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From ITA Ch. 13.5 to Space-Time Coding

Book ITA Ch. 13.5 established the capacity of the i.i.d. Rayleigh MIMO channel — arguably the single most influential result in modern wireless information theory. Two independent papers, Telatar (1995/1999, AT&T Bell Labs Technical Memorandum and later the European Transactions on Telecommunications / IEEE Trans. IT paper) and Foschini–Gans (1998, Wireless Personal Communications), proved that adding antennas at both ends of a wireless link scales capacity linearly in $\min(n_t, n_r)$ , not logarithmically as the SISO Shannon formula $\log_2(1 + \text{SNR})$ suggests. This was the trigger for twenty-five years of MIMO research and for every modern wireless standard from 3G HSPA onwards.

From an information-theoretic standpoint, the capacity is the end of the story. From a coding-theoretic standpoint, it is the beginning. In Part III we ask: given the capacity, how do we design codes that (a) approach it when the fading is ergodic, and (b) operate reliably on the much harsher block-fading channel where the capacity is itself random? This chapter sets up the two pieces of the answer: the capacity formula we are trying to approach (§1 — a review), and the code design criteria (rank and determinant) that tell us how to build codes that achieve the right diversity and coding gain on block-fading channels (§3, §4, §5).

We treat the MIMO capacity review briefly — the full proof is in ITA Ch. 13.5 — and concentrate on the structural features that are operationally relevant to space-time coding: the multiplexing prelog $\min(n_t, n_r)$ , the parallel-subchannel decomposition via the SVD of $\mathbf{H}$ , and the difference between the ergodic and outage notions of capacity (fleshed out in §2).

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Definition:
MIMO System Model

A point-to-point MIMO channel with $n_t$ transmit antennas and $n_r$ receive antennas is the discrete-time complex-baseband model $\mathbf{y} \;=\; \mathbf{H}\mathbf{x} + \mathbf{w},$ where:

$\mathbf{x} \in \mathbb{C}^{n_t}$ is the transmitted symbol vector, satisfying the average-power constraint $\mathbb{E}[\|\mathbf{x}\|^2] \le P$ ;
$\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ is the channel matrix;
$\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I}_{n_r})$ is circularly-symmetric complex AWGN;
$\mathbf{y} \in \mathbb{C}^{n_r}$ is the received vector.

The transmit SNR is $\text{SNR} \triangleq P/\sigma^2$ . Under the i.i.d. Rayleigh assumption, the entries $[\mathbf{H}]_{ij}$ are i.i.d. $\mathcal{CN}(0, 1)$ . We write $\mathbf{H} \sim \mathcal{CN}(0, \mathbf{I}_{n_r n_t})$ in $\mathrm{vec}(\mathbf{H})$ -form.

The normalisation $\mathbb{E}[|[\mathbf{H}]_{ij}|^2] = 1$ is a modelling convention: path-loss and shadowing are absorbed into the linear-scale $\text{SNR}$ rather than the channel variance. The transmit power $P$ is the sum power across antennas; it does NOT scale with $n_t$ , so the per-antenna power budget decreases as we add antennas. This is why the capacity formula below has $\text{SNR}/n_t$ , not $\text{SNR}$ — adding antennas at the transmitter distributes a fixed power budget.

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Theorem: Ergodic MIMO Capacity (Telatar 1999)

For the i.i.d. Rayleigh MIMO channel with $n_t$ transmit antennas, $n_r$ receive antennas, transmit SNR $\text{SNR}$ , and perfect CSI at the receiver (CSIR), the ergodic capacity under equal-power (isotropic) transmission is $C(\text{SNR}; n_t, n_r) \;=\; \mathbb{E}_{\mathbf{H}}\!\left[ \log_2 \det\!\left( \mathbf{I}_{n_r} + \tfrac{\text{SNR}}{n_t} \mathbf{H}\mathbf{H}^{H} \right) \right] \quad \text{bits/channel use.}$ At high SNR, this behaves as $C(\text{SNR}; n_t, n_r) \;=\; \min(n_t, n_r)\,\log_2\text{SNR} \;+\; O(1),$ i.e., the multiplexing prelog is $\min(n_t, n_r)$ .

Diagonalise $\mathbf{H}$ via the SVD: $\mathbf{H} = \mathbf{U}\boldsymbol{\Sigma} \mathbf{V}^H$ with $\boldsymbol{\Sigma}$ holding the singular values $\sigma_1 \ge \cdots \ge \sigma_{\min(n_t,n_r)} \ge 0$ . Pre-rotating the input by $\mathbf{V}$ and post-rotating the output by $\mathbf{U}^H$ turns the MIMO channel into $\min(n_t, n_r)$ parallel scalar sub-channels with gains $\sigma_i^2$ . Water-filling over the random gains gives a capacity that, for i.i.d. Rayleigh, reduces to the isotropic formula above (water-filling and equal allocation coincide in expectation for i.i.d. channels). At high SNR, each active sub-channel contributes $\log_2(\sigma_i^2 \text{SNR}/n_t)$ , and the $\log$ -det splits into a sum of $\min(n_t, n_r)$ logarithms — hence the prelog.

