Ferkans — Interactive Telecom Tutor

Multiple Antennas: The Spatial Dimension

So far we have studied scalar fading channels with a single transmit and receive antenna. The capacity scales as $\frac{1}{2}\log(\text{SNR})$ at high SNR — one additional bit per 3 dB increase. Can we do better without more bandwidth or more power?

The answer is a resounding yes: by using $n_t$ transmit and $n_r$ receive antennas, the high-SNR capacity scales as $\min(n_t, n_r) \cdot \log(\text{SNR})$ . This is the spatial multiplexing gain — we get $\min(n_t, n_r)$ independent parallel channels "for free" by exploiting the spatial dimension. This discovery, due to Foschini (1996) and Telatar (1999), revolutionized wireless communications and motivated the massive MIMO systems deployed in 5G networks today.

Definition:
MIMO Channel Model

The MIMO (multiple-input multiple-output) channel with $n_t$ transmit antennas and $n_r$ receive antennas is described by

$\mathbf{y} = \mathbf{H}\,\mathbf{x} + \mathbf{z},$

where:

$\mathbf{x} \in \mathbb{C}^{n_t}$ is the transmitted signal vector with covariance constraint $\mathbb{E}[\mathbf{x}\mathbf{x}^H] = \mathbf{K}_x$ and total power constraint $\text{tr}(\mathbf{K}_x) \leq P$ ,
$\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ is the channel matrix,
$\mathbf{z} \sim \mathcal{CN}(\mathbf{0}, N\mathbf{I}_{n_r})$ is additive Gaussian noise,
$\mathbf{y} \in \mathbb{C}^{n_r}$ is the received signal vector.

Each entry $[\mathbf{H}]_{ij}$ represents the complex channel gain from transmit antenna $j$ to receive antenna $i$ . The noise is spatially white with power $N$ per receive antenna. The total transmit power constraint is $\text{tr}(\mathbf{K}_x) \leq P$ .

MIMO channel

A wireless channel with $n_t$ transmit and $n_r$ receive antennas, modeled as $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{z}$ where $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ is the channel matrix.

Related: Spatial multiplexing gain

Theorem: MIMO Capacity with CSIR (Fixed Channel)

For a fixed (deterministic) MIMO channel $\mathbf{H}$ with CSIR, the capacity is

$C(\mathbf{H}) = \max_{\substack{\mathbf{K}_x \succeq \mathbf{0} \\ \text{tr}(\mathbf{K}_x) \leq P}} \log\det\!\left(\mathbf{I}_{n_r} + \frac{1}{N}\mathbf{H}\,\mathbf{K}_x\,\mathbf{H}^{H}\right).$

If the transmitter also knows $\mathbf{H}$ (CSIT), the optimal input covariance is obtained by water-filling over the singular values of $\mathbf{H}$ .

The mutual information for Gaussian inputs is $I(\mathbf{x}; \mathbf{y}) = h(\mathbf{y}) - h(\mathbf{z})$ , and Gaussian inputs maximize $h(\mathbf{y})$ under the covariance constraint. The $\log\det$ formula is the natural matrix extension of the scalar $\frac{1}{2}\log(1 + \text{SNR})$ .

Show Hint

Write $I(\mathbf{x}; \mathbf{y} | \mathbf{H}) = h(\mathbf{y} | \mathbf{H}) - h(\mathbf{z})$ and compute each term.

Use the fact that $\mathbf{y} | \mathbf{H}$ is Gaussian with covariance $\mathbf{H}\mathbf{K}_x\mathbf{H}^{H} + N\mathbf{I}$ .

Recall that $h(\mathbf{w}) = \log\det(\pi e \mathbf{K}_w)$ for complex Gaussian.

