Ferkans — Interactive Telecom Tutor

From SISO to MIMO: The Spatial Dimension

A single-input single-output (SISO) channel is characterised by a single complex gain $h$ . By deploying $n_t$ transmit and $n_r$ receive antennas, we open a spatial dimension that can carry multiple independent data streams simultaneously. The channel is no longer a scalar but a matrix $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ , and the rich linear algebra of this matrix governs every aspect of MIMO performance: capacity, diversity, beamforming, and spatial multiplexing.

This section establishes the fundamental input-output model and the key matrix properties that determine what a MIMO channel can do.

Historical Note: The MIMO Revolution: Foschini and Telatar

1990s

The capacity potential of MIMO channels was independently discovered by Gerard Foschini at Bell Labs (1996) and Emre Telatar at AT&T (1995, published 1999). Foschini's BLAST architecture showed that capacity scales linearly with $\min(n_t, n_r)$ --- a revolutionary result that implied wireless spectral efficiency could be increased by simply adding antennas, without additional bandwidth or power.

Telatar's elegant information-theoretic analysis derived the exact capacity formula using random matrix theory. Together, these papers launched the modern era of MIMO communications and led directly to the multi-antenna technologies in Wi-Fi (802.11n/ac/ax), 4G LTE, and 5G NR.

,

Definition:
MIMO Channel Matrix

Consider a narrowband MIMO system with $n_t$ transmit antennas and $n_r$ receive antennas. The MIMO channel matrix $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ has entry

$[\mathbf{H}]_{ij} = h_{ij}$

where $h_{ij}$ is the complex channel gain from transmit antenna $j$ to receive antenna $i$ . The baseband input-output relationship is

$\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$

where $\mathbf{x} \in \mathbb{C}^{n_t}$ is the transmitted signal vector subject to a power constraint $\mathbb{E}[\|\mathbf{x}\|^2] \leq P$ , the received vector is $\mathbf{y} \in \mathbb{C}^{n_r}$ , and the noise $\mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I}_{n_r})$ is circularly symmetric complex Gaussian.

This is the flat-fading (narrowband) model. For frequency-selective MIMO channels, each OFDM subcarrier has its own $\mathbf{H}[k]$ , and the same framework applies per subcarrier.

Definition:
Channel Rank

The rank of the MIMO channel is

$\mathrm{rank}(\mathbf{H}) = \text{number of nonzero singular values of } \mathbf{H}$

The rank determines the maximum number of independent spatial streams (parallel data pipes) that the channel can support. Since $\mathbf{H}$ is $n_r \times n_t$ , we always have

$\mathrm{rank}(\mathbf{H}) \leq \min(n_t, n_r)$

A channel with $\mathrm{rank}(\mathbf{H}) = \min(n_t, n_r)$ is called full-rank. Rich scattering environments (many multipath components, no dominant LOS) tend to produce full-rank channels.

Definition:
Condition Number

The condition number of the channel matrix is

$\kappa(\mathbf{H}) = \frac{\sigma_{\max}}{\sigma_{\min}}$

where $\sigma_{\max}$ and $\sigma_{\min}$ are the largest and smallest nonzero singular values of $\mathbf{H}$ .

$\kappa = 1$ : all singular values equal (perfectly conditioned). The channel supports equally strong parallel streams. An i.i.d. Rayleigh channel has $\kappa$ close to 1 on average for large arrays.
$\kappa \gg 1$ : ill-conditioned. Some spatial streams carry much less signal than others. LOS channels and highly correlated channels tend to have large $\kappa$ .

A large condition number does not mean the channel is "bad" --- it means it favours beamforming over spatial multiplexing. The optimal strategy depends on both the channel and the SNR regime.

MIMO System Model — The $n_t \times n_r$ MIMO system: $n_t$ transmit antennas send the signal vector $\mathbf{x}$ , the channel matrix $\mathbf{H}$ maps each transmit antenna to each receive antenna, and the receiver observes $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$ .

Definition:
Spatial Multiplexing

Spatial multiplexing is the transmission of multiple independent data streams over the same time-frequency resource using multiple antennas. If the channel has rank $r$ , up to $r$ streams can be multiplexed, each seeing an effective SNR determined by the corresponding singular value $\sigma_i$ of $\mathbf{H}$ .

The total rate achieved by spatial multiplexing is

$R = \sum_{i=1}^{r} \log_2\!\left(1 + \frac{p_i \sigma_i^2}{\sigma_n^2}\right) \quad \text{bits/s/Hz}$

where $p_i$ is the power allocated to stream $i$ .

Theorem: Channel Rank Determines Spatial Degrees of Freedom

For a MIMO channel $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ with SVD $\mathbf{H} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$ , the number of independent spatial sub-channels (degrees of freedom) equals $\mathrm{rank}(\mathbf{H})$ . Each sub-channel has gain $\sigma_i$ (the $i$ -th singular value), and the sub-channels are mutually non-interfering.

