Ferkans — Interactive Telecom Tutor

Why PEP is the Right Lens

On a block-fading MIMO channel, the capacity is random and the operational benchmark is the outage probability $P_{\mathrm{out}}(R)$ (§2). The code-design question then becomes: given a target rate $R$ , what space-time codebook $\mathcal{C} \subset \mathbb{C}^{n_t \times T}$ minimises the error probability across channel realisations?

A union bound on the codeword error probability decomposes into pairwise error probabilities (PEPs): $P_e \;\le\; \frac{1}{|\mathcal{C}|}\sum_{\mathbf{X} \in \mathcal{C}} \sum_{\hat{\mathbf{X}} \ne \mathbf{X}} P(\mathbf{X} \to \hat{\mathbf{X}}).$ Analysing every summand is the space-time analogue of the AWGN PEP analysis of Ch. 2 and the fading PEP analysis of Ch. 6 — and the machinery is remarkably similar: conditional Gaussian PEP on the random channel, then average the resulting $Q$ -function over the fading distribution using the Chernoff bound and the MGF of a $\chi^2$ .

What changes is the $r$ -dimensional diversity branch structure: the error-matrix $\boldsymbol{\Delta} = \mathbf{X} - \hat{\mathbf{X}}$ has rank $r \le \min(n_t, T)$ , and the PEP is ruled by the $r$ nonzero eigenvalues $\lambda_i(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ — one per effective diversity branch. The central PEP bound of this section (Theorem TSpace-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999)) is the object from which the rank criterion (§4) and determinant criterion (§5) are derived as simple high-SNR corollaries.

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Definition:
Space-Time Codebook and Error Matrix

A space-time code of block length $T$ with $n_t$ transmit antennas is a finite set $\mathcal{C} = \{\mathbf{X}^{(1)}, \ldots, \mathbf{X}^{(M)}\} \subset \mathbb{C}^{n_t \times T}$ of codeword matrices satisfying the average-power constraint $\frac{1}{M} \sum_{m = 1}^M \|\mathbf{X}^{(m)}\|_F^2 \;\le\; P T,$ equivalently $\mathbb{E}[\|\mathbf{x}_t\|^2] \le P$ per channel use under uniform codeword selection. The spectral efficiency (bits/channel use) of $\mathcal{C}$ is $\eta = (\log_2 M)/T$ .

Given two distinct codewords $\mathbf{X}, \hat{\mathbf{X}} \in \mathcal{C}$ , the error matrix is $\boldsymbol{\Delta} \;\triangleq\; \mathbf{X} - \hat{\mathbf{X}} \in \mathbb{C}^{n_t \times T},$ with rank $r = \mathrm{rank}(\boldsymbol{\Delta}) \le \min(n_t, T)$ . The $n_t \times n_t$ matrix $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$ is Hermitian PSD of rank $r$ , with nonzero eigenvalues $\lambda_1(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) \ge \cdots \ge \lambda_r(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) > 0$ .

We will usually assume $T \ge n_t$ (codeword matrices at least as wide as they are tall), so that the rank $r$ can be as large as $n_t$ . This is the natural setting for full-diversity codes. For $T < n_t$ the error matrix cannot be full-rank, so a fundamental ceiling $r \le T$ on the diversity order already exists at the dimension-counting level — this is why Alamouti ( $T = n_t = 2$ ) and the other OSTBCs of Ch. 11 are "square" in $n_t$ .

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Theorem: Space-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999)

For a quasi-static i.i.d. Rayleigh MIMO channel with $n_t$ transmit and $n_r$ receive antennas, CSIR, and total transmit SNR $\text{SNR}$ , the pairwise error probability between two space-time codewords $\mathbf{X}, \hat{\mathbf{X}} \in \mathbb{C}^{n_t \times T}$ under ML decoding satisfies $P(\mathbf{X} \to \hat{\mathbf{X}}) \;\le\; \prod_{i=1}^{r} \left( 1 + \frac{\text{SNR}}{4 n_t}\, \lambda_i(\boldsymbol{\Delta}\boldsymbol{\Delta}^H) \right)^{-n_r},$ where $\boldsymbol{\Delta} = \mathbf{X} - \hat{\mathbf{X}}$ , $r = \mathrm{rank}(\boldsymbol{\Delta})$ , and $\lambda_1, \ldots, \lambda_r$ are the nonzero eigenvalues of the $n_t \times n_t$ PSD matrix $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$ .

