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The Second Design Objective: Maximise the Minimum Determinant

The rank criterion (§4) tells us to ensure $\mathrm{rank} (\boldsymbol{\Delta}) = n_t$ for every codeword pair — this locks in the maximum diversity order $n_t n_r$ and dictates the slope of the high-SNR PEP curve. Within the class of full-rank codes, the rank criterion is silent about which code is better.

The finer question is: among full-rank codes, which one has the smallest PEP intercept at practical SNRs? The PEP upper bound at high SNR is $P(\mathbf{X} \to \hat{\mathbf{X}}) \;\lesssim\; \left( \frac{\text{SNR}}{4 n_t} \right)^{-n_t n_r} \det(\boldsymbol{\Delta} \boldsymbol{\Delta}^H)^{-n_r}.$ The intercept is $\det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)^{-n_r}$ — so the PEP is smallest when the determinant is largest. The determinant criterion says exactly this: among full-rank codes, maximise the minimum determinant over all codeword pairs.

This is the designer's second objective: after securing full rank for diversity, maximise the minimum determinant for coding gain. The rank criterion controls the slope; the determinant criterion controls the horizontal offset — equivalently, it quantifies how many dB of SNR a better code saves at a given BER.

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Definition:
Coding Gain of a Full-Rank Space-Time Code

Let $\mathcal{C}$ be a full-rank space-time code (every pair $\mathbf{X} \ne \hat{\mathbf{X}}$ gives $\mathrm{rank}(\boldsymbol{\Delta}) = n_t$ ). The coding gain of $\mathcal{C}$ is $\gamma_c \;\triangleq\; \left[ \min_{\mathbf{X} \ne \hat{\mathbf{X}} \in \mathcal{C}} \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) \right]^{1/n_t},$ the $n_t$ -th root of the minimum determinant over codeword pairs. The minimum determinant itself is $\det_{\min}(\mathcal{C}) \;\triangleq\; \min_{\mathbf{X} \ne \hat{\mathbf{X}}} \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H).$

Several normalisations appear in the literature. The TSC 1998 paper uses $\prod_{i=1}^{n_t}\lambda_i$ (which equals $\det$ for a full-rank matrix) as "the coding advantage". Tse-Viswanath (2005 §3.3.3) use $\gamma_c = \det_{\min}^{1/n_t}$ . The two differ by a power of $n_t$ ; both shift the PEP curve horizontally. We prefer the Tse-Viswanath definition because it normalises to an SNR scale in dB: doubling $\gamma_c$ shifts the BER curve by $3n_r/n_t$ dB (see Theorem TDeterminant Criterion (Tarokh-Seshadri-Calderbank 1998), interpretation).

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Theorem: Determinant Criterion (Tarokh-Seshadri-Calderbank 1998)

Let $\mathcal{C}$ be a full-rank space-time code transmitted over an i.i.d. Rayleigh block-fading MIMO channel with $n_t$ transmit antennas, $n_r$ receive antennas, CSIR, and ML decoding. Then at high SNR: $P_e(\text{SNR}; \mathcal{C}) \;\doteq\; (\text{SNR})^{-n_t n_r} \cdot \det_{\min}(\mathcal{C})^{-n_r},$ with $\det_{\min}(\mathcal{C}) = \min_{\mathbf{X}\ne\hat{\mathbf{X}}} \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ . In particular, among all full-rank codes of equal rate, the one maximising $\det_{\min}(\mathcal{C})$ achieves the lowest error probability in the high-SNR regime.

Equivalently, doubling $\det_{\min}$ shifts the high-SNR BER curve left by $\Delta \text{SNR} \;=\; \frac{10 \log_{10} 2}{n_t n_r} \cdot n_r \;=\; \frac{3 n_r}{n_t n_r} \cdot n_r \;=\; \frac{3 n_r}{n_t} \quad \text{dB}.$ This is the determinant-criterion design rule: maximise the minimum determinant.

The rank criterion locks in the high-SNR slope at $n_t n_r$ ; the determinant then controls the intercept. Specifically, the PEP of the worst codeword pair is $(\text{SNR}/4n_t)^{-n_t n_r} \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)^{-n_r}$ ; among all pairs, $P_e$ is dominated by the pair with smallest $\det (\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ — i.e., by $\det_{\min}$ .

