BER: Alamouti vs. MRC vs. SISO

Bit error rate of Alamouti $2\times 1$ (blue) versus the classic benchmarks: $1\times 2$ MRC (green) and SISO $1\times 1$ (red). All three curves use QPSK; the fading is i.i.d. Rayleigh; an average-energy constraint is applied. You should observe:

Alamouti and $1\times 2$ MRC both achieve diversity order 2 — their slopes match at high SNR.
Alamouti sits $\sim 3$ dB to the right of MRC — the power-splitting penalty of Thm. $2 n_r$ $2 n_{r}$ with Linear Decoding" data-ref-type="theorem">TAlamouti Achieves Full Diversity $2 n_r$ with Linear Decoding.
The SISO curve is one order of magnitude flatter (diversity 1) and quickly dominates the error floor.

This is the figure that made Alamouti famous: 3 dB is a small price for the freedom to put the diversity burden on the transmitter. In a cellular context that meant two antennas on the base station rather than every handset.

Parameters

Show SISO reference

Show

1\times 2

MRC

Alamouti Signal-Space Projections

The Alamouti codeword $\mathbf{X}_A(s_1, s_2) \in \mathbb{C}^{2\times 2}$ lives in a complex 4-dimensional signal space (8 real dimensions). This figure shows two 2-D projections of the codeword cloud under a noisy channel — the $(s_1, s_2)$ plane after matched-filter decoupling, with the noise clouds overlayed. Because of Thm. $2 n_r$ $2 n_{r}$ with Linear Decoding" data-ref-type="theorem">TAlamouti Achieves Full Diversity $2 n_r$ with Linear Decoding, the decoupled output has diagonal noise covariance and the decision regions look exactly like those of the underlying constellation — no rotated decision regions, no joint ML search, just two scalar decisions.

Vary the SNR and the constellation ( $E_s/N_0$ slider plus QPSK / 8-PSK / 16-QAM picker) to see the noise cloud shrink as $\text{SNR}$ grows; each projection is independent because the post-MF noise is white.

Parameters

Constellation

SNR [dB]12

Common Mistake: Alamouti Does Not Match MRC — It Matches MRC Minus 3 dB

Mistake:

Reading "Alamouti achieves the same diversity as $1\times 2$ MRC" as the claim that Alamouti is equivalent to MRC in BER. Plotting Alamouti on the same axes as $1\times 2$ MRC and concluding they overlap.

Correction:

Alamouti and $1\times 2$ MRC have the same slope at high SNR (diversity 2), but Alamouti is offset by exactly $10\log_{10} 2 \approx 3$ dB to the right. The reason is that Alamouti must split the total transmit energy across two antennas, each radiating $E_s/2$ , while the MRC receiver collects an antenna that radiated the full $E_s$ .

The only way to recover the 3 dB is transmit beamforming — which requires knowledge of the channel at the transmitter (CSIT). Alamouti is the best open-loop rate-1 transmit-diversity scheme for $n_t = 2$ , not the best transmit-diversity scheme unconditionally. When CSIT is available, an MRT (maximum-ratio-transmission) beamformer closes the 3 dB gap.

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Historical Note: Alamouti 1998: The Landmark Paper

1998

Siavash Alamouti, then at Cadence Design Systems, published "A simple transmit diversity technique for wireless communications" in IEEE JSAC vol. 16 (October 1998, pp. 1451–1458). The paper is only 8 pages long — remarkably short for a landmark — and it has two sections of substance: one that describes the $2\times 2$ codeword matrix, and one that derives the decoupled ML decoder. The punchline is the orthogonality identity $\mathbf{X}_A\mathbf{X}_A^H = (|s_1|^2 + |s_2|^2)\mathbf{I}_2$ , which makes every subsequent analysis a two-line calculation.

At the time, the alternative transmit-diversity schemes were delay diversity (Wittneben 1991, Seshadri-Winters 1993 — send the same symbol on both antennas with a one-symbol delay, then use a Viterbi decoder on the resulting ISI) and the more general trellis space-time codes (Tarokh-Seshadri-Calderbank, which appeared in IEEE Trans. IT March 1998, six months before Alamouti). The Tarokh-Seshadri-Calderbank paper gave the rank and determinant design criteria of Ch. 10 and constructed several $n_t = 2, 4$ -state trellis codes. But the resulting codes required Viterbi decoding over a trellis whose state space grew with the code memory — expensive, and never adopted commercially in that form.

