Ferkans — Interactive Telecom Tutor

When Capacity Becomes Random

The ergodic capacity of §1 is the right answer only if the code spans enough independent fades to invoke the ergodic theorem on $\mathbb{E}[\log\det(\cdot)]$ . That is a strong assumption. In practice a typical wireless codeword (an LTE transport block, a 5G NR slot, a Wi-Fi packet) has duration well below the channel's coherence time: the fading is constant over the codeword and changes only between codewords. The channel that matters to the designer is the quasi-static (or block-fading) model.

On a block-fading channel, the conditional capacity $C(\mathbf{H}) = \log\det(\mathbf{I} + (\text{SNR}/n_t)\mathbf{H}\mathbf{H}^{H})$ is itself a random variable. There is no rate $R$ that is achievable with probability one — for any $R > 0$ , the event $\{C(\mathbf{H}) < R\}$ has strictly positive probability, and on that event no code whatsoever can decode reliably. This forces a shift in perspective: from "what is the capacity?" to "what codes minimise the probability of the channel being too weak to support the target rate?". That probability is the outage probability, and the matching rate metric is the outage capacity.

The operational link to space-time code design is direct: a code whose PEP is large on every channel realisation in the outage set is a bad code; a code whose PEP is small on every non-outage realisation is a good code. Sections §3–§5 make this link quantitative via the PEP analysis. This section sets up the outage machinery itself.

,

Definition:
Quasi-Static (Block-Fading) MIMO Channel

The quasi-static (or block-fading) MIMO channel is the model $\mathbf{y}_t \;=\; \mathbf{H}\mathbf{x}_t + \mathbf{w}_{t}, \quad t = 1, \ldots, T,$ where the channel matrix $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ is drawn from a fading distribution (typically i.i.d. Rayleigh) and held constant across all $T$ time samples of a codeword. Between codewords, $\mathbf{H}$ is redrawn independently. The receiver has perfect CSIR; the transmitter has no CSI (CSIT).

The conditional capacity for a given realisation $\mathbf{H}$ is $C(\mathbf{H}) \;=\; \log_2 \det\!\left( \mathbf{I}_{n_r} + \tfrac{\text{SNR}}{n_t} \mathbf{H}\mathbf{H}^{H} \right),$ a function of the random matrix $\mathbf{H}$ only.

Two limiting regimes bracket the block-fading model. On one side, fast fading (the channel changes every symbol, $T \to \infty$ independent fades) makes the ergodic capacity attainable and renders classical AWGN-style capacity-achieving codes near-optimal. On the other side, the pure AWGN channel is the degenerate case where $\mathbf{H}$ is deterministic. Block fading is the operationally relevant middle ground and the setting of all of Part III.

,

Definition:
Outage Probability and $\epsilon$ -Outage Capacity

For a block-fading MIMO channel with random conditional capacity $C(\mathbf{H})$ and a fixed target rate $R$ (bits/channel use), the outage probability at rate $R$ is $P_{\mathrm{out}}(R) \;\triangleq\; \Pr\!\left[ C(\mathbf{H}) < R \right] \;=\; \Pr\!\left[ \log_2 \det\!\left( \mathbf{I}_{n_r} + \tfrac{\text{SNR}}{n_t} \mathbf{H}\mathbf{H}^{H} \right) < R \right].$

For a fixed outage tolerance $\epsilon \in (0, 1)$ , the $\epsilon$ -outage capacity is the largest rate such that the outage probability stays below $\epsilon$ : $C_\epsilon \;\triangleq\; \sup\{ R : P_{\mathrm{out}}(R) \le \epsilon \}.$ Equivalently, $C_\epsilon$ is the $\epsilon$ -quantile of the random variable $C(\mathbf{H})$ .

Operationally, $P_{\mathrm{out}}(R)$ is the lower bound on the error probability of any code of rate $R$ on the block-fading channel — no matter how clever the code, the channel sometimes hands you a realisation that cannot support rate $R$ . The $\epsilon$ -outage capacity is the designer's knob: systems that can tolerate a 10% outage (speech, best-effort data) operate at $C_{0.1}$ ; systems that need $10^{-4}$ outage (URLLC, control channels) operate at $C_{10^{-4}}$ , which is dramatically smaller.