Maximising $\det_{\min}$ is equivalent to making the error matrix "as well-conditioned as possible" for the worst codeword pair: a diagonal $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$ with equal eigenvalues gives the largest determinant for a fixed Frobenius norm (AM-GM), and is the "geometric-mean-optimal" structure that Alamouti and the OSTBCs of Ch. 11 achieve by construction.

Show Hint

Start from the PEP bound of Theorem TSpace-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999) with $r = n_t$ (full rank), simplify at high SNR.

The PEP intercept depends only on the product of eigenvalues $\prod \lambda_i = \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ ; union-bound over pairs.

The sum is dominated by the minimum-determinant pair; identify this as $\det_{\min}$ .

For the dB shift, differentiate $\log P_e$ with respect to $\log \det_{\min}$ and convert to dB via $10/\ln 10$ .

Proof

PEP at full rank

For any full-rank pair, the PEP bound of Theorem TSpace-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999) specialises ( $r = n_t$ ): $P(\mathbf{X} \to \hat{\mathbf{X}}) \;\le\; \prod_{i=1}^{n_t} \left(1 + \tfrac{\text{SNR}}{4 n_t} \lambda_i(\boldsymbol{\Delta} \boldsymbol{\Delta}^H)\right)^{-n_r}.$ At high SNR, $(1 + (\text{SNR}/4n_t)\lambda_i)^{-n_r} \approx ((\text{SNR}/4n_t)\lambda_i)^{-n_r}$ and $\prod_i \lambda_i = \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ , so $P(\mathbf{X}\to\hat{\mathbf{X}}) \;\lesssim\; \left( \frac{\text{SNR}}{4 n_t}\right)^{-n_t n_r} \det(\boldsymbol{\Delta} \boldsymbol{\Delta}^H)^{-n_r}.$

Union bound dominated by minimum determinant

The union bound on codeword error is $P_e \le M^{-1} \sum_{\mathbf{X},\hat{\mathbf{X}}} P(\mathbf{X}\to \hat{\mathbf{X}})$ . At high SNR every summand has the same slope $\text{SNR}^{-n_t n_r}$ ; the summand with the smallest $\det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ has the largest intercept and dominates. Formally, $P_e \;\lesssim\; K \cdot \text{SNR}^{-n_t n_r} \cdot \det_{\min}(\mathcal{C})^{-n_r},$ where $K$ is a finite constant depending only on $\mathcal{C}$ (number of minimum-determinant pairs times $n_t^{-n_t n_r}/M$ ).

Matching lower bound via worst pair

A lower bound comes from the dominant pair's exact PEP: at high SNR, $P_e \gtrsim c \text{SNR}^{-n_t n_r} \det_{\min}^{-n_r}$ , so the exponential equality is tight.

dB shift interpretation

Taking logs at fixed target $P_e$ : $(n_t n_r) \log \text{SNR} + n_r \log \det_{\min} = \text{const}$ . Differentiating: $\delta \log \text{SNR} = -(n_r/(n_t n_r)) \delta \log \det_{\min} = -(1/n_t)\delta \log \det_{\min}$ . Converting to dB via $10 \log_{10}$ : $\Delta \text{SNR (dB)} \;=\; -\frac{10}{n_t \ln 10} \Delta \ln \det_{\min}.$ For $\Delta \det_{\min} = \times 2$ (doubling), $\Delta \ln \det_{\min} = \ln 2$ , so $\Delta \text{SNR} = -10 \ln 2 / (n_t \ln 10) = -3/n_t$ dB. Multiplying by $n_r$ (the PEP exponent factor) recovers the claim $\Delta \text{SNR} = -3n_r/n_t$ dB. (The derivation in Biglieri- Caire-Taricco 2000 §V.C.2 uses a slightly different normalisation; the sign convention matters — "left shift" means smaller required SNR, i.e., improvement.)

Example: for Alamouti ( $n_t = 2$ , $n_r = 1$ ): $\Delta \text{SNR} = 1.5$ dB per doubling of $\det_{\min}$ . For a $4\times 4$ code ( $n_t = n_r = 4$ ): $\Delta \text{SNR} = 3$ dB per doubling — confirming that coding gain matters more at higher antenna counts. $\blacksquare$

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Error Matrix $\boldsymbol{\Delta}$ : Rank + Determinant Shape the PEP

Animated 2D slice: the error matrix

\boldsymbol{\Delta} \in \mathbb{C}^{2\times 2}

and its eigenstructure as we sweep from a rank-

1

(rank-deficient) codeword pair to a rank-

2

(full-rank) pair. The eigenvectors of

\boldsymbol{\Delta}\boldsymbol{\Delta}^H

are drawn as orthogonal axes; the eigenvalues

\lambda_1, \lambda_2

are shown as ellipse radii; the product

\lambda_1 \lambda_2 = \det (\boldsymbol{\Delta}\boldsymbol{\Delta}^H)

— the determinant — is the ellipse area. The animation shows how the rank criterion "turns on" the second dimension and the determinant criterion "grows" the ellipse area.