Conditional on $\mathbf{H}$ , the ML decoder makes a Gaussian decision between two hypotheses with distance $\|\mathbf{H}\boldsymbol{\Delta}\|_F$ . The conditional PEP is a Q-function; its Chernoff bound $Q(x) \le \tfrac{1}{2}e^{-x^2/2}$ gives $\Pr(\mathbf{X} \to \hat{\mathbf{X}} \mid \mathbf{H}) \le \tfrac{1}{2} \exp(-\|\mathbf{H}\boldsymbol{\Delta}\|_F^2 \text{SNR}/(4 n_t))$ .

Averaging this exponential over the i.i.d. Rayleigh distribution of $\mathbf{H}$ is an MGF of a $\chi^2$ computation: per receive antenna, $\|[\mathbf{H}]_i \boldsymbol{\Delta}\|^2$ is a weighted sum of central $\chi^2_2$ random variables whose weights are the eigenvalues $\lambda_i(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ . The MGF produces the product form $\prod_i (1 + (\text{SNR}/(4n_t))\lambda_i)^{-1}$ , and the $n_r$ receive antennas multiply the exponents — hence the $-n_r$ exponent.

The pattern is exactly the Chernoff + MGF template of Ch. 2 (PEP on AWGN) and Ch. 6 (PEP on Rayleigh fading), generalised to $r$ parallel diversity branches.

Show Hint

Condition on $\mathbf{H}$ : show the conditional PEP is $Q\big(\sqrt{\text{SNR}\|\mathbf{H}\boldsymbol{\Delta}\|_F^2/(2 n_t)}\big)$ .

Apply the Chernoff bound $Q(x) \le \tfrac{1}{2}e^{-x^2/2}$ , then identify $\|\mathbf{H}\boldsymbol{\Delta}\|_F^2 = \sum_{j=1}^{n_r} \mathbf{h}_j^H (\boldsymbol{\Delta}\boldsymbol{\Delta}^H)\mathbf{h}_j$ with $\mathbf{h}_j$ the rows of $\mathbf{H}$ .

Each quadratic form $\mathbf{h}_j^H \mathbf{A} \mathbf{h}_j$ with $\mathbf{h}_j \sim \mathcal{CN}(0, \mathbf{I}_{n_t})$ and $\mathbf{A}$ PSD of rank $r$ has MGF $\prod_{i=1}^r (1 - t \lambda_i(\mathbf{A}))^{-1}$ .

Independence across rows $j$ gives $\prod_j M(t) = \prod_i (1 - t\lambda_i)^{-n_r}$ ; set $t = -\text{SNR}/(4 n_t)$ (negative because we want $M(-t)$ ) and minimise.

Proof

Conditional ML decoder

Given the channel realisation $\mathbf{H}$ , the received sequence over $T$ time samples is $\mathbf{Y} = \mathbf{H}\mathbf{X} + \mathbf{W}$ , where $\mathbf{Y}, \mathbf{W} \in \mathbb{C}^{n_r \times T}$ and the noise entries are i.i.d. $\mathcal{CN}(0, \sigma^2)$ . The ML decoder chooses $\hat{\mathbf{X}}_{\mathrm{ML}} = \arg\min_{\mathbf{X}'\in\mathcal{C}} \|\mathbf{Y} - \mathbf{H}\mathbf{X}'\|_F^2$ . Assume w.l.o.g. transmit power $P = 1$ so that $\text{SNR} = 1/\sigma^2$ and the codeword constraint is $\|\mathbf{X}\|_F^2 \le T$ (equivalently $\mathbb{E}[\|\mathbf{x}_t\|^2] \le 1$ under uniform codeword selection; the general case scales by $P$ ).

Conditional PEP via the Q-function

Given $\mathbf{H}$ , the PEP between two specific codewords is the probability that the ML metric picks $\hat{\mathbf{X}}$ over $\mathbf{X}$ : $P(\mathbf{X} \to \hat{\mathbf{X}} \mid \mathbf{H}) \;=\; \Pr\!\left[ \|\mathbf{Y} - \mathbf{H}\hat{\mathbf{X}}\|_F^2 \le \|\mathbf{Y} - \mathbf{H}\mathbf{X}\|_F^2 \,\Big|\, \mathbf{H}, \mathbf{X} \right].$ A standard Gaussian computation (the difference of the two metrics is Gaussian under $\mathbf{X}$ ) yields $P(\mathbf{X} \to \hat{\mathbf{X}} \mid \mathbf{H}) \;=\; Q\!\left( \sqrt{ \tfrac{\text{SNR}}{2 n_t} \|\mathbf{H}\boldsymbol{\Delta}\|_F^2} \right),$ where the factor $\text{SNR}/n_t$ comes from normalising the transmit power per antenna (isotropic Gaussian input), and $\boldsymbol{\Delta} = \mathbf{X} - \hat{\mathbf{X}}$ .

