## Spatial Multiplexing

Spatial multiplexing is a MIMO technique that increases the data rate in wireless communications by transmitting different data streams simultaneously from multiple transmit antennas to multiple receive antennas. The receiver separates these streams, allowing for an increase in the data rate without requiring additional bandwidth or transmit power. This technique exploits the spatial dimension and the independent fading paths in a rich scattering environment to improve throughput.

Spatial multiplexing methods include precoding (as applied for single user and multi-user systems), channel sounding, and spherical decoding.

### Precoding

Precoding preadjusts the phases and amplitudes of the signals being transmitted from the
multiple antennas to manage interference and optimize the signal power at the receiver. The
method discussed here focuses on linear precoding, which achieves good performance with much
lower complexity than nonlinear precoding. Technical literature often refers to precoder
outputs as *spatial streams* and precoding as *spatial
mapping*.

Precoding is particularly useful in scenarios where the transmitter knows the channel
conditions. Knowledge of the channel condition, known as *channel state
information* (CSI), makes it possible to precode the transmitted signals so
that the receiver can separate each desired signal from interference caused by other
signals. A part of spatial multiplexing, precoding is a foundational method of current MIMO
wireless systems, such as 5G and WLAN.

#### SU-MIMO Precoding

Precoding codes data symbols per stream and maps them to a number of transmit chains.
For example, represent the channel matrix of an SU-MIMO system, *H*, as

$$H=\left\{\begin{array}{cccc}{h}_{11}& {h}_{12}& \mathrm{...}& {h}_{1{N}_{R}}\\ {h}_{21}& {h}_{22}& \mathrm{...}& {h}_{2{N}_{R}}\\ {h}_{31}& {h}_{32}& \mathrm{...}& {h}_{3{N}_{R}}\\ \begin{array}{l}\mathrm{...}\\ {h}_{{N}_{T}1}\end{array}& \begin{array}{l}\mathrm{...}\\ {h}_{{N}_{T}2}\end{array}& \begin{array}{l}\mathrm{...}\\ \mathrm{...}\end{array}& \begin{array}{l}\mathrm{...}\\ {h}_{{N}_{T}{N}_{R}}\end{array}\end{array}\right\}$$

In the SU-MIMO system, assume that
*N*_{T} =
*N*_{R}, resulting in a square channel matrix,
*H*.

The received signal at antenna *RX 1* is a linear combination of the
individual signal from each transmit antenna, weighted by the respective channel
coefficient. In this case, it is not possible to extract just the signal from any specific
transmit antenna because the signals received from other antennas corrupt the desired
signal. However, if the channel matrix were able to be diagonalized, then each signal can
be received without corruption. This spatial mapping can happen if *H* is
full rank, which occurs when a rich scattering channel is present between the transmitter
and receiver. Now, decompose *H* using singular value decomposition (SVD)
to form

$$H=US{V}^{\prime}$$

where *V'* is the Hermitian of *V*.
SVD splits *H* into two unitary matrices *U* and
*V*, and a diagonal matrix *S*. The diagonal elements
of *S* comprise the eigenvalues of *H*. Model the
transmission as

$$y=Hx+n=US{V}^{\prime}x+n$$

where *y* is the received signal vector of length
*N*_{R}, *x* is a transmitted
signal vector of length *N*_{T}, and
*n* is the noise vector of length
*N*_{R}. By premultiplying *x*
with *V* (precoding) and premultiplying *y* with
*U'* (combining), the received signal is

$${U}^{\prime}y={U}^{\prime}US{V}^{\prime}Vx+{U}^{\prime}n$$

Since *U* and *V* are unitary,
*U'U = V'V = I* and what remains is *U'y = Sx +
U'n*.

In this case, *V* is the precoding matrix and *U'*
is the combining matrix. In practice, combining is done as part of receiver
equalization.

This figure shows the effective separated channel,
*H*_{eff}.

Represent *H*_{eff} as

$${H}_{eff}=\left[\begin{array}{l}{\tilde{h}}_{11}\\ {\tilde{h}}_{22}\\ \vdots \\ {\tilde{h}}_{{N}_{T}{N}_{R}}\end{array}\right]$$

The recovered signal *x* is scaled by the diagonal elements of
*S*, which are the eigenvalues of *H*. The richness of
the channel determines the signal powers of each stream. Even if *H* is
full-rank, *S* might have some small eigenvalues. Specifically, the
spatial stream for that path has small signal power and, consequently, low signal-to-noise
ratio

For an example that demonstrates precoding of a SU-MIMO system, see 802.11be Packet Error Rate Simulation for an EHT MU Single-User Packet Format (WLAN Toolbox).

#### MU-MIMO Precoding

A base station with multiple antennas transmitting to multiple receivers can use MU-MIMO precoding methods, such as zero-forcing (ZF) and minimum mean square error (MMSE) precoding. These MU-MIMO precoding methods form transmit antenna nulls at interfering users so that each stream assigned to that receiver is free of interference from the other streams.

For examples that demonstrate precoding of MU-MIMO systems, see:

TDD Reciprocity-Based PDSCH MU-MIMO Using SRS (5G Toolbox) implements a downlink MU-MIMO system by exploiting channel reciprocity in a time division duplex (TDD) scenario.

NR Cell Performance Evaluation with MIMO (5G Toolbox) models a 5G New Radio (NR) cell with a MIMO antenna configuration and evaluates the network performance.

### Channel Sounding

Channel sounding transmits a known signal from the transmitter. The receiver estimates the channel properties by comparing the received signal to the expected known signal by using derived information (such as path loss, fading, delay spread, and other environmental factors) to construct or estimate the channel matrix in the MIMO system. The channel matrix represents the gain and phase shift introduced by the channel between each transmitter-receiver antenna pair.

For an example that demonstrates channel sounding, see Massive MIMO Hybrid Beamforming. This example determines the channel state information at the transmitter by using full channel sounding.

### Spherical Decoding

Spherical decoding searches for the transmitted signal vector within a spherical region in the signal space, rather than examining the entire space. Spherical decoding approaches the optimal maximum likelihood (ML) decoding performance, offering significant gains in error rates compared to simpler decoding methods. It addresses the complexity and performance challenges associated with detecting signals in MIMO systems, offering a balance between performance and computational complexity.