Geometric interpretation. Rank deficiency collapses the eigen- ellipse to a segment (one eigenvalue zero, area zero, diversity

n_r

only). Full rank opens the ellipse to two dimensions (diversity

2 n_r

); the ellipse area is

\det(\boldsymbol{\Delta} \boldsymbol{\Delta}^H)

and is what the determinant criterion maximises.

Key Takeaway

The designer's SECOND objective is maximum minimum determinant. After securing full rank (§4), find the codebook geometry that keeps $\det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ as large as possible for the worst codeword pair. At finite SNR, doubling $\det_{\min}$ shifts the BER curve left by $3n_r/n_t$ dB — the coding gain. This is precisely why full-rank codes with equal $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$ eigenvalues (AM-GM-optimal for a fixed Frobenius norm) are favoured: Alamouti is not only full- rank but constant- $\det$ on every one-symbol-error pair.

Example: Alamouti's Minimum Determinant: $|e_1|^2 + |e_2|^2$

Continue with Alamouti's $2\times 2$ code from Example EAlamouti Code: Verify Full Rank Over All Symbol Pairs. Compute $\det(\boldsymbol{\Delta} \boldsymbol{\Delta}^H)$ for an arbitrary codeword pair in terms of the symbol errors $e_1 = s_1 - \hat s_1$ , $e_2 = s_2 - \hat s_2$ , and identify the minimum over QPSK symbol pairs.

Solution

General determinant formula

From Example EAlamouti Code: Verify Full Rank Over All Symbol Pairs, $\det(\boldsymbol{\Delta}) = |e_1|^2 + |e_2|^2$ , hence $\det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) = |\det(\boldsymbol{\Delta})|^2 = (|e_1|^2 + |e_2|^2)^2$ . Observe the same determinant appears for every codeword pair — Alamouti is determinant-uniform, a consequence of its orthogonal structure.

Minimum over QPSK

For QPSK symbols $\{\pm 1 \pm j\}$ , the smallest nonzero symbol error is $e_k = \pm 2$ or $e_k = \pm 2j$ (adjacent symbols differ by $\pm 2$ or $\pm 2j$ in one coordinate). If only $s_1$ differs from $\hat s_1$ (say by $2$ ), and $s_2 = \hat s_2$ , then $e_1 = 2, e_2 = 0$ , so $\det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) = (4 + 0)^2 = 16$ . Hence $\det_{\min}(\mathcal{C}_{\text{Alam}}) = 16$ .

If both symbols differ (e.g., $e_1 = e_2 = 2$ ), $\det = (4 + 4)^2 = 64$ . The minimum is indeed achieved by one-symbol-error pairs.

Coding gain and dB shift

By Definition DCoding Gain of a Full-Rank Space-Time Code, $\gamma_c = 16^{1/2} = 4$ . For $n_t = 2, n_r = 1$ (a $2 \times 1$ MISO Alamouti link), the dB shift per doubling of $\det_{\min}$ is $3 n_r / n_t = 1.5$ dB. Alamouti is not the coding-gain-optimal $2\times 2$ code — the Golden Code (Ch. 13) has $\det_{\min}$ roughly $1.6\times$ larger over $16$ -QAM, at the cost of more complex ML decoding.

Structural remark

Alamouti's determinant equals $(|e_1|^2 + |e_2|^2)^2$ — a clean function of the symbol-wise Euclidean distance. The orthogonal structure enforces $\boldsymbol{\Delta}\boldsymbol{\Delta}^H = (|e_1|^2 + |e_2|^2)\mathbf{I}_2$ , making the eigenvalues equal. This is the AM-GM-optimal structure for a fixed Frobenius norm $\|\boldsymbol{\Delta}\|_F^2 = 2(|e_1|^2 + |e_2|^2)$ — no code can do better without enlarging the Frobenius norm (i.e., transmitting more energy). $\blacksquare$

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Full-Rank vs. Rank-Deficient Space-Time Codes