Specifically, the transformation $\tilde{\mathbf{y}} = \mathbf{U}^H \mathbf{y}$ and $\tilde{\mathbf{x}} = \mathbf{V}^H \mathbf{x}$ decomposes the MIMO channel into $r = \mathrm{rank}(\mathbf{H})$ parallel scalar channels:

$\tilde{y}_i = \sigma_i \tilde{x}_i + \tilde{n}_i, \qquad i = 1, \ldots, r$

The SVD finds the "natural coordinate system" of the channel. Transmitting along right singular vectors $\mathbf{v}_{i}$ and receiving along left singular vectors $\mathbf{u}_i$ creates non-interfering pipes through the channel, each with gain $\sigma_i$ . This is the MIMO analogue of diagonalising a symmetric matrix.

Proof

Apply SVD

Write $\mathbf{H} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$ where $\mathbf{U} \in \mathbb{C}^{n_r \times n_r}$ and $\mathbf{V} \in \mathbb{C}^{n_t \times n_t}$ are unitary, and $\boldsymbol{\Sigma} \in \mathbb{R}^{n_r \times n_t}$ is diagonal with entries $\sigma_1 \geq \sigma_2 \geq \cdots \geq 0$ .

Transform input and output

Define $\tilde{\mathbf{x}} = \mathbf{V}^H \mathbf{x}$ and $\tilde{\mathbf{y}} = \mathbf{U}^H \mathbf{y}$ . Substituting into $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$ :

$\tilde{\mathbf{y}} = \mathbf{U}^H \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H \mathbf{x} + \mathbf{U}^H \mathbf{n} = \boldsymbol{\Sigma}\tilde{\mathbf{x}} + \tilde{\mathbf{n}}$

Since $\mathbf{U}$ is unitary, $\tilde{\mathbf{n}} = \mathbf{U}^H \mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$ (unitary transformations preserve the i.i.d. Gaussian distribution).

Read off parallel channels

Because $\boldsymbol{\Sigma}$ is diagonal, the $i$ -th component decouples:

$\tilde{y}_i = \sigma_i \tilde{x}_i + \tilde{n}_i$

There are exactly $r = \mathrm{rank}(\mathbf{H})$ nonzero singular values, hence $r$ independent scalar sub-channels. $\blacksquare$

MIMO Capacity vs. SNR

Explore how MIMO capacity scales with SNR for different antenna configurations. Compare $1\times 1$ (SISO), $2\times 2$ , $4\times 4$ , and $8\times 8$ systems with i.i.d. Rayleigh fading. The capacity is averaged over many channel realisations (ergodic capacity).

Parameters

n_t

4

Number of transmit antennas

n_r

4

Number of receive antennas

SNR max (dB)30

Maximum SNR on the x-axis

Example: $2 \times 2$ MIMO Channel Analysis

Consider a $2 \times 2$ MIMO channel with

$\mathbf{H} = \begin{bmatrix} 1 & 0.5 \\ 0.3 & 0.8 \end{bmatrix}$

(a) Find the singular values and condition number.

(b) Decompose into parallel channels and compute the capacity at $\text{SNR} = 20$ dB with equal power allocation.

Solution

Compute singular values

We need the eigenvalues of $\mathbf{H}^{H} \mathbf{H}$ :

$\mathbf{H}^{H} \mathbf{H} = \begin{bmatrix} 1.09 & 0.74 \\ 0.74 & 0.89 \end{bmatrix}$

The eigenvalues are $\lambda_1 \approx 1.66$ and $\lambda_2 \approx 0.32$ , so the singular values are $\sigma_1 \approx 1.29$ and $\sigma_2 \approx 0.57$ .

Condition number

$\kappa(\mathbf{H}) = \frac{\sigma_1}{\sigma_2} = \frac{1.29}{0.57} \approx 2.26$ $

The channel is moderately well-conditioned --- both streams carry useful signal, though one is weaker.

Capacity with equal power allocation

At $\text{SNR} = 20$ dB ( $= 100$ linear), equal power gives $p_1 = p_2 = P/2 = 50$ . With $\sigma^2 = 1$ :

$C = \log_2\!\left(1 + 50 \times 1.66\right) + \log_2\!\left(1 + 50 \times 0.32\right)$

$= \log_2(84) + \log_2(17) \approx 6.39 + 4.09 = 10.48 \;\text{bits/s/Hz}$

For comparison, a SISO channel at the same total SNR gives $C_{\mathrm{SISO}} = \log_2(1 + 100) \approx 6.66$ bits/s/Hz. The $2 \times 2$ MIMO system provides a $58\%$ capacity increase. $\blacksquare$

Quick Check

A $4 \times 2$ MIMO channel ( $n_r = 4$ , $n_t = 2$ ) has channel matrix $\mathbf{H} \in \mathbb{C}^{4 \times 2}$ . What is the maximum number of independent spatial streams this channel can support?