, ,

Outage Probability $P_{\mathrm{out}}(R)$ vs. SNR at Fixed Rate

The MIMO outage probability at a fixed target rate $R$ , computed by Monte-Carlo over i.i.d. Rayleigh channel realisations. At high SNR the slope of the curve (in log-log scale) equals $n_t n_r$ — the full MIMO diversity order that is achievable by an outage-optimal code (forward reference to the DMT of Ch. 12).

Parameters

n_t

2

n_r

2

Target rate [bits/ch. use]4

Theorem: High-SNR Scaling of $P_{\mathrm{out}}(R)$ (DMT Preview)

For the $n_t \times n_r$ i.i.d. Rayleigh block-fading MIMO channel, if the target rate scales as $R(\text{SNR}) = r \log_2 \text{SNR}$ with fixed multiplexing gain $r \in [0, \min(n_t, n_r)]$ , then the outage probability decays polynomially in SNR: $P_{\mathrm{out}}(r \log_2 \text{SNR}) \;\doteq\; \text{SNR}^{-d^\star(r)},$ where $\doteq$ denotes exponential equality on the $\log \text{SNR}$ scale and the outage exponent $d^\star(r)$ is the piecewise-linear interpolation of $(k, (n_t - k)(n_r - k))$ for $k = 0, 1, \ldots, \min(n_t, n_r)$ . At $r = 0$ (fixed target rate), $d^\star(0) = n_t n_r$ — the maximum diversity order.

Outage occurs when the channel's ordered squared singular values $\lambda_i = \sigma_i^2(\mathbf{H})$ collapse jointly enough that $\sum_i \log_2(1 + (\text{SNR}/n_t)\lambda_i) < R$ . The joint probability of such a collapse is dominated, at high SNR, by the large-deviation rate of the smallest $n_t n_r - d^\star(r)$ eigenvalue exponents. Laplace's method on the Wishart density gives the piecewise-linear exponent — this is the Zheng-Tse DMT, covered in full in Ch. 12.

Show Hint

Parametrise the singular values as $\sigma_i^2 = \text{SNR}^{-\alpha_i}$ with $\alpha_i \ge 0$ ; the outage event is $\{\sum_i (1-\alpha_i)^+ < r\}$ .

The Wishart joint density of the $\alpha_i$ has the Laplace-method exponent $\sum_i (2i-1+|n_r-n_t|)\alpha_i$ ; minimise this subject to the outage constraint.

The solution is piecewise linear in $r$ ; verify the endpoints $(0, n_t n_r)$ and $(\min(n_t, n_r), 0)$ .

Proof

Scaling ansatz

By the SVD, the outage event $\{C(\mathbf{H}) < R\}$ is equivalent to $\{\sum_{i=1}^{\min(n_t, n_r)} \log_2(1 + (\text{SNR}/n_t)\sigma_i^2) < R\}$ . Substitute $\sigma_i^2 = \text{SNR}^{-\alpha_i}$ : at high $\text{SNR}$ , the $i$ -th log is $\log_2(1 + \text{SNR}^{1-\alpha_i}/n_t) \doteq (1 - \alpha_i)^+ \log_2 \text{SNR}$ . The outage event becomes $\{\sum_i (1 - \alpha_i)^+ < r\}$ , in terms of the scaled exponents $\alpha_i \ge 0$ .

Large-deviation exponent of the joint eigenvalue law

The joint probability density of $(\alpha_1, \ldots, \alpha_{\min(n_t, n_r)})$ under i.i.d. Rayleigh Wishart is, by Laplace's method on the Marchenko–Pastur weighting of singular values, $p(\boldsymbol{\alpha}) \doteq \text{SNR}^{-\sum_i (2i - 1 + |n_r - n_t|)\alpha_i}$ for $\alpha_1 \ge \cdots \ge \alpha_{\min(n_t, n_r)} \ge 0$ . Standard references: Zheng-Tse 2003 §II, Tse-Viswanath 2005 §9.1.