Chernoff bound on the Q-function

Apply the Chernoff bound $Q(x) \le \tfrac{1}{2} e^{-x^2/2}$ for $x \ge 0$ : $P(\mathbf{X} \to \hat{\mathbf{X}} \mid \mathbf{H}) \;\le\; \tfrac{1}{2} \exp\!\left( - \tfrac{\text{SNR}}{4 n_t} \|\mathbf{H}\boldsymbol{\Delta}\|_F^2 \right).$ This is the exponent-level version of the PEP: taking expectations over $\mathbf{H}$ reduces to computing the MGF of the random variable $\|\mathbf{H}\boldsymbol{\Delta}\|_F^2$ at $t = -\text{SNR}/(4 n_t)$ .

MGF of a weighted $\chi^2$ sum

Decompose $\|\mathbf{H}\boldsymbol{\Delta}\|_F^2 = \sum_{j=1}^{n_r} \mathbf{h}_j^H \boldsymbol{\Delta}\boldsymbol{\Delta}^H \mathbf{h}_j$ , where $\mathbf{h}_j^H \in \mathbb{C}^{1 \times n_t}$ is the $j$ -th row of $\mathbf{H}$ . Under the i.i.d. Rayleigh assumption, $\mathbf{h}_j \sim \mathcal{CN}(0, \mathbf{I}_{n_t})$ i.i.d. across $j$ . The matrix $\mathbf{A} \triangleq \boldsymbol{\Delta} \boldsymbol{\Delta}^H$ is Hermitian PSD of rank $r$ with nonzero eigenvalues $\lambda_1, \ldots, \lambda_r$ ; by eigendecomposition $\mathbf{h}_j^H \mathbf{A} \mathbf{h}_j = \sum_{i=1}^{r} \lambda_i |\tilde{h}_{ji}|^2$ , with $\tilde{h}_{ji}$ i.i.d. $\mathcal{CN}(0,1)$ .

Each $|\tilde{h}_{ji}|^2 \sim \chi^2_2/2$ is an exponential with mean $1$ , with MGF $\mathbb{E}[e^{t|\tilde{h}_{ji}|^2}] = (1 - t)^{-1}$ for $t < 1$ . Independence of the $r$ terms within row $j$ and across rows gives $\mathbb{E}\big[e^{t\|\mathbf{H}\boldsymbol{\Delta}\|_F^2}\big] \;=\; \prod_{j=1}^{n_r} \mathbb{E}\big[e^{t \mathbf{h}_j^H \mathbf{A} \mathbf{h}_j}\big] \;=\; \prod_{i=1}^{r} (1 - t \lambda_i)^{-n_r}.$

Assemble the unconditional PEP bound

Take expectation of the conditional Chernoff bound over $\mathbf{H}$ and substitute $t = -\text{SNR}/(4 n_t)$ : $P(\mathbf{X} \to \hat{\mathbf{X}}) \;\le\; \tfrac{1}{2} \mathbb{E}\!\left[\exp\!\left(-\tfrac{\text{SNR}}{4 n_t} \|\mathbf{H}\boldsymbol{\Delta}\|_F^2\right)\right] \;=\; \tfrac{1}{2} \prod_{i=1}^{r} \left(1 + \tfrac{\text{SNR}}{4 n_t} \lambda_i(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)\right)^{-n_r}.$ Dropping the $1/2$ for notational cleanliness (it does not affect the asymptotic rate analysis) gives the form stated. $\blacksquare$

High-SNR simplification

At large $\text{SNR}$ , $(1 + (\text{SNR}/4 n_t)\lambda_i)^{-n_r} \approx ((\text{SNR}/4 n_t)\lambda_i)^{-n_r}$ . Multiplying across the $r$ eigenvalues, $P(\mathbf{X} \to \hat{\mathbf{X}}) \;\lesssim\; \left( \frac{\text{SNR}}{4 n_t} \right)^{-r n_r} \prod_{i=1}^r \lambda_i^{-n_r}.$ Two structural parameters now emerge as the drivers of high-SNR performance: the rank $r$ (controls the diversity order — §4) and the product $\prod_i \lambda_i$ (controls the coding gain — §5).

