Spatial multiplexing is a technique used in wireless communication systems to transmit multiple data streams simultaneously over the same radio frequency channel. Instead of using different frequencies or time slots, spatial multiplexing equips multiple antennas at both the transmitter and receiver to create independent communication links. By exploiting the spatial dimension of the wireless channel, spatial multiplexing significantly increases data throughput and spectral efficiency. It is a fundamental component of modern wireless standards, particularly in Multiple Input Multiple Output (MIMO) systems.

Assuming the knowledge of the channel state information (CSI) at the transmitter, we can convert the MIMO channel into multiple non-interfering parallel SISO channels via singular value decomposition (SVD) based pre-processing and post-processing at the transmitter and receiver, respectively.

Consider a MIMO system with $N_t$ antennas at the transmitter and $N_r$ antennas at the receiver. Let the signal received at the receiver be

$$\mathbf{y} = \mathbf{Hx} + \mathbf{n}$$

where $\mathbf{x}\in\mathbb{C}^{N_t\times1}$ is the transmitted signal vector, $\mathbf{H}\in\mathbb{C}^{N_r\times N_t}$ is the MIMO channel matrix, and $\mathbf{n}\in\mathbb{C}^{N_r\times1}$ is the AWGN noise with variance $\sigma_N^2$. Without loss of generality, we assume unit transmission power, i.e. $\mathbb{E}[\mathbf{x}^T\mathbf{x}]=1$.

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Structure of the MIMO Channel Matrix

The effectiveness of spatial multiplexing fundamentally depends on the structure and properties of the MIMO channel matrix. In particular, the rank of the channel matrix determines how many independent spatial data streams can be supported. This concept is closely related to the spatial Degrees of Freedom (DoF), which quantifies the multiplexing capability of the system. The channel matrix represents the spatial coupling between transmit and receive antennas, where each element denotes the complex channel gain from transmit antenna to receive antenna.

The structure of this matrix depends on antenna spacing, array geometry, propagation environment, scattering richness, and the presence or absence of Line-of-Sight (LoS) components. These factors directly influence the rank of the channel matrix and therefore the achievable spatial multiplexing performance.

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Channel Decomposition and Linear Precoding in Spatial Multiplexing

The received signal components are coupled due to interference, commonly referred to as crosstalk.

Each receive antenna recieves data $y_i$ as

$$y_i = \sum_{j=1}^{N_t} h_{ij} x_j + n_i,$$

where $h_{ij}$ is the channel coefficient for between the $i^{th}$ receiver antenna and the $j^{th}$ transmitter antenna, $x_j$ is the information transmitted by the $j^{th}$ antenna, and $n_i$ is the noise component affecting the $i^{th}$ transmitter antenna. This shows that multiple transmitted streams interfere with one another. This crosstalk prevents direct separation of spatial streams without appropriate signal processing.

From matrix theory, the channel matrix can be decomposed using SVD as

$$\mathbf{H} = \mathbf{USV}^H$$

where $\mathbf{U}\in\mathbb{C}^{N_r\times N_r}$ and $\mathbf{V}\in\mathbb{C}^{N_t\times N_t}$ are unitary matrices (i.e $\mathbf{U}^H\mathbf{U}=1$ and $\mathbf{V}^H\mathbf{V}=1$) and $\mathbf{S}\in\mathbb{C}^{N_r\times N_t}$ is a diagonal matrix with singular values as its entries. There exist $R$ singular values where $R$ is the rank of the matrix $\mathbf{H}$. The received signal can now be written as

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

1. Pre-processing at the transmitter

   a) Let $\mathbf{x}=\mathbf{V\bar{x}}$

   b) The received signal is now modified as

$$\mathbf{y} = \mathbf{US}\mathbf{\bar{x}} + \mathbf{n}$$

2. Post-processing at the receiver

   a) Let $\mathbf{\bar{y}}=\mathbf{U}^H\mathbf{y}$

   b) The received signal is now modified as

$$\mathbf{\bar{y}} = \mathbf{S}\mathbf{\bar{x}} + \mathbf{\bar{n}}$$

The above transformation diagonalizes the channel matrix and eliminates crosstalk between spatial streams. The MIMO system is therefore converted into $R$ parallel, non-interfering SISO channels, where $R$ is given as

$$R = \mathrm{rank}(\mathbf{H}),$$

It determines the number of independent spatial data streams that can be transmitted simultaneously. It satisfies $R \leq \min(N_t, N_r)$. The spatial Degrees of Freedom (DoF) of the MIMO system are equal to the rank of $\mathbf{H}$. If $R = \min(N_t,N_r)$, full spatial multiplexing gain is achieved. If $R=1$, the system effectively behaves as a SISO channel regardless of the number of antennas. Therefore, spatial multiplexing capability is fundamentally limited by the rank of the channel matrix rather than merely by the number of antennas deployed.

Finally, each precoded symbol $\bar{y}_i$ at the receiver is given as

$$\bar{y}_i = \lambda_i \bar{x}_i + \bar{n}_i, \quad i = 1, \dots, R.$$

where $\lambda_i$ is the $i^{th}$ singular value of $\mathbf{S}$. It can be observed that the transmit precoding and receiver shaping transform the MIMO channel into $R$ non-interfering parallel single-input single-output (SISO) channels with input $\mathbf{\bar{x}}$ and output $\mathbf{\bar{y}}$.

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MIMO Capacity

The capacity of a MIMO communication system can be expressed as

$$\mathrm{C} = \sum_{i=1}^{R}B\log_2\left(1+\frac{P_t\lambda_i^2}{R\sigma_n^2}\right)$$

where $P_t$ is the total transmission power and $\sigma_n^2$ being the noise power.

The capacity expression explicitly depends on $R$. At high SNRs, the capacity scales approximately as

$$\mathrm{C} \approx R \log_2(\mathrm{SNR}),$$

which demonstrates that the spatial DoF directly determine the multiplexing gain and the slope of capacity growth with SNR. Therefore, the rank of the MIMO channel matrix governs the number of parallel data streams, the achievable multiplexing gain, and the overall spatial efficiency of the system.