Alamouti's contribution was to show that a rate-1 transmit-diversity scheme could be decoded with zero trellis — just two scalar matched filters. The structure was so simple that within two years every standards body was adopting it: WCDMA (3GPP Release 99, 2000), UMTS, and later LTE all include Alamouti-style transmit diversity modes. The paper has now been cited more than 25,000 times; it is among the most cited papers in all of wireless communications. Alamouti's career moved to industry soon after — he subsequently co-founded Vivato (the first commercial 802.11 smart-antenna AP maker) and went on to a long career at Intel and at startups. The textbook treatment of transmit diversity starts, still, with this 1998 paper.

Quick Check

Which of the following is the reason Alamouti achieves diversity $2 n_r$ with a linear (matched-filter) decoder?

The codeword matrix is orthogonal: $\mathbf{X}_A\mathbf{X}_A^H = (|s_1|^2 + |s_2|^2)\mathbf{I}_2$ for every $(s_1, s_2)$

The channel is i.i.d. Rayleigh with zero mean and unit variance

The receiver knows the channel coefficients $h_1, h_2$ and inverts $\tilde{\mathbf{H}}$

The two transmit antennas are jointly beamforming in the direction of the receiver

Correction:

The codeword matrix is orthogonal:

\mathbf{X}_A\mathbf{X}_A^H = (|s_1|^2 + |s_2|^2)\mathbf{I}_2

for every

(s_1, s_2)

This is the one structural fact that makes the proof work. Orthogonality of the codeword matrix implies that after stacking the two channel uses into a virtual $2\times 2$ channel $\tilde{\mathbf{H}}$ , the columns of $\tilde{\mathbf{H}}$ are orthogonal and the matched filter $\tilde{\mathbf{H}}^H$ decouples the decision on $s_1$ from the decision on $s_2$ . Each scalar decision sees the sum of $|h_1|^2 + |h_2|^2$ (per receive antenna) — the $2 n_r$ -fold diversity sum.

Quick Check

An Alamouti $2\times n_r$ link operates at average total transmit energy $E_s$ per channel use and the BER at target reliability is achieved at $\text{SNR} = 15$ dB. A comparable $1\times n_r$ MRC link (with the same total transmit energy and the same $n_r$ ) meets the same BER target at roughly what SNR?

$15 - 3 = 12$ dB (MRC is 3 dB better because it does not split transmit power)

Exactly 15 dB — they have the same diversity order, so the curves overlap

$15 + 3 = 18$ dB (MRC is worse because it has only one transmit antenna)

15 dB if $n_r = 1$ , but 12 dB if $n_r \ge 2$

Correction:

15 - 3 = 12

dB (MRC is 3 dB better because it does not split transmit power)

This is the fundamental 3 dB gap between Alamouti and same- $n_r$ MRC at the same total transmit power. Alamouti splits $E_s$ equally across its two transmit antennas, so each antenna radiates $E_s/2$ ; the per-antenna-to-receiver path therefore carries half the energy of the corresponding MRC antenna at the same total budget.

Alamouti Code

The rate-1 space-time block code for $n_t = 2$ defined by the orthogonal codeword matrix $\mathbf{X}_A(s_1, s_2) = \begin{pmatrix} s_1 & -s_2^* \\ s_2 & s_1^* \end{pmatrix}$ (Alamouti 1998). Achieves full diversity $2 n_r$ with linear matched-filter decoding at a 3 dB penalty relative to $1\times 2 n_r$ MRC.

Space-Time Codeword

A matrix $\mathbf{X} \in \mathbb{C}^{n_t \times T}$ whose rows index transmit antennas and whose columns index time slots within a coherence block of length $T$ . The MIMO received block is $\mathbf{Y} = \mathbf{H} \mathbf{X} + \mathbf{w}$ .