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PEP Upper Bound vs. SNR, Varying Rank and Determinant

The PEP upper bound $(1 + (\text{SNR}/4n_t) \lambda_i)^{-n_r}$ -product curve, parametrised by the rank $r$ of the error matrix and the eigenvalue product $\prod_i \lambda_i$ (assumed equal-eigenvalue for visualisation: $\lambda_i = (\det)^{1/r}$ ). Observe how the slope of the curve at high SNR is ruled by $r n_r$ (rank criterion) and the intercept by $\prod_i \lambda_i$ (determinant criterion).

Parameters

Error matrix rank

r

2

Determinant

\prod\lambda_i

4

n_t

2

n_r

2

Pattern-Aware: Same Chernoff + MGF Template as Ch. 2 and Ch. 6

The PEP analysis of Theorem TSpace-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999) re-uses the Chernoff + MGF technique of Ch. 2 (TCM on AWGN) and Ch. 6 (BICM on Rayleigh) with one structural change: the exponent $\|\mathbf{H}\boldsymbol{\Delta}\|_F^2$ is a quadratic form in the channel rather than a scalar $|h|^2$ times a squared distance. The eigendecomposition $\boldsymbol{\Delta}\boldsymbol{\Delta}^H = \mathbf{V}\mathbf{\Lambda}\mathbf{V}^H$ turns this quadratic form into $r$ independent scalar fading branches, one per nonzero eigenvalue.

This is the operational sense in which "space-time coding transforms a correlated MIMO channel into $r$ parallel diversity branches indexed by the eigenvalues of $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$ ". The code designer controls $r$ and the eigenvalues; the channel provides the random fades on each branch. Ch. 11 will construct codes (Alamouti, OSTBCs) where the eigenvalues are equal and fixed across all codeword pairs — a particularly clean and analytically tractable design.

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Common Mistake: PEP Is a Union Bound — Often Loose at Low SNR

Mistake:

A student applies the PEP bound of Theorem TSpace-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999) at $\text{SNR} = 0$ dB for a code with $|\mathcal{C}| = 256$ codewords, sums $\binom{256}{2} = 32640$ PEP terms, and concludes that the BER is greater than $1$ — which is nonsense, and then concludes that the code is bad.

Correction:

The PEP bound and its union-bound aggregate $P_e \le \sum_{\mathbf{X},\hat{\mathbf{X}}} P(\mathbf{X} \to \hat{\mathbf{X}})/|\mathcal{C}|$ are upper bounds on the pairwise probabilities, and the union bound is well known to be loose at low SNR (much larger than $1$ is possible and meaningless). The bound becomes tight only at high SNR, where the dominant pair(s) drive the error probability. The rank and determinant criteria that follow are asymptotic statements — valid in the high-SNR limit.

Example: Eigenvalues of $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$ for an Alamouti Codeword Pair

The Alamouti code transmits two QPSK symbols $s_1, s_2 \in \{\pm 1 \pm j\}$ over $n_t = 2$ antennas across $T = 2$ channel uses as the codeword matrix $\mathbf{X}(s_1, s_2) \;=\; \begin{pmatrix} s_1 & -s_2^* \\ s_2 & s_1^* \end{pmatrix}.$ Compute the error matrix $\boldsymbol{\Delta}$ and the eigenvalues of $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$ for the pair $(s_1, s_2) = (1+j, 1+j)$ versus $(\hat s_1, \hat s_2) = (-1-j, 1+j)$ (differing in $s_1$ only).

Solution

Error matrix

$\boldsymbol{\Delta} \;=\; \mathbf{X}(1+j, 1+j) - \mathbf{X}(-1-j, 1+j) \;=\; \begin{pmatrix} 2(1+j) & 0 \\ 0 & 2(1-j) \end{pmatrix}.$ $This$ \boldsymbol{\Delta} $is diagonal with entries of equal magnitude$ |2(1\pm j)| = 2\sqrt{2} $, and is clearly rank$ 2$.

Product $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$

$\boldsymbol{\Delta}\boldsymbol{\Delta}^H \;=\; \begin{pmatrix} |2(1+j)|^2 & 0 \\ 0 & |2(1-j)|^2 \end{pmatrix} \;=\; 8 \mathbf{I}_2.$ $The eigenvalues are$ \lambda_1 = \lambda_2 = 8 $. The matrix is **full-rank** ($ r = n_t = 2$) and has equal eigenvalues.