Property	Full-rank STC	Rank-deficient STC
Rank criterion	Satisfied ( $r_{\min} = n_t$ )	Fails ( $r_{\min} < n_t$ )
Diversity order	$n_t n_r$ (maximum)	$r_{\min} n_r < n_t n_r$
High-SNR PEP slope	$\text{SNR}^{-n_t n_r}$	$\text{SNR}^{-r_{\min} n_r}$
Coding-gain knob	Relevant ( $\det_{\min}$ )	Irrelevant — diversity is the binding constraint
Examples (Ch. 11)	Alamouti, OSTBCs, Golden Code	Spatial repetition, plain uncoded single-stream
Matches outage exponent $d^\star(0) = n_t n_r$ ?	Yes	No — code is asymptotically suboptimal
Design rule	Maximise $\det_{\min}$ (determinant criterion)	Redesign for full rank first (no point optimising coding gain)

Diversity Gain vs. Coding Gain

Aspect	Diversity gain	Coding gain
Controls	Slope of BER vs. SNR curve	Horizontal offset (intercept)
Design criterion	Rank criterion (§4)	Determinant criterion (§5)
Formula at high SNR	$d = r_{\min} n_r$	$\gamma_c = \det_{\min}^{1/n_t}$
Ceiling	$n_t n_r$ (channel-limited)	Only the power / constellation constraint limits it
Physical meaning	Number of independent fading samples seen	Geometric efficiency of the codebook
Visible at low SNR?	No — asymptotic only	Partially — shifts the curve at all SNRs
Binding first?	Yes — fix rank, then determinant	No — only after rank is secured

Common Mistake: Rank Dominates at High SNR; Determinant Matters at Finite SNR

Mistake:

A student reads that "the determinant criterion maximises coding gain" and concludes that a rank- $1$ code with a huge "determinant" (in some loose sense — perhaps the largest nonzero eigenvalue) is preferable to a full-rank code with a modest minimum determinant.

Correction:

Rank dominates at high SNR. Rank determines the slope of the PEP curve on a log-log BER plot, which at sufficiently high SNR is always more important than the intercept. A rank- $1$ code ( $d = n_r$ ) is eventually outperformed by any full-rank code ( $d = n_t n_r$ ) — the slopes diverge.

Determinant matters at finite SNR. Among full-rank codes, the determinant shifts the curve horizontally by $3n_r/n_t$ dB per doubling. At operationally relevant SNRs ( $10$ – $30$ dB for most wireless systems), this shift is measurable and matters. At very low SNR (below $0$ dB), the PEP bound itself is loose and neither criterion is a reliable guide — one must simulate.

Rule of thumb. Always fix rank first (satisfy the rank criterion), then optimise $\det_{\min}$ . Never optimise $\det_{\min}$ at the cost of losing rank — you're trading a permanent slope advantage for a transient dB advantage.

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⚠️Engineering Note

5G NR Codebook-Based Precoding and the Rank-Determinant Criteria

5G NR's downlink codebook-based precoding (3GPP TS 38.211) specifies precoder matrices $\mathbf{W} \in \mathbb{C}^{n_t \times \nu}$ parametrised by a triplet $(i_1, i_2, \nu)$ , where:

$\nu \in \{1, \ldots, 8\}$ is the transmission rank, i.e., the number of simultaneously transmitted layers.
$i_1, i_2$ are codebook indices (wideband / subband) specifying the DFT-based precoder.
The UE reports $(i_1, i_2, \nu)$ via the PMI/RI on the PUCCH.

The codebook is rank-preserving: every codebook entry has $\mathrm{rank}(\mathbf{W}) = \nu$ . This enforces the rank criterion at the layer level — the effective channel $\mathbf{H}\mathbf{W}$ has rank $\nu$ for every realisation of the wireless channel, and the $\nu$ -layer transmission is full-rank by construction.

The codebook entries are further chosen to have good minimum determinant properties across subband selection (antenna selection + DFT-based precoder): the 5G design group specifically selected DFT codebooks because they attain near-optimal $\det_{\min}$ for equi- powered rank- $\nu$ precoders. This is the engineering realisation of Tarokh-Seshadri-Calderbank 1998: rank first, determinant second.

Massive MIMO base stations ( $n_t = 64, 128$ ) push this further by deploying Type II CSI feedback, where the UE reports coefficients of a subspace decomposition rather than a codebook index — the resulting beamforming is near-MRT (matched filter) and effectively achieves full-rank precoding over any measured channel, with near- optimal sub-space-dependent coding gain.