Proof

Mutual information with Gaussian inputs

For $\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, \mathbf{K}_x)$ and given $\mathbf{H}$ :

$\mathbf{y} | \mathbf{H} \sim \mathcal{CN}(\mathbf{0}, \mathbf{H}\mathbf{K}_x\mathbf{H}^{H} + N\mathbf{I}).$

The mutual information is

$I(\mathbf{x}; \mathbf{y} | \mathbf{H}) = h(\mathbf{y} | \mathbf{H}) - h(\mathbf{z}).$

Compute the differential entropies

Using the complex Gaussian entropy formula $h(\mathbf{w}) = \log\det(\pi e \mathbf{K}_w)$ :

$h(\mathbf{y} | \mathbf{H}) = \log\det(\pi e (\mathbf{H}\mathbf{K}_x\mathbf{H}^{H} + N\mathbf{I})),$ $h(\mathbf{z}) = \log\det(\pi e \cdot N\mathbf{I}).$

Therefore

$I(\mathbf{x}; \mathbf{y} | \mathbf{H}) = \log\det\!\left(\mathbf{I} + \frac{1}{N}\mathbf{H}\mathbf{K}_x\mathbf{H}^{H}\right).$

Optimality of Gaussian inputs

For any input distribution with covariance $\mathbf{K}_x$ ,

$I(\mathbf{x}; \mathbf{y} | \mathbf{H}) \leq h(\mathbf{y} | \mathbf{H}) - h(\mathbf{z}).$

Since Gaussian $\mathbf{y}$ maximizes $h(\mathbf{y})$ under a fixed covariance (Chapter 5), and $\mathbf{y}$ is Gaussian when $\mathbf{x}$ is Gaussian, the maximum is achieved by Gaussian inputs.

Optimization over $\mathbf{K}_x$

The capacity is the maximum over all valid $\mathbf{K}_x$ :

$C(\mathbf{H}) = \max_{\mathbf{K}_x \succeq \mathbf{0},\, \text{tr}(\mathbf{K}_x) \leq P} \log\det\!\left(\mathbf{I} + \frac{1}{N}\mathbf{H}\mathbf{K}_x\mathbf{H}^{H}\right).$

The objective is concave in $\mathbf{K}_x$ (log-det is concave on positive definite matrices), and the constraint set is convex, so this is a convex optimization problem. $\blacksquare$

,

Theorem: SVD Decomposition into Parallel Channels

Let the SVD of $\mathbf{H}$ be $\mathbf{H} = \mathbf{U}\boldsymbol{\Lambda}\mathbf{V}^H$ , where $\mathbf{U} \in \mathbb{C}^{n_r \times n_r}$ , $\mathbf{V} \in \mathbb{C}^{n_t \times n_t}$ are unitary, and $\boldsymbol{\Lambda} \in \mathbb{R}^{n_r \times n_t}$ has singular values $\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_r > 0$ with $r = \text{rank}(\mathbf{H}) \leq \min(n_t, n_r)$ .

With CSIT, the MIMO channel decomposes into $r$ parallel independent sub-channels:

$\tilde{y}_k = \sigma_k \tilde{x}_k + \tilde{z}_k, \quad k = 1, \ldots, r,$

where $\tilde{\mathbf{y}} = \mathbf{U}^H \mathbf{y}$ , $\tilde{\mathbf{x}} = \mathbf{V}^H \mathbf{x}$ , $\tilde{\mathbf{z}} = \mathbf{U}^H \mathbf{z}$ . The capacity is

$C(\mathbf{H}) = \sum_{k=1}^r \log\!\left(1 + \frac{\sigma_k^2 P_k}{N}\right),$

with water-filling power allocation

$P_k = \left(\nu - \frac{N}{\sigma_k^2}\right)^{\!+}, \quad \sum_{k=1}^r P_k \leq P.$

The SVD rotates the input and output spaces so that the MIMO channel becomes $r$ independent scalar channels, one per singular value. The $k$ -th sub-channel has gain $\sigma_k$ and receives power $P_k$ . The strongest sub-channel ( $\sigma_1$ ) carries the most data; weak sub-channels may receive zero power. This is the water-filling solution from Chapter 12, applied to the singular values of $\mathbf{H}$ .

Proof

Apply the SVD transformation

Substitute $\mathbf{H} = \mathbf{U}\boldsymbol{\Lambda}\mathbf{V}^H$ into the channel model and left-multiply by $\mathbf{U}^H$ :

$\mathbf{U}^H \mathbf{y} = \boldsymbol{\Lambda}\mathbf{V}^H \mathbf{x} + \mathbf{U}^H \mathbf{z}.$

Define $\tilde{\mathbf{y}} = \mathbf{U}^H\mathbf{y}$ , $\tilde{\mathbf{x}} = \mathbf{V}^H\mathbf{x}$ , $\tilde{\mathbf{z}} = \mathbf{U}^H\mathbf{z}$ . Since $\mathbf{U}$ is unitary, $\tilde{\mathbf{z}} \sim \mathcal{CN}(\mathbf{0}, N\mathbf{I})$ (noise statistics are preserved). The channel becomes

$\tilde{\mathbf{y}} = \boldsymbol{\Lambda}\tilde{\mathbf{x}} + \tilde{\mathbf{z}},$

which decouples into $\tilde{y}_k = \sigma_k \tilde{x}_k + \tilde{z}_k$ .

