Ferkans — Interactive Telecom Tutor

From Deterministic to Random Channels

In practice, the channel matrix $\mathbf{H}$ is random --- it changes with time, frequency, and user location. Section 15.3 gave the capacity for a known, fixed $\mathbf{H}$ . Now we ask: what rate can we sustain when $\mathbf{H}$ is drawn from a random distribution?

The answer depends critically on the delay constraint and the channel state information (CSI) available at the transmitter. Two complementary capacity metrics emerge: ergodic capacity (long codewords that experience all channel states) and outage capacity (short codewords that see a single channel realisation).

Definition:
Ergodic MIMO Capacity

The ergodic capacity of a fading MIMO channel is the capacity averaged over all channel realisations:

$C_{\mathrm{erg}} = \mathbb{E}_{\mathbf{H}}\!\left[\max_{\mathbf{Q}(\mathbf{H}):\; \mathrm{tr}(\mathbf{Q}) \leq P} \log_2 \det\!\left(\mathbf{I} + \frac{1}{\sigma^2}\mathbf{H}\mathbf{Q}\mathbf{H}^{H}\right)\right]$

This is achievable when codewords span many independent fading realisations (ergodic regime), i.e., the coding block length is much larger than the coherence time.

When the transmitter has no CSI (only the receiver knows $\mathbf{H}$ ), the optimal input is $\mathbf{Q} = (P/n_t)\mathbf{I}_{n_t}$ (equal power, isotropic transmission), and the capacity simplifies to

$C_{\mathrm{erg}} = \mathbb{E}_{\mathbf{H}}\!\left[\log_2 \det\!\left(\mathbf{I}_{n_r} + \frac{\text{SNR}}{n_t}\mathbf{H}\mathbf{H}^{H}\right)\right]$

In the no-CSIT case, transmitting isotropically is optimal because without knowledge of $\mathbf{H}$ , no direction is preferred. This is the most common scenario in practice (e.g., downlink in FDD systems without channel feedback).

Theorem: Telatar's Formula for Ergodic MIMO Capacity

For an i.i.d. Rayleigh fading MIMO channel with $n_t$ transmit and $n_r$ receive antennas, no CSIT, and $\text{SNR} = P/\sigma^2$ , the ergodic capacity is

$C_{\mathrm{erg}} = \mathbb{E}\!\left[\sum_{i=1}^{m} \log_2\!\left(1 + \frac{\text{SNR}}{n_t}\lambda_i\right)\right]$

where $m = \min(n_t, n_r)$ , and $\lambda_1, \ldots, \lambda_m$ are the $m$ nonzero eigenvalues of the Wishart matrix $\mathbf{W}$ :

$\mathbf{W} = \begin{cases} \mathbf{H}\mathbf{H}^{H} & \text{if } n_r \leq n_t \\ \mathbf{H}^{H}\mathbf{H} & \text{if } n_r > n_t \end{cases}$

The joint density of $\{\lambda_i\}$ is the unordered eigenvalue distribution of a complex Wishart matrix with parameters $(m, n)$ where $n = \max(n_t, n_r)$ :

$f(\lambda_1, \ldots, \lambda_m) = K_{m,n} \prod_{i<j}(\lambda_i - \lambda_j)^2 \prod_{i=1}^{m} \lambda_i^{n-m} e^{-\lambda_i}$

where $K_{m,n}$ is a normalisation constant.

The capacity is the sum of rates across $\min(n_t, n_r)$ spatial sub-channels, each with a random gain $\lambda_i$ drawn from the Wishart distribution. The Vandermonde term $\prod_{i<j}(\lambda_i - \lambda_j)^2$ ensures eigenvalue repulsion --- the eigenvalues tend to spread out, which is good for spatial multiplexing.

Proof

Optimal input with no CSIT

Without CSIT, the capacity-achieving input distribution is $\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, (P/n_t)\mathbf{I})$ . This follows from the symmetry of the i.i.d. Rayleigh model: for any unitary $\mathbf{V}$ , $\mathbf{H}\mathbf{V}$ has the same distribution as $\mathbf{H}$ , so no input direction is preferred.

Eigenvalue decomposition

The mutual information for a given realisation is

$I = \log_2\det\!\left(\mathbf{I} + \frac{\text{SNR}}{n_t}\mathbf{H}\mathbf{H}^{H}\right) = \sum_{i=1}^{m}\log_2\!\left(1 + \frac{\text{SNR}}{n_t}\lambda_i\right)$

where $\lambda_i$ are eigenvalues of $\mathbf{H}\mathbf{H}^{H}$ .