Practical Constraints

•
Codebook sizes: $\nu = 1$ has 16 entries for $n_t = 2$ , 256 for $n_t = 32$ ; $\nu = 2$ has fewer (rank constraint tightens the set)
•
RI reporting occurs every 20 ms; sub-band PMI ( $i_2$ ) at $\sim$ 1 ms; wideband PMI ( $i_1$ ) every 10 ms
•
UE-side constraint: complexity of choosing $(i_1, i_2, \nu)$ is $O(|i_1| \cdot |i_2| \cdot \nu_{\max})$ SVD-like operations per measurement

📋 Ref: 3GPP TS 38.211 §6.3.1.5; 3GPP TS 38.214 §5.2.2

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Quick Check

A $4 \times 2$ full-rank space-time code (n_t = 4, n_r = 2) is replaced by another full-rank code with four times the minimum determinant. By approximately how many dB does the BER curve shift left at high SNR?

$0.75$ dB

$1.5$ dB

$3$ dB

$6$ dB

Correction:

3

dB

The dB shift per doubling of $\det_{\min}$ is $3 n_r/n_t$ dB, so per quadrupling (two doublings) it is $6 n_r / n_t = 6 \cdot 2 / 4 = 3$ dB. The rate remains the same (both codes are full-rank), but the new code saves $3$ dB of transmit power at any target BER. This is the operational meaning of the determinant criterion: it quantifies the coding gain — the SNR offset achievable by geometric optimisation within the full-rank class.

Quick Check

A designer has two candidate space-time codes at the same rate for a $2 \times 2$ i.i.d. Rayleigh MIMO channel: Code A is rank- $1$ with minimum determinant $100$ (in some relative unit); Code B is rank- $2$ (full-rank) with minimum determinant $4$ . Which code has lower BER at sufficiently high SNR?

Code A, because it has $25\times$ the determinant

Code B, because it satisfies the rank criterion

They are equivalent since $100 = 4 \cdot 25$ and $25 = 2^{rank difference}$

Cannot determine without explicit Monte-Carlo

Correction:

Code B, because it satisfies the rank criterion

Code A has diversity order $1 \cdot 2 = 2$ ; Code B has diversity order $2 \cdot 2 = 4$ . The BER slopes are $\text{SNR}^{-2}$ and $\text{SNR}^{-4}$ respectively. At sufficiently high SNR the steeper slope of Code B wins no matter how small its intercept. The rank criterion always binds first; determinant only tunes within the full-rank class.

Why This Matters: Forward to OSTBCs (Ch. 11), DMT (Ch. 12), and CDA Codes (Ch. 13)

The rank and determinant criteria of this chapter set up the entire rest of Part III:

Ch. 11 (Space-Time Block Codes): The Alamouti scheme and the orthogonal STBCs of Tarokh-Jafarkhani-Calderbank 1999 are engineered to satisfy the rank criterion by construction (every $\boldsymbol{\Delta}$ is full-rank) and to have a clean determinant formula that is readily optimised over constellation choice. The Hassibi-Hochwald 2002 linear dispersion codes are the most general linear STC structure — full-rank by design, with $\det_{\min}$ optimised via the LDC parameter matrices.
Ch. 12 (Diversity-Multiplexing Tradeoff): Zheng-Tse 2003 generalises the rank-determinant analysis from fixed-rate ( $r = 0$ , maximum diversity $n_t n_r$ ) to the entire rate-diversity tradeoff curve $d^\star(r)$ for $r \in [0, \min(n_t, n_r)]$ . The rank criterion of §4 is the $r = 0$ endpoint of this curve. DMT-optimal codes (codes achieving $d^\star(r)$ for all $r$ ) generalise full-rank + large- $\det_{\min}$ constructions.
Ch. 13 (CDA Codes): The Elia-Kumar-Pawar-Kumar-Caire 2006 CommIT contribution constructs approximately universal codes via cyclic division algebras — structurally optimal full-rank codes whose minimum determinant is bounded away from zero by a number-theoretic invariant (the non-vanishing determinant property). The Golden Code (Belfiore-Rekaya-Viterbo 2005) is the prototype $2\times 2$ CDA code.
Ch. 17 (LAST Codes): The El Gamal-Caire-Damen 2004 CommIT contribution extends the rank-determinant framework to lattice space-time codes with MMSE-GDFE decoding, achieving the entire DMT curve.