Apply water-filling

Each sub-channel is an independent AWGN channel with gain $\sigma_k$ . The optimal input covariance in the rotated basis is $\mathbf{K}_{\tilde{x}} = \text{diag}(P_1, \ldots, P_r, 0, \ldots, 0)$ . From Chapter 12, the optimal power allocation is water-filling:

$P_k = \left(\nu - \frac{N}{\sigma_k^2}\right)^{\!+}, \quad \sum_k P_k = P,$

and the capacity is the sum of sub-channel capacities. $\blacksquare$

Theorem: Optimal Input Without CSIT for i.i.d. Rayleigh

For an ergodic MIMO channel with i.i.d. Rayleigh fading ( $[\mathbf{H}]_{ij} \sim \mathcal{CN}(0, 1)$ independently) and CSIR but no CSIT, the capacity-achieving input covariance is

$\mathbf{K}_x^* = \frac{P}{n_t}\,\mathbf{I}_{n_t}.$

The ergodic capacity is

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

where $\text{SNR} = P/N$ .

Without CSIT, the transmitter cannot align its signal with the channel. Since the channel is isotropically distributed (i.i.d. entries), no direction in the transmit space is preferred. The best the transmitter can do is spread power equally across all antennas — the isotropic input $\mathbf{K}_x = (P/n_t)\mathbf{I}$ . Any other allocation would waste power by favoring directions that are no better on average.

Proof

Symmetry argument

For i.i.d. Rayleigh $\mathbf{H}$ , the distribution of $\mathbf{H}$ is unitarily invariant: $\mathbf{U}\mathbf{H}\mathbf{V}$ has the same distribution as $\mathbf{H}$ for any unitary $\mathbf{U}$ and $\mathbf{V}$ .

Suppose $\mathbf{K}_x^*$ is optimal. For any unitary $\mathbf{V}$ , $\mathbf{V}\mathbf{K}_x^*\mathbf{V}^H$ achieves the same ergodic rate (by the unitary invariance of $\mathbf{H}$ ). By the concavity of $\log\det$ in $\mathbf{K}_x$ , the average over all unitaries $\mathbb{E}_{\mathbf{V}}[\mathbf{V}\mathbf{K}_x^*\mathbf{V}^H] = \frac{\text{tr}(\mathbf{K}_x^*)}{n_t}\mathbf{I} = \frac{P}{n_t}\mathbf{I}$ achieves at least as high a rate. Therefore $\mathbf{K}_x^* = (P/n_t)\mathbf{I}$ is optimal. $\blacksquare$

Theorem: Telatar's Linear Capacity Scaling

For the i.i.d. Rayleigh MIMO channel with $n_t$ transmit and $n_r$ receive antennas, the ergodic capacity at high SNR scales as

$C_{\text{erg}} \approx \min(n_t, n_r) \cdot \log\!\left(\frac{\text{SNR}}{n_t}\right) + O(1) \quad \text{as } \text{SNR} \to \infty.$

The factor $\min(n_t, n_r)$ is the spatial multiplexing gain: the MIMO channel supports $\min(n_t, n_r)$ independent data streams, each contributing $\log(\text{SNR}/n_t)$ bits per channel use.

The channel matrix $\mathbf{H}$ has at most $\min(n_t, n_r)$ nonzero singular values. At high SNR, each singular value contributes approximately $\log(\text{SNR})$ to the capacity (the water-filling level is high enough that all sub-channels are active). So the total capacity grows $\min(n_t, n_r)$ times faster than the scalar channel. Doubling the number of antennas (on both sides) roughly doubles the high-SNR capacity without any increase in bandwidth or transmit power.