Implication for PEP

With $\text{SNR}$ the total transmit SNR and $n_r$ receive antennas, the PEP upper bound of Theorem TSpace-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999) is $P(\mathbf{X} \to \hat{\mathbf{X}}) \;\le\; \left(1 + \tfrac{8\text{SNR}}{4 \cdot 2}\right)^{-2 n_r} \;=\; (1 + \text{SNR})^{-2 n_r}.$ The slope is $2 n_r = n_t n_r$ — full MIMO diversity. Alamouti's minimum determinant over all codeword pairs is $\det (8\mathbf{I}_2) = 64$ , and this value is achieved on every one-symbol-error pair (not just the pair above). Ch. 11 will show this is why Alamouti is a full-rank + constant-determinant code — and why it is the canonical $2\times 2$ space-time code. $\blacksquare$

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Quick Check

A space-time code for $n_t = 4, n_r = 2$ has error matrices with minimum rank $r_{\min} = 3$ over all codeword pairs. What is the high-SNR slope of the PEP upper bound (bits of decay per doubling of SNR)?

$2$ (i.e., $2$ orders per $6$ dB)

$3$

$6$

$8$ (full $n_t n_r$ )

Correction:

6

At high SNR, the PEP bound is $(\text{SNR}/(4 n_t))^{-r n_r}$ to leading order (Theorem TSpace-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999)). The slope in log-log is $r n_r = 3 \cdot 2 = 6$ — i.e., the PEP decays by a factor of $2^6 = 64$ per $3$ dB increase in SNR, equivalently $6$ decades per $30$ dB. The code achieves $r_{\min} n_r$ diversity order, not the full MIMO diversity of $n_t n_r = 8$ .

Key Takeaway

The PEP bound is the central object. Via Chernoff + MGF-of- $\chi^2$ it decomposes into a product over the eigenvalues of $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$ . Two structural properties of the code fall out at high SNR: the rank $r$ of the minimum-rank error matrix controls the diversity order $r n_r$ (slope of the PEP curve — rank criterion, §4), and the minimum determinant $\prod_i \lambda_i$ controls the coding gain (intercept of the PEP curve — determinant criterion, §5).

Historical Note: Tarokh, Seshadri, Calderbank 1998 — The Paper that Started STC

1998–1999

Tarokh, Seshadri, and Calderbank's 1998 IEEE Trans. IT paper "Space-time codes for high data rate wireless communication: Performance criterion and code construction" is the origin of space-time coding as a formal discipline. The paper established the PEP bound of Theorem TSpace-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999), derived the rank and determinant criteria, and gave the first systematic trellis-based constructions (Tarokh-Seshadri-Calderbank trellis codes).

Simultaneously, Siavash Alamouti's 1998 IEEE JSAC paper "A simple transmit diversity technique for wireless communications" introduced the $2 \times 2$ Alamouti scheme — the simplest and most practical full-diversity full-rate $2 \times n_r$ space-time code. Alamouti was not an information theorist: his motivation was engineering (build a receiver-side SIMO-equivalent using only transmit-side processing). The two papers — Tarokh et al. (theory) and Alamouti (engineering) — together defined the field in 1998 and every subsequent development (OSTBCs, DMT, LAST, CDA codes) builds on them. Guey-Fitz-Bell-Kuo 1999 provided a concurrent and equivalent derivation of the PEP bound and design criteria.

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Pairwise Error Probability for Space-Time Codes

Why PEP is the Right Lens

Definition: Space-Time Codebook and Error Matrix

Theorem: Space-Time PEP Bound (Tarokh-Seshadri-Calderbank 1998 / Guey-Fitz-Bell-Kuo 1999)

Conditional ML decoder

Conditional PEP via the Q-function

Chernoff bound on the Q-function

MGF of a weighted $\chi^2$ sum

Assemble the unconditional PEP bound

High-SNR simplification

PEP Upper Bound vs. SNR, Varying Rank and Determinant

Parameters

Pattern-Aware: Same Chernoff + MGF Template as Ch. 2 and Ch. 6

Common Mistake: PEP Is a Union Bound — Often Loose at Low SNR

Example: Eigenvalues of ΔΔH\boldsymbol{\Delta}\boldsymbol{\Delta}^HΔΔH for an Alamouti Codeword Pair

Error matrix

Product $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$

Implication for PEP

Quick Check

Key Takeaway

Historical Note: Tarokh, Seshadri, Calderbank 1998 — The Paper that Started STC

Definition:
Space-Time Codebook and Error Matrix

Example: Eigenvalues of $\boldsymbol{\Delta}\boldsymbol{\Delta}^H$ for an Alamouti Codeword Pair