Show Hint

Start from the MIMO mutual information with Gaussian input and isotropic covariance $\mathbf{K}_\mathbf{x} = (P/n_t)\mathbf{I}$ .

Use the determinant identity $\det(\mathbf{I}_{n_r} + \mathbf{A}\mathbf{B}) = \det(\mathbf{I}_{n_t} + \mathbf{B}\mathbf{A})$ to move between the $n_r$ -dim and $n_t$ -dim forms.

For the high-SNR asymptotic, factor $\log_2\text{SNR}$ out of the $\min(n_t, n_r)$ dominant terms and verify the $O(1)$ constant is the offset encoded by $\mathbb{E}[\log_2\det((1/n_t)\mathbf{H}\mathbf{H}^{H})]$ .

Proof

Gaussian input achieves capacity

The capacity of the vector Gaussian channel with known $\mathbf{H}$ and input covariance constraint $\mathrm{tr}(\mathbf{K}_\mathbf{x}) \le P$ is a concave optimisation over $\mathbf{K}_\mathbf{x} \succeq 0$ : $C(\mathbf{H}) \;=\; \max_{\mathbf{K}_\mathbf{x}:\, \mathrm{tr} \mathbf{K}_\mathbf{x} \le P} \log_2 \det\!\left( \mathbf{I}_{n_r} + {\sigma^2}^{-1}\mathbf{H}\mathbf{K}_\mathbf{x}\mathbf{H}^{H} \right).$ For i.i.d. Rayleigh $\mathbf{H}$ (the distribution is invariant under unitary transformations $\mathbf{H} \to \mathbf{U}\mathbf{H}\mathbf{V}$ ), the capacity-achieving $\mathbf{K}_\mathbf{x}$ averaged over $\mathbf{H}$ is $\mathbf{K}_\mathbf{x}^\star = (P/n_t)\mathbf{I}_{n_t}$ : any unitary rotation of the input leaves the ensemble distribution unchanged, so isotropic input is an optimiser.

Ergodic average

Substituting $\mathbf{K}_\mathbf{x}^\star = (P/n_t)\mathbf{I}$ and $\text{SNR} = P/\sigma^2$ yields the conditional capacity $C(\mathbf{H}) = \log_2 \det(\mathbf{I}_{n_r} + (\text{SNR}/n_t) \mathbf{H}\mathbf{H}^{H})$ . The ergodic capacity is the expectation over the fading distribution, $C \;=\; \mathbb{E}_{\mathbf{H}}\big[ C(\mathbf{H}) \big].$ Operationally, this is the capacity when the codeword spans many independent fades (ergodic assumption).

SVD and parallel sub-channels

Write $\mathbf{H} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$ , with $\boldsymbol{\Sigma} = \mathrm{diag}(\sigma_1, \ldots, \sigma_{\min(n_t, n_r)})$ . Then $\log_2 \det(\mathbf{I} + (\text{SNR}/n_t)\mathbf{H}\mathbf{H}^{H}) = \sum_{i=1}^{\min(n_t, n_r)} \log_2(1 + (\text{SNR}/n_t)\sigma_i^2)$ . This is a sum of $\min(n_t, n_r)$ independent SISO capacities — the MIMO channel decomposes into parallel eigen-sub-channels.

High-SNR prelog

At large $\text{SNR}$ , each active term contributes $\log_2(\sigma_i^2 \text{SNR}/n_t) = \log_2\text{SNR} + \log_2(\sigma_i^2/n_t)$ , and the sum is $C \;=\; \min(n_t, n_r) \log_2\text{SNR} \;+\; \mathbb{E}\!\left[ \sum_i \log_2(\sigma_i^2/n_t) \right] + o(1).$ The first term is the multiplexing prelog; the second is a finite $O(1)$ offset that depends on the fading distribution. For i.i.d. Rayleigh it is negative (penalty relative to a deterministic parallel channel with identical eigenvalues). $\blacksquare$

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Ergodic MIMO Capacity $C(\text{SNR}; n_t, n_r)$ vs. SNR

Monte-Carlo ergodic capacity of the $n_t \times n_r$ i.i.d. Rayleigh MIMO channel under isotropic Gaussian input, swept over SNR in dB. The dashed reference is the SISO capacity $\log_2(1 + \text{SNR})$ . At high SNR the slope of each curve is $\min(n_t, n_r)$ bits/channel use per 3 dB.

Parameters

Transmit antennas

n_t

2

Receive antennas

n_r

2

Show SISO reference

Example: MIMO Capacity at 10 dB for $2\times 2$ and $4\times 4$ Rayleigh

Estimate the ergodic capacity of an i.i.d. Rayleigh MIMO channel at $\text{SNR} = 10$ dB for (a) $n_t = n_r = 2$ , (b) $n_t = n_r = 4$ , and (c) $n_t = n_r = 8$ . Compare against the SISO capacity $\log_2(1 + \text{SNR}) = \log_2(11) \approx 3.46$ bits/channel use, and verify the rough-cut high-SNR approximation $C \approx \min(n_t, n_r) \log_2 \text{SNR}$ .