Wishart distribution

Since the entries of $\mathbf{H}$ are i.i.d. $\mathcal{CN}(0,1)$ , the matrix $\mathbf{H}\mathbf{H}^{H}$ follows a complex Wishart distribution $\mathcal{W}_{n_r}(n_t, \mathbf{I})$ . The joint eigenvalue density is known from random matrix theory (James 1964, Telatar 1999). Taking the expectation over this distribution yields the ergodic capacity formula. $\blacksquare$

,

Definition:
Outage Capacity

When codewords cannot span multiple fading realisations (non-ergodic regime, e.g., slow fading), the instantaneous mutual information $I(\mathbf{H})$ is a random variable. The $\varepsilon$ -outage capacity $C_\varepsilon$ is defined as

$\Pr\!\left[\log_2 \det\!\left(\mathbf{I} + \frac{\text{SNR}}{n_t}\mathbf{H}\mathbf{H}^{H}\right) < C_\varepsilon\right] = \varepsilon$

i.e., $C_\varepsilon$ is the rate that can be supported with probability $1 - \varepsilon$ . Typical values are $\varepsilon = 0.01$ or $\varepsilon = 0.05$ (1% or 5% outage).

Outage capacity is the relevant metric for delay-sensitive applications (voice, real-time video) where the codeword length is comparable to or shorter than the coherence time. MIMO diversity helps by making the outage probability decay faster with SNR.

Ergodic Capacity and Capacity CDF

Plot the CDF of the instantaneous mutual information for different antenna configurations. The ergodic capacity is the mean, and the outage capacity is read from the CDF at a given outage probability.

Parameters

n_t

4

n_r

4

SNR (dB)20

\\varepsilon

0.05

Outage probability (vertical line on CDF)

MIMO Capacity Scaling with Antenna Count

Animate how the ergodic capacity grows as the number of antennas increases from 1 to $n_{\max}$ (with $n_t = n_r$ ). Observe the approximately linear scaling $C \approx \min(n_t, n_r) \log_2(1 + \text{SNR}/n_t)$ for i.i.d. Rayleigh channels.

Parameters

n_{\\max}

8

Maximum number of antennas (both Tx and Rx)

SNR (dB)20

Example: Ergodic Capacity of $4 \times 4$ i.i.d. Rayleigh Channel

Compute the ergodic capacity of a $4 \times 4$ MIMO system at $\text{SNR} = 20$ dB with i.i.d. Rayleigh fading and no CSIT. Compare with: (a) a SISO channel at the same SNR, (b) the deterministic capacity with a perfectly conditioned channel ( $\sigma_i = 1$ for all $i$ ).

Solution

Apply Telatar formula

For $n_t = n_r = 4$ with $\text{SNR} = 100$ (20 dB):

$C_{\mathrm{erg}} = \mathbb{E}\!\left[\sum_{i=1}^{4} \log_2\!\left(1 + \frac{100}{4}\lambda_i\right)\right] = \mathbb{E}\!\left[\sum_{i=1}^{4} \log_2(1 + 25\lambda_i)\right]$

The expected eigenvalues of a $4 \times 4$ Wishart matrix $\mathbf{H}\mathbf{H}^{H}$ (with $n_t = 4$ ) are approximately $\bar{\lambda}_1 \approx 7.47$ , $\bar{\lambda}_2 \approx 3.53$ , $\bar{\lambda}_3 \approx 1.73$ , $\bar{\lambda}_4 \approx 0.56$ (from the Marchenko-Pastur distribution moments).

Numerical evaluation

Using the expected eigenvalues as an approximation (Jensen's inequality makes this a lower bound):

$C_{\mathrm{erg}} \gtrsim \log_2(1 + 25 \times 7.47) + \log_2(1 + 25 \times 3.53) + \log_2(1 + 25 \times 1.73) + \log_2(1 + 25 \times 0.56)$

$\approx \log_2(187.8) + \log_2(89.3) + \log_2(44.3) + \log_2(15.0)$

$\approx 7.55 + 6.48 + 5.47 + 3.91 = 23.4 \;\text{bits/s/Hz}$

Monte Carlo simulation gives $C_{\mathrm{erg}} \approx 22.0$ bits/s/Hz (the Jensen approximation overestimates slightly).

Comparisons

(a) SISO: $C_{\mathrm{SISO}} = \mathbb{E}[\log_2(1 + 100|h|^2)] \approx 5.0$ bits/s/Hz. The $4 \times 4$ MIMO provides a $\sim 4.4\times$ capacity increase.