Proof

Eigenvalue decomposition

Let $m = \min(n_t, n_r)$ and let $\lambda_1, \ldots, \lambda_m$ be the nonzero eigenvalues of $\mathbf{H}\mathbf{H}^{H}$ (or equivalently of $\mathbf{H}^{H}\mathbf{H}$ ). The ergodic capacity is

$C_{\text{erg}} = \mathbb{E}\!\left[\sum_{k=1}^m \log\!\left(1 + \frac{\text{SNR}}{n_t}\lambda_k\right)\right].$

High-SNR expansion

At high SNR, $\log(1 + \frac{\text{SNR}}{n_t}\lambda_k) \approx \log(\frac{\text{SNR}}{n_t}) + \log(\lambda_k)$ for all $k$ such that $\lambda_k > 0$ . Since all $m$ eigenvalues are positive with probability 1 (the matrix $\mathbf{H}\mathbf{H}^{H}$ is full rank a.s. when $n_t, n_r \geq m$ ):

$C_{\text{erg}} \approx m \cdot \log\!\left(\frac{\text{SNR}}{n_t}\right) + \sum_{k=1}^m \mathbb{E}[\log \lambda_k].$

The second term is a constant independent of $\text{SNR}$ , so the capacity grows as $m \cdot \log(\text{SNR})$ at high SNR. $\blacksquare$

Spatial multiplexing gain

The pre-log factor in the high-SNR capacity of a MIMO channel. For an $n_r \times n_t$ MIMO channel with i.i.d. Rayleigh fading, the spatial multiplexing gain is $\min(n_t, n_r)$ , meaning the capacity scales as $\min(n_t, n_r) \cdot \log(\text{SNR})$ .

Related: MIMO channel

Example: Capacity of a $2 \times 2$ MIMO Channel

Consider a $2 \times 2$ MIMO channel with

$\mathbf{H} = \begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \end{pmatrix}$

and noise power $N = 1$ . Compute the capacity with CSIT (water-filling over singular values) for total power $P = 10$ .

Solution

Compute the SVD

The eigenvalues of $\mathbf{H}\mathbf{H}^{H} = \mathbf{H}^{H}\mathbf{H}$ (since $\mathbf{H}$ is symmetric here) are found from the characteristic equation:

$(1 - \lambda)^2 - 0.25 = 0 \implies \lambda_1 = 1.5, \; \lambda_2 = 0.5.$

The singular values are $\sigma_1 = \sqrt{1.5} \approx 1.225$ and $\sigma_2 = \sqrt{0.5} \approx 0.707$ .

Water-filling power allocation

The sub-channel gains are $\sigma_1^2 = 1.5$ and $\sigma_2^2 = 0.5$ . The water-filling equations are:

$P_k = \left(\nu - \frac{N}{\sigma_k^2}\right)^+, \quad P_1 + P_2 = P = 10.$

Both channels are active (since $N/\sigma_2^2 = 2 < P = 10$ ). From $\nu - 1/1.5 + \nu - 1/0.5 = 10$ :

$2\nu - 0.667 - 2 = 10 \implies \nu = 6.333.$

$P_1 = 6.333 - 0.667 = 5.667, \quad P_2 = 6.333 - 2 = 4.333.$

Compute capacity

$C = \log(1 + 1.5 \cdot 5.667) + \log(1 + 0.5 \cdot 4.333)KATEXPLACEHOLDER0END= \log(9.5) + \log(3.167) \approx 3.248 + 1.663 = 4.911 \text{ bits/use}.$ $Compare with equal power allocation ($ P_1 = P_2 = 5 $):$ C_{\text{equal}} = \log(8.5) + \log(3.5) \approx 3.087 + 1.807 = 4.894$ bits/use.

Water-filling provides only a marginal gain here (0.017 bits) because the sub-channel gains are not vastly different.

MIMO Capacity Scaling with Antenna Number

Observe how the ergodic capacity of an i.i.d. Rayleigh MIMO channel scales with the number of antennas. At high SNR, the capacity grows linearly with $\min(n_t, n_r)$ . Adjust the SNR and antenna configuration to explore the spatial multiplexing gain.

Parameters

\text{SNR}

(dB)15

Max antennas8

Plot $n_t = n_r = 1, 2, \ldots, n_{\max}$

Antenna configuration

MIMO Ergodic Capacity vs. SNR

Compare the ergodic capacity of different MIMO configurations as a function of SNR. The slopes at high SNR reveal the spatial multiplexing gains: a $4 \times 4$ system gains capacity 4 times faster per dB than a $1 \times 1$ system.