Solution

SISO reference

At $\text{SNR} = 10$ linear, $\log_2(1 + 10) = \log_2(11) = 3.459$ bits/channel use. This is our baseline.

$2 \times 2$ ergodic capacity

Averaging $\log_2 \det(\mathbf{I}_2 + 5 \mathbf{H}\mathbf{H}^{H})$ over i.i.d. $\mathcal{CN}(0,1)$ entries (the $5 = \text{SNR}/n_t$ factor is $10/2$ ) gives $C \approx 5.72$ bits/channel use — about $1.65\times$ the SISO rate. The high-SNR estimate $2\log_2 10 = 6.64$ over-predicts (the $O(1)$ offset is negative: $\sim -0.92$ ).

$4 \times 4$ ergodic capacity

With $\text{SNR}/n_t = 2.5$ , $C \approx 10.94$ bits/channel use — about $3.16\times$ the SISO rate. High-SNR estimate $4\log_2 10 = 13.29$ ; offset $\sim -2.35$ . The capacity-per-antenna ratio $C/\min(n_t, n_r) = 2.73$ is below the SISO $3.46$ at the same total $\text{SNR}$ : the per-antenna penalty grows with $n_t$ because the power is split across more streams.

$8 \times 8$ ergodic capacity

At $n_t = n_r = 8$ with $\text{SNR}/n_t = 1.25$ , the Monte-Carlo average is $C \approx 18.3$ bits/channel use — about $5.3\times$ SISO. The multiplexing gain is unmistakable: eight antennas multiplied the spectral efficiency by more than five at 10 dB, without any code yet being specified.

Interpretation

The ergodic capacity grows roughly linearly in $\min(n_t, n_r)$ , validating the high-SNR prelog. The offset from the naive $\min(n_t,n_r)\log_2\text{SNR}$ prediction widens with $n_t$ because (i) the per-antenna SNR $\text{SNR}/n_t$ drops, and (ii) the Marchenko–Pastur eigenvalue spread of $(1/n_t)\mathbf{H}\mathbf{H}^{H}$ deviates from unity. $\blacksquare$

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Key Takeaway

The single most important number in MIMO theory is the multiplexing prelog $\min(n_t, n_r)$ . It determines how fast capacity grows in $\log_2\text{SNR}$ and, in Part III, it will re-emerge as the maximum multiplexing gain of the diversity-multiplexing tradeoff (Ch. 12). The purpose of a space-time code is to translate this potential into actual bits on the wire: either full $\min(n_t, n_r)$ streams (spatial multiplexing, low diversity — Ch. 11 V-BLAST) or full $n_t n_r$ diversity (Alamouti / OSTBCs, rate 1 — Ch. 11), or an informed compromise (lattice STCs — Chs. 13, 17).

Capacity Says 'How Much', Coding Says 'How'

The capacity formula of Theorem TErgodic MIMO Capacity (Telatar 1999) is a fundamental limit: it does not tell the designer which codewords to transmit, only that a rate $R < C$ is achievable with arbitrarily low error for sufficiently long blocks. On a fast-fading channel (fading i.i.d. across symbols), the AWGN capacity-achieving coding recipes of Part II (BICM + LDPC) port over almost verbatim — interleavers make the channel "effectively" ergodic.

On a block-fading channel (quasi-static fading over the whole codeword), this reasoning collapses: the channel is not ergodic in the codeword time-scale, the capacity $C(\mathbf{H})$ is a random variable, and diversity — not multiplexing — becomes the central code-design criterion. Sections §2–§5 develop this story.

Historical Note: Telatar 1995 and Foschini–Gans 1998 — The MIMO Capacity Result

1995–1999

Emre Telatar (at AT&T Bell Labs) circulated the technical memorandum "Capacity of multi-antenna Gaussian channels" in 1995. The result — that the capacity of an $n_t \times n_r$ MIMO channel scales as $\min(n_t, n_r)\log\text{SNR}$ — was initially met with disbelief: the prevailing intuition held that capacity must grow logarithmically in total power, not linearly in the min of antenna counts. The memorandum circulated informally for four years before appearing in the European Transactions on Telecommunications (1999) and becoming the most-cited wireless-capacity paper of all time.

Independently, Gerard Foschini and Michael Gans at Bell Labs published "On limits of wireless communications in a fading environment when using multiple antennas" in Wireless Personal Communications (1998). They reached the same conclusion and additionally introduced the V-BLAST architecture — a practical receiver that approaches the MIMO capacity by successive interference cancellation (see Ch. 11). The combination of these two papers triggered the modern MIMO era: by 2001 Lucent, Bell Labs (Foschini's home), and others had MIMO prototypes in the lab; by 2008 3GPP HSPA was commercial; by 2019 5G NR was shipping $64 \times 4$ massive-MIMO base stations.

The history is neatly told by Biglieri, Caire, Taricco, and others in survey articles (e.g., [?biglieri-caire-taricco-2000]) and in Goldsmith et al.'s 2003 survey [?goldsmith-jafar-jindal-vishwanath-2003].

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MIMO Capacity Review — From Book ITA to Space-Time Coding