Optimise over the outage set

The outage probability is, exponentially, $P_{\mathrm{out}} \;\doteq\; \int_{\sum_i (1-\alpha_i)^+ < r} \text{SNR}^{-\sum_i (2i - 1 + |n_r - n_t|)\alpha_i} d\boldsymbol{\alpha} \;\doteq\; \text{SNR}^{-d^\star(r)},$ with $d^\star(r) = \inf_{\boldsymbol\alpha : \sum_i (1-\alpha_i)^+ < r} \sum_i (2i - 1 + |n_r - n_t|)\alpha_i$ . This is a linear program; its optimiser is piecewise-linear in $r$ , passing through the breakpoints $(k, (n_t-k)(n_r-k))$ for $k = 0, \ldots, \min(n_t, n_r)$ . Details in Ch. 12.

Full diversity at $r = 0$

At $r = 0$ (fixed target rate), the infimum is achieved at $\alpha_i = 1$ for all $i$ , giving $d^\star(0) = \sum_i (2i - 1 + |n_r - n_t|) = n_t n_r$ . This is the maximum diversity order of the MIMO channel — the number that the rank criterion of §4 will also produce as the maximum diversity achievable by a space-time code. $\blacksquare$

,

Outage Capacity $C_\epsilon$ vs. SNR

The $\epsilon$ -outage capacity of the $n_t \times n_r$ i.i.d. Rayleigh MIMO channel, computed as the $\epsilon$ -quantile of $\log_2\det(\mathbf{I} + (\text{SNR}/n_t)\mathbf{H}\mathbf{H}^{H})$ by Monte-Carlo. Curves are plotted for $\epsilon \in \{0.01, 0.1, 0.5\}$ and compared against the ergodic capacity $C$ . At low outage ( $\epsilon = 0.01$ ), $C_\epsilon$ is dramatically smaller than $C$ — the quantitative cost of the block-fading assumption.

Parameters

n_t

2

n_r

2

Example: Outage Capacity of $2\times 2$ Rayleigh at 10 dB, $\epsilon = 0.1$

For a $2 \times 2$ i.i.d. Rayleigh MIMO channel at $\text{SNR} = 10$ dB, estimate the $10\%$ -outage capacity $C_{0.1}$ and compare against the ergodic capacity $C \approx 5.72$ bits/channel use computed in Example $2\times 2$ $2 \times 2$ and $4\times 4$ Rayleigh" data-ref-type="example">EMIMO Capacity at 10 dB for $2\times 2$ and $4\times 4$ Rayleigh.

Solution

Monte-Carlo distribution of $\ntn{cap}(\ntn{ch})$

Generate $N = 10^5$ i.i.d. $2\times 2$ Rayleigh realisations. Compute the conditional capacity $C(\mathbf{H}) = \log_2 \det(\mathbf{I}_2 + 5 \mathbf{H}\mathbf{H}^{H})$ for each, yielding an empirical distribution of capacities.

Read the $10\%$-quantile

The $10\%$ -quantile of this distribution gives $C_{0.1} \approx 3.9$ bits/channel use. The median (quantile $0.5$ ) gives $C_{0.5} \approx 5.7$ , matching the ergodic $C$ .

Interpretation

At $\epsilon = 0.1$ , the outage capacity is about $68\%$ of the ergodic: a $2 \times 2$ system at $10$ dB with a $10\%$ outage tolerance can reliably transmit at only $3.9$ bits/channel use, though its ergodic mean is $5.7$ . The gap grows as $\epsilon$ shrinks: at $\epsilon = 0.01$ , $C_{0.01} \approx 2.5$ bits — less than half the ergodic.

A code that operates at $R < C_\epsilon$ must avoid PEPs that contribute to the $\epsilon$ -outage event. This is exactly what the rank and determinant criteria (§§4, 5) achieve: they ensure that the PEP's high-SNR behaviour is ruled by the full $n_t n_r$ diversity order, matching the outage exponent $d^\star(0)$ of Theorem $P_{\mathrm{out}}(R)$ $P_{out} (R)$ (DMT Preview)" data-ref-type="theorem">THigh-SNR Scaling of $P_{\mathrm{out}}(R)$ (DMT Preview). $\blacksquare$

Common Mistake: Do Not Quote Ergodic Capacity for a Slow-Fading System

Mistake:

A report or textbook cites "the MIMO capacity at 10 dB is $5.7$ bits/channel use" in the context of a system whose coherence time exceeds the packet duration, and then proceeds to design codes assuming this rate is achievable with probability close to one.