(b) Deterministic with $\sigma_i = 1$ : $C_{\mathrm{det}} = 4\log_2(1 + 25) \approx 4 \times 4.70 = 18.8$ bits/s/Hz. The random channel actually achieves higher capacity because the expected eigenvalue spread of the Wishart distribution pushes some eigenvalues above 1, compensating for those below 1. $\blacksquare$

Common Mistake: Ergodic Capacity Is Not Achievable in Slow Fading

Mistake:

Quoting ergodic capacity as the "MIMO capacity" for a system operating in a slow-fading environment where the channel is approximately constant over the entire codeword duration.

Correction:

In slow fading, the relevant metric is outage capacity, which is always lower than ergodic capacity. For example, a $2 \times 2$ system at 20 dB SNR might have ergodic capacity of 12 bits/s/Hz but only 7 bits/s/Hz at 1% outage. The gap depends on the diversity order: more antennas (higher diversity) make the outage CDF steeper, bringing outage capacity closer to ergodic capacity.

Common Mistake: Capacity Scales Linearly with $\min(n_t, n_r)$ , Not $n_t + n_r$

Mistake:

Claiming that doubling the total number of antennas doubles the capacity. For example, expecting a $4 \times 1$ system (5 total antennas) to have higher capacity than a $2 \times 2$ system (4 total antennas).

Correction:

At high SNR, capacity scales as $\min(n_t, n_r) \log_2(\text{SNR}/n_t)$ . The $2 \times 2$ system has $\min(2,2) = 2$ multiplexing streams, while the $4 \times 1$ (MISO) system has only $\min(4,1) = 1$ stream. The MISO system gets a beamforming (array) gain of $\log_2(n_t) = 2$ bits, but the $2\times 2$ gets a multiplexing gain of $2\times$ the high-SNR slope. At 20 dB, the $2 \times 2$ vastly outperforms the $4 \times 1$ in capacity.

Quick Check

For an i.i.d. Rayleigh fading $n_t \times n_r$ MIMO channel with no CSIT, what is the optimal transmit strategy?

Isotropic transmission: $\mathbf{Q} = (P/n_t)\mathbf{I}_{n_t}$

Beamforming along the strongest eigenvector of $\mathbb{E}[\mathbf{H}^{H}\mathbf{H}]$

Concentrate all power on one antenna

Water-filling based on the channel statistics $\mathbf{R}_{t}$

Correction:

Isotropic transmission:

\mathbf{Q} = (P/n_t)\mathbf{I}_{n_t}

Without knowledge of $\mathbf{H}$ , no transmit direction is preferred (the channel distribution is unitarily invariant). Equal power across all transmit antennas is optimal.

Ergodic capacity

The capacity of a fading channel averaged over the channel distribution, achievable when codewords span many independent fading realisations.

Related: Outage capacity, MIMO capacity

Outage capacity

The rate $C_\varepsilon$ supportable with probability at least $1 - \varepsilon$ over a fading channel. Relevant for delay-constrained communication in slow fading.

Related: Ergodic capacity

Wishart matrix

A random matrix of the form $\mathbf{W} = \mathbf{H}\mathbf{H}^{H}$ where $\mathbf{H}$ has i.i.d. Gaussian entries. The eigenvalue distribution of Wishart matrices governs MIMO capacity statistics.

Related: Ergodic capacity

MIMO Capacity: Fading Channels

From Deterministic to Random Channels

Definition: Ergodic MIMO Capacity

Theorem: Telatar's Formula for Ergodic MIMO Capacity

Optimal input with no CSIT

Eigenvalue decomposition

Wishart distribution

Definition: Outage Capacity

Ergodic Capacity and Capacity CDF

Parameters

MIMO Capacity Scaling with Antenna Count

Parameters

Example: Ergodic Capacity of 4×44 \times 44×4 i.i.d. Rayleigh Channel

Apply Telatar formula

Numerical evaluation

Comparisons

Common Mistake: Ergodic Capacity Is Not Achievable in Slow Fading

Common Mistake: Capacity Scales Linearly with min⁡(nt,nr)\min(n_t, n_r)min(nt​,nr​), Not nt+nrn_t + n_rnt​+nr​

Quick Check

Ergodic capacity

Outage capacity

Wishart matrix

Definition:
Ergodic MIMO Capacity

Definition:
Outage Capacity

Example: Ergodic Capacity of $4 \times 4$ i.i.d. Rayleigh Channel

Common Mistake: Capacity Scales Linearly with $\min(n_t, n_r)$ , Not $n_t + n_r$