Parameters

Transmit antennas

n_t

4

Receive antennas

n_r

4

Max SNR (dB)30

Show CSIT (water-filling) capacity

Historical Note: The MIMO Revolution: Foschini and Telatar

1996-1999

The idea that multiple antennas could provide a linear capacity increase was first demonstrated by Gerard Foschini at Bell Labs in his 1996 paper introducing the BLAST (Bell Labs Layered Space-Time) architecture. Foschini showed through simulation that a $10 \times 10$ MIMO system in Rayleigh fading could achieve spectral efficiencies of 40 bits/s/Hz at 24 dB SNR — a tenfold improvement over SISO.

Emre Telatar, also at Bell Labs, provided the rigorous information-theoretic foundation in his landmark 1999 paper "Capacity of Multi-Antenna Gaussian Channels." Telatar derived the ergodic capacity in closed form (as an expectation over random eigenvalues), proved the optimality of isotropic input for i.i.d. Rayleigh fading without CSIT, and established the $\min(n_t, n_r) \cdot \log(\text{SNR})$ scaling law.

These results were independently corroborated by the work of Marzetta and Hochwald at Bell Labs. Together, they launched the field of MIMO communications, which led to MIMO being incorporated into every modern wireless standard: 802.11n (Wi-Fi 4), LTE, LTE-Advanced, and 5G NR.

🎓CommIT Contribution(2006)

MIMO Broadcast Channel Capacity

H. Weingarten, Y. Steinberg, S. Shamai (Shitz), G. Caire — IEEE Trans. Information Theory, vol. 52, no. 9

While this chapter treats the point-to-point MIMO channel, the CommIT group has made fundamental contributions to the multiuser MIMO setting. Weingarten, Steinberg, and Shamai (with contributions from Caire) established the capacity region of the MIMO broadcast channel (downlink), proving that dirty paper coding (DPC) achieves the capacity region. This result extended the single-user MIMO capacity theory to the multiuser setting and provided the theoretical foundation for MU-MIMO precoding techniques used in LTE-Advanced and 5G NR.

The duality between the MIMO broadcast channel and the multiple access channel (uplink), established in companion work, showed that the capacity-achieving strategy can be computed via a dual MAC optimization — a computationally tractable approach. This duality is covered in the MIMO book (Book MIMO, Chapters 8-10).

MIMObroadcast-channelDPCcapacity-regionView Paper →

SISO vs. MIMO Capacity

Property	SISO ( $1 \times 1$ )	MIMO ( $n_t \times n_r$ )
Channel model	$Y = HX + Z$	$\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{z}$
Degrees of freedom	1	$\min(n_t, n_r)$
High-SNR capacity	$\frac{1}{2}\log(\text{SNR})$	$\min(n_t, n_r) \cdot \log(\text{SNR}/n_t)$
Optimal input (no CSIT, Rayleigh)	$X \sim \mathcal{N}(0, P)$	$\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, (P/n_t)\mathbf{I})$
Benefit from CSIT	Water-filling over fading states	Water-filling over singular values + fading states
Diversity order (quasi-static)	1	Up to $n_t n_r$
Array gain	None	$n_r$ (receive beamforming) or $n_t$ (transmit beamforming with CSIT)

Common Mistake: MIMO Multiplexing Gain Requires Rich Scattering

Mistake:

Assuming the $\min(n_t, n_r) \cdot \log(\text{SNR})$ capacity scaling holds for any MIMO channel, regardless of the propagation environment.

Correction:

The linear scaling requires the channel matrix $\mathbf{H}$ to be full rank, which happens with probability 1 for i.i.d. Rayleigh fading (rich scattering). In a pure line-of-sight (LoS) environment, $\mathbf{H} = \mathbf{a}_r \mathbf{a}_t^H$ is rank-1 regardless of the number of antennas, and the capacity is

$C_{\text{LoS}} = \log(1 + n_t n_r \cdot \text{SNR}),$

which provides an array gain ( $n_t n_r$ ) but only $1 \times \log(\text{SNR})$ growth — no spatial multiplexing. In practice, mixed LoS + scattered environments (Rician fading) provide intermediate behavior.