Correction:

Ergodic capacity is the time average of $C(\mathbf{H})$ under the assumption that a codeword spans many independent fades. On a quasi-static channel, a codeword is on a single fade realisation; the meaningful benchmark is the outage capacity $C_\epsilon$ , which at $\epsilon = 0.1$ is typically $50$ – $70\%$ of the ergodic capacity and at $\epsilon = 10^{-4}$ can be $20\%$ or less. The gap closes only when the code is designed to exploit full MIMO diversity (full-rank $\mathbf{\Delta}$ — see §4), in which case the BER slope matches the outage slope $n_t n_r$ .

Why This Matters: From Outage to Space-Time Codes and the DMT

The outage probability is the target that space-time codes try to approach. Two follow-ups sharpen this into concrete code design:

Ch. 11 (Space-Time Block Codes): Alamouti and orthogonal STBCs are codes that achieve full diversity $n_t n_r$ at the cost of rate bounded above by $1$ (or $3/4$ for $n_t > 2$ with complex symbols). Their PEP slope matches the $r = 0$ outage exponent $d^\star(0) = n_t n_r$ .
Ch. 12 (Diversity-Multiplexing Tradeoff): The Zheng-Tse framework characterises the entire $(r, d)$ tradeoff curve that space-time codes can approach. The rank and determinant criteria introduced in §§4, 5 are the high-SNR asymptotic expression of the endpoint $r = 0$ of this curve.
Ch. 13 (CDA / Perfect Codes): The Elia-Kumar-Pawar-Kumar-Caire 2006 CommIT contribution constructs space-time codes that achieve the entire DMT curve for all $(n_t, n_r)$ , via cyclic division algebras — structurally optimal lattice constructions over number fields.

See full treatment in Motivating the Tradeoff: Diversity vs Multiplexing

Key Takeaway

Outage is the right benchmark. On a block-fading MIMO channel, the meaningful capacity notion is $C_\epsilon$ , not the ergodic $C$ . At high SNR, $P_{\mathrm{out}}$ decays polynomially in SNR with exponent $d^\star(r)$ (Zheng-Tse DMT), and the maximum diversity order $d^\star(0) = n_t n_r$ is the target that space-time codes chase. The rank criterion of §4 says what the code must achieve; the determinant criterion of §5 says how to optimise within the full-diversity class.

Block-Fading Model, Outage Probability, and Outage Capacity

When Capacity Becomes Random

Definition: Quasi-Static (Block-Fading) MIMO Channel

Definition: Outage Probability and ϵ\epsilonϵ-Outage Capacity

Outage Probability Pout(R)P_{\mathrm{out}}(R)Pout​(R) vs. SNR at Fixed Rate

Parameters

Theorem: High-SNR Scaling of Pout(R)P_{\mathrm{out}}(R)Pout​(R) (DMT Preview)

Scaling ansatz

Large-deviation exponent of the joint eigenvalue law

Optimise over the outage set

Full diversity at $r = 0$

Outage Capacity CϵC_\epsilonCϵ​ vs. SNR

Parameters

Example: Outage Capacity of 2×22\times 22×2 Rayleigh at 10 dB, ϵ=0.1\epsilon = 0.1ϵ=0.1

Monte-Carlo distribution of $\ntn{cap}(\ntn{ch})$

Read the $10\%$-quantile

Interpretation

Common Mistake: Do Not Quote Ergodic Capacity for a Slow-Fading System

Why This Matters: From Outage to Space-Time Codes and the DMT

Key Takeaway

Definition:
Quasi-Static (Block-Fading) MIMO Channel

Definition:
Outage Probability and $\epsilon$ -Outage Capacity

Outage Probability $P_{\mathrm{out}}(R)$ vs. SNR at Fixed Rate

Theorem: High-SNR Scaling of $P_{\mathrm{out}}(R)$ (DMT Preview)

Outage Capacity $C_\epsilon$ vs. SNR

Example: Outage Capacity of $2\times 2$ Rayleigh at 10 dB, $\epsilon = 0.1$