2

4

6

8

Correction:

2

The rank is at most $\min(n_r, n_t) = \min(4, 2) = 2$ , so at most 2 independent streams can be supported.

Common Mistake: More Antennas Does Not Always Mean More Streams

Mistake:

Assuming that adding more transmit antennas always increases the number of spatial streams. For example, expecting a $2 \times 8$ system to support 8 streams.

Correction:

The number of spatial streams is limited by $\min(n_t, n_r)$ . A $2 \times 8$ system can support at most 2 streams (limited by the 2 receive antennas). The extra transmit antennas provide beamforming gain (array gain), not additional multiplexing. To increase multiplexing, you must add antennas on both sides of the link.

Key Takeaway

The SVD is the master key to MIMO: it decomposes any channel $\mathbf{H} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$ into $\mathrm{rank}(\mathbf{H})$ non-interfering parallel sub-channels, each with gain $\sigma_i$ . Every MIMO result in this chapter — capacity, degrees of freedom, diversity — ultimately traces back to this decomposition.

⚠️Engineering Note

Channel Estimation Overhead in Practical MIMO

The theoretical MIMO capacity assumes perfect CSI at the receiver (CSIR). In practice, estimating the $n_r \times n_t$ channel matrix requires transmitting orthogonal pilot sequences of length at least $n_t$ symbols. This creates a tension:

Pilot overhead: for a coherence block of $T$ symbols, $n_t$ symbols are spent on pilots, leaving $T - n_t$ for data. The effective rate is reduced by a factor $(1 - n_t/T)$ .
Estimation noise: with limited pilot energy, the estimated $\hat{\mathbf{H}}$ has error $\Delta\mathbf{H}$ , causing residual interference that limits capacity.
Scaling problem: for massive MIMO ( $n_t = 64$ -- $256$ ), the pilot overhead becomes prohibitive in FDD (where downlink and uplink channels differ). This motivates TDD operation (channel reciprocity) and compressed CSI feedback.

At 5G NR frequencies with $T \approx 14$ OFDM symbols per slot and $n_t = 32$ antenna ports, the system uses only 4--8 orthogonal pilot symbols via CSI-RS beamforming, exploiting angular-domain sparsity to reduce overhead.

Practical Constraints

•
Minimum pilot length: $n_t$ symbols per coherence block
•
FDD feedback overhead grows with $n_t$ ; TDD uses reciprocity
•
5G NR CSI-RS uses beamformed pilots (4-8 ports) even with 32-64 physical antennas

📋 Ref: 3GPP TS 38.214, §5.2

MIMO (Multiple-Input Multiple-Output)

A communication system with $n_t > 1$ transmit antennas and $n_r > 1$ receive antennas, exploiting the spatial dimension for multiplexing, diversity, or beamforming gains.

Channel matrix

The matrix $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ whose $(i,j)$ entry is the complex gain from transmit antenna $j$ to receive antenna $i$ .

Channel rank

The number of nonzero singular values of $\mathbf{H}$ , equal to the number of independent spatial sub-channels available for communication.

Condition number

The ratio $\kappa(\mathbf{H}) = \sigma_{\max}/\sigma_{\min}$ measuring how evenly the channel distributes signal energy across spatial sub-channels. $\kappa = 1$ is ideal for multiplexing; $\kappa \gg 1$ favours beamforming.

Related: Channel matrix

Spatial multiplexing

Transmitting multiple independent data streams over the same time-frequency resource by exploiting the spatial dimensions provided by multiple antennas.

The MIMO Channel Matrix

From SISO to MIMO: The Spatial Dimension

Historical Note: The MIMO Revolution: Foschini and Telatar

Definition: MIMO Channel Matrix

Definition: Channel Rank

Definition: Condition Number

MIMO System Model

Definition: Spatial Multiplexing

Theorem: Channel Rank Determines Spatial Degrees of Freedom

Apply SVD

Transform input and output

Read off parallel channels

MIMO Capacity vs. SNR

Parameters

Example: 2×22 \times 22×2 MIMO Channel Analysis

Compute singular values

Condition number

Capacity with equal power allocation

Quick Check

Common Mistake: More Antennas Does Not Always Mean More Streams

Key Takeaway

Channel Estimation Overhead in Practical MIMO

MIMO (Multiple-Input Multiple-Output)

Channel matrix

Channel rank

Condition number

Spatial multiplexing

Definition:
MIMO Channel Matrix

Definition:
Channel Rank

Definition:
Condition Number

Definition:
Spatial Multiplexing

Example: $2 \times 2$ MIMO Channel Analysis