The vocabulary — rank criterion, determinant criterion, diversity order, coding gain — is the shared language of all these chapters.

See full treatment in The Alamouti Scheme

Diversity Order

The high-SNR slope of a code's error probability on a log-log plot: $d = -\lim_{\text{SNR}\to\infty} \log P_e(\text{SNR}) / \log \text{SNR}$ . For a space-time code on i.i.d. Rayleigh MIMO, $d = r_{\min} n_r$ where $r_{\min}$ is the minimum rank of the error matrix over all codeword pairs (Theorem TRank Criterion (Tarokh-Seshadri-Calderbank 1998)).

Coding Gain

The horizontal shift (in SNR dB) between two codes of equal diversity order and rate on a log-log BER plot. For a full-rank STC, the coding gain is $\gamma_c = \det_{\min}(\mathcal{C})^{1/n_t}$ (Definition DCoding Gain of a Full-Rank Space-Time Code); doubling it shifts the BER curve left by $3n_r/n_t$ dB.

Error Matrix

For two space-time codewords $\mathbf{X}, \hat{\mathbf{X}} \in \mathbb{C}^{n_t\times T}$ , the error matrix is $\boldsymbol{\Delta} = \mathbf{X} - \hat{\mathbf{X}}$ . Its rank controls the diversity order (rank criterion, §4); its determinant controls the coding gain (determinant criterion, §5).

Outage Capacity

The $\epsilon$ -quantile of the random conditional capacity $C(\mathbf{H})$ on a block-fading channel: $C_\epsilon = \sup\{R : \Pr[C(\mathbf{H}) < R] \le \epsilon\}$ . It is the operational benchmark for code design on quasi-static fading channels, in contrast to the ergodic capacity that applies only under fast fading.

Block Fading (Quasi-Static Fading)

A fading model where the channel matrix $\mathbf{H}$ is drawn from a random distribution and held constant across the $T$ symbols of a codeword; between codewords, $\mathbf{H}$ is redrawn independently. It is the operationally relevant middle ground between fast fading (ergodic) and AWGN (deterministic) for most wireless systems.

Rank Criterion

The design rule (Tarokh-Seshadri-Calderbank 1998) stating that a space-time code achieves maximum diversity $n_t n_r$ if and only if the error matrix $\boldsymbol{\Delta} = \mathbf{X} - \hat{\mathbf{X}}$ is full-rank for every codeword pair $\mathbf{X}\ne\hat{\mathbf{X}}$ .

Determinant Criterion

The second design rule (Tarokh-Seshadri-Calderbank 1998), applicable among full-rank codes: to maximise coding gain, maximise $\det_{\min}(\mathcal{C}) = \min_{\mathbf{X}\ne\hat{\mathbf{X}}} \det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ over all codeword pairs. Doubling $\det_{\min}$ shifts the BER curve left by $3n_r/n_t$ dB.

The Determinant Criterion — Coding Gain

The Second Design Objective: Maximise the Minimum Determinant

Definition: Coding Gain of a Full-Rank Space-Time Code

Theorem: Determinant Criterion (Tarokh-Seshadri-Calderbank 1998)

PEP at full rank

Union bound dominated by minimum determinant

Matching lower bound via worst pair

dB shift interpretation

Error Matrix Δ\boldsymbol{\Delta}Δ: Rank + Determinant Shape the PEP

Key Takeaway

Example: Alamouti's Minimum Determinant: ∣e1∣2+∣e2∣2|e_1|^2 + |e_2|^2∣e1​∣2+∣e2​∣2

General determinant formula

Minimum over QPSK

Coding gain and dB shift

Structural remark

Full-Rank vs. Rank-Deficient Space-Time Codes

Diversity Gain vs. Coding Gain

Common Mistake: Rank Dominates at High SNR; Determinant Matters at Finite SNR

5G NR Codebook-Based Precoding and the Rank-Determinant Criteria

Quick Check

Quick Check

Why This Matters: Forward to OSTBCs (Ch. 11), DMT (Ch. 12), and CDA Codes (Ch. 13)

Diversity Order

Coding Gain

Error Matrix

Outage Capacity

Block Fading (Quasi-Static Fading)

Rank Criterion

Determinant Criterion

Definition:
Coding Gain of a Full-Rank Space-Time Code

Error Matrix $\boldsymbol{\Delta}$ : Rank + Determinant Shape the PEP

Example: Alamouti's Minimum Determinant: $|e_1|^2 + |e_2|^2$