⚠️Engineering Note

From MIMO Theory to Massive MIMO Practice

Telatar's capacity formula suggests that capacity grows linearly with the number of antennas. In theory, a base station with $n_t = 128$ antennas serving a single user with $n_r = 1$ antenna achieves an array gain of 128 (21 dB) but no multiplexing gain. If instead it serves $K = 16$ single-antenna users simultaneously (MU-MIMO), the system achieves a multiplexing gain of 16.

In practice, the scaling is limited by:

Channel estimation overhead: estimating $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ requires at least $n_t$ pilot symbols, consuming resources.
Pilot contamination: in multi-cell systems, users in neighboring cells reuse pilot sequences, creating interference that does not vanish as $n_t \to \infty$ (Marzetta 2010).
Hardware impairments: phase noise and quantization errors in low-cost RF chains degrade the effective channel.
Spatial correlation: real channels are rarely i.i.d.; correlation reduces the effective rank and the multiplexing gain.

Practical Constraints

•
In 5G NR, the maximum number of antenna ports per TRP is 32 (Rel-15) or 64 (Rel-17)
•
CSI-RS overhead scales linearly with the number of antenna ports in FDD
•
TDD massive MIMO exploits channel reciprocity to avoid downlink pilot scaling

📋 Ref: 3GPP TS 38.214, Section 5.2

Quick Check

For a $4 \times 4$ i.i.d. Rayleigh MIMO channel at high SNR, how does the capacity scale compared to a SISO channel at the same SNR?

4 dB higher (array gain only)

4 times the capacity (4x multiplexing gain)

16 times the capacity ( $n_t \cdot n_r$ gain)

Same capacity but more reliable

Correction:

4 times the capacity (4x multiplexing gain)

With $\\min(n_t, n_r) = 4$ and i.i.d. Rayleigh fading, the high-SNR capacity is $4 \\cdot \\log(\\text{SNR}/4)$ , which is approximately 4 times the SISO capacity $\\log(\\text{SNR})$ at high SNR.

Key Takeaway

The MIMO channel $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{z}$ decomposes via SVD into $\min(n_t, n_r)$ parallel sub-channels. With CSIT, water-filling over singular values is optimal. Without CSIT but with i.i.d. Rayleigh fading, isotropic input $(P/n_t)\mathbf{I}$ is optimal. Telatar's result shows that ergodic capacity scales as $\min(n_t, n_r) \cdot \log(\text{SNR})$ at high SNR — the spatial multiplexing gain that launched the MIMO revolution.

MIMO Capacity Scaling with Antennas

MIMO ergodic capacity grows approximately linearly with

\min(n_t, n_r)

at high SNR — spatial multiplexing. The animation shows capacity computed via Monte Carlo for i.i.d. Rayleigh channels alongside the linear scaling reference.

MIMO Channel Capacity

Multiple Antennas: The Spatial Dimension

Definition: MIMO Channel Model

MIMO channel

Theorem: MIMO Capacity with CSIR (Fixed Channel)

Mutual information with Gaussian inputs

Compute the differential entropies

Optimality of Gaussian inputs

Optimization over $\mathbf{K}_x$

Theorem: SVD Decomposition into Parallel Channels

Apply the SVD transformation

Apply water-filling

Theorem: Optimal Input Without CSIT for i.i.d. Rayleigh

Symmetry argument

Theorem: Telatar's Linear Capacity Scaling

Eigenvalue decomposition

High-SNR expansion

Spatial multiplexing gain

Example: Capacity of a 2×22 \times 22×2 MIMO Channel

Compute the SVD

Water-filling power allocation

Compute capacity

MIMO Capacity Scaling with Antenna Number

Parameters

MIMO Ergodic Capacity vs. SNR

Parameters

Historical Note: The MIMO Revolution: Foschini and Telatar

MIMO Broadcast Channel Capacity

SISO vs. MIMO Capacity

Common Mistake: MIMO Multiplexing Gain Requires Rich Scattering

From MIMO Theory to Massive MIMO Practice

Quick Check

Key Takeaway

MIMO Capacity Scaling with Antennas

Definition:
MIMO Channel Model

Example: Capacity of a $2 \times 2$ MIMO Channel