⚠️Engineering Note

Alamouti in Cellular Standards (WCDMA, LTE, 5G NR)

Alamouti is the simplest and most widely deployed space-time code in cellular history. Specifically:

3GPP Release 99 / WCDMA (2000): introduced space-time transmit diversity (STTD) based on the Alamouti matrix for the downlink (2 Tx at the base station). It is the first standardised STBC.
LTE (3GPP Release 8, 2008, TS 36.211): includes Alamouti-based transmit diversity (TM2) as a fallback mode when channel state information at the transmitter is unreliable — typically for cell-edge or high-mobility users.
5G NR (3GPP Release 15, 2018, TS 38.211): uses closed-loop codebook-based precoding as the default, but the Alamouti structure still appears implicitly in the $2\times 2$ rank-1 precoding codebook, and some broadcast / control channels use Alamouti-style transmit diversity in certain fallback modes.

Practical Constraints

•
Requires at least 2 Tx antennas at the transmitter (base station or UE with 2 Tx)
•
Requires coherence over $T = 2$ channel uses — i.e., the channel must be quasi-static for at least two consecutive OFDM resource elements
•
Implementation complexity is dominated by the matched-filter multiplication, which is $O(n_r)$ per symbol — negligible
•
3 dB transmit-power-splitting penalty versus coherent beamforming (MRT); acceptable when CSIT is unreliable

📋 Ref: 3GPP TS 36.211, 3GPP TS 38.211

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Why This Matters: Forward: Alamouti and the Diversity-Multiplexing Tradeoff

Alamouti achieves full diversity $2 n_r$ at rate 1 (symbol/channel use). From the Chapter 12 Zheng-Tse diversity-multiplexing tradeoff (DMT) viewpoint, this means Alamouti sits at $(r, d) = (1, 2 n_r)$ in the DMT plane — it is one of the points on the DMT curve but only one. For $r > 1$ (multiplexing gains larger than 1) Alamouti has nothing to offer: its rate is fixed at 1 by construction. Chapter 12 will show that the DMT-optimal curve for $n_t = n_r = 2$ is a piecewise-linear interpolation of $(0, 4), (1, 1), (2, 0)$ , and Alamouti achieves only the first two corner points.

For $n_t > 2$ , Alamouti does not even apply — it is specifically a $2$ -antenna code. Orthogonal STBCs for $n_t > 2$ (§2) cover rate $\le 3/4$ at full diversity; Chapter 13's Golden code and Perfect codes achieve the full DMT curve for $n_t = 2$ and $n_t \ge 2$ respectively, at the cost of requiring a lattice decoder. Alamouti is the first corner of a large story.

The Alamouti Scheme

The Simplest Non-Trivial Space-Time Code

Definition: Space-Time Codeword Matrix

Definition: Alamouti Codeword Matrix

Definition: Diversity Order of an STBC

Alamouti Scheme: XA\mathbf{X}_AXA​ Encoding and Orthogonal Decoding

Theorem: Alamouti Achieves Full Diversity 2nr2 n_r2nr​ with Linear Decoding

System model and column-wise expansion

The effective channel has orthogonal columns

Matched-filter decoupling

Per-symbol effective SNR and diversity

Key Takeaway

Alamouti vs. MRC: Why the 3 dB?

BER: Alamouti vs. MRC vs. SISO

Parameters

Alamouti Signal-Space Projections

Parameters

Common Mistake: Alamouti Does Not Match MRC — It Matches MRC Minus 3 dB

Historical Note: Alamouti 1998: The Landmark Paper

Quick Check

Quick Check

Alamouti Code

Space-Time Codeword

Alamouti in Cellular Standards (WCDMA, LTE, 5G NR)

Why This Matters: Forward: Alamouti and the Diversity-Multiplexing Tradeoff

Definition:
Space-Time Codeword Matrix

Definition:
Alamouti Codeword Matrix

Definition:
Diversity Order of an STBC

Alamouti Scheme: $\mathbf{X}_A$ Encoding and Orthogonal Decoding

Theorem: Alamouti Achieves Full Diversity $2 n_r$ with Linear Decoding