Ferkans — Interactive Telecom Tutor

MIMO Channels: The Foundation of Modern Wireless

Multiple-input multiple-output (MIMO) uses multiple antennas at both transmitter and receiver. Generating realistic MIMO channel matrices with proper spatial correlation is essential for simulating 4G, 5G, and Wi-Fi MIMO systems.

Definition:
MIMO Channel Matrix

The $N_r \times N_t$ MIMO channel matrix $\mathbf{H}$ has entry $h_{ij}$ representing the complex gain from transmit antenna $j$ to receive antenna $i$ :

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

For i.i.d. Rayleigh fading: $h_{ij} \sim \mathcal{CN}(0, 1)$ .

def iid_mimo_channel(Nr, Nt, rng=None):
    if rng is None:
        rng = np.random.default_rng()
    return (rng.standard_normal((Nr, Nt)) + 1j*rng.standard_normal((Nr, Nt))) / np.sqrt(2)

Definition:
Kronecker Spatial Correlation Model

The Kronecker model separates transmit and receive correlations:

$\mathbf{H} = \mathbf{R}_r^{1/2} \mathbf{H}_w (\mathbf{R}_t^{1/2})^T$

where $\mathbf{H}_w$ has i.i.d. $\mathcal{CN}(0,1)$ entries, $\mathbf{R}_r$ is the $N_r \times N_r$ receive correlation, and $\mathbf{R}_t$ is the $N_t \times N_t$ transmit correlation.

def kronecker_mimo(Nr, Nt, Rr, Rt, rng=None):
    if rng is None:
        rng = np.random.default_rng()
    Lr = np.linalg.cholesky(Rr)
    Lt = np.linalg.cholesky(Rt)
    Hw = (rng.standard_normal((Nr, Nt)) + 1j*rng.standard_normal((Nr, Nt))) / np.sqrt(2)
    return Lr @ Hw @ Lt.T

The Kronecker model assumes separable TX/RX correlations. This is a good approximation for moderate correlations but breaks down for keyhole channels.

Definition:
Exponential Correlation Matrix

For a uniform linear array (ULA) with correlation coefficient $\rho$ :

$[\mathbf{R}]_{ij} = \rho^{|i-j|}$

This Toeplitz matrix is positive definite for $|\rho| < 1$ and captures the exponential decay of correlation with antenna separation.

def exp_corr_matrix(N, rho):
    return rho ** np.abs(np.arange(N)[:, None] - np.arange(N)[None, :])

Theorem: MIMO Channel Capacity

The ergodic capacity of an $N_r \times N_t$ MIMO channel with equal power allocation is:

$C = E\!\left[\log_2 \det\!\left(\mathbf{I} + \frac{\text{SNR}}{N_t}\mathbf{H}\mathbf{H}^H\right)\right]$

For i.i.d. Rayleigh fading at high SNR, $C$ grows linearly with $\min(N_t, N_r)$ : each spatial stream adds approximately $\log_2(1 + \text{SNR}/N_t)$ bits/s/Hz.

MIMO creates parallel spatial channels. The number of independent channels equals the rank of $\mathbf{H}$ , which is $\min(N_t, N_r)$ for i.i.d. fading.

Theorem: Correlation Reduces MIMO Capacity

Spatial correlation always reduces MIMO capacity:

$C_{\text{corr}} \le C_{\text{i.i.d.}}$

with equality only when $\mathbf{R}_t = \mathbf{I}$ and $\mathbf{R}_r = \mathbf{I}$ . High correlation ( $|\rho| \to 1$ ) makes the channel matrix rank-deficient, reducing the number of effective spatial streams.

Correlated antennas see similar channels, reducing the spatial multiplexing gain. This is why antenna spacing of $\lambda/2$ or more is preferred.

Example: Generating Correlated MIMO Channels

Generate 4x4 MIMO channels with exponential correlation $\rho = 0.7$ at both TX and RX. Verify the correlation structure.

Solution

Implementation

Nr, Nt, rho = 4, 4, 0.7
Rr = rho ** np.abs(np.arange(Nr)[:,None] - np.arange(Nr)[None,:])
Rt = rho ** np.abs(np.arange(Nt)[:,None] - np.arange(Nt)[None,:])
Lr = np.linalg.cholesky(Rr)
Lt = np.linalg.cholesky(Rt)

N_ch = 10000
channels = np.zeros((N_ch, Nr, Nt), dtype=complex)
rng = np.random.default_rng(42)
for i in range(N_ch):
    Hw = (rng.standard_normal((Nr,Nt)) + 1j*rng.standard_normal((Nr,Nt))) / np.sqrt(2)
    channels[i] = Lr @ Hw @ Lt.T

Verify correlation

# Check receive correlation: E[h_{i,0} h_{j,0}*] should equal Rr[i,j]
h_col0 = channels[:, :, 0]
R_est = np.mean(h_col0[:,:,None] * np.conj(h_col0[:,None,:]), axis=0)
print("Estimated Rr:\n", np.abs(R_est).round(2))

MIMO Channel Matrix Visualization

Visualize the magnitude and phase of MIMO channel matrices with adjustable correlation and antenna configuration.

Parameters

Common Mistake: Using Real Cholesky for Complex Correlation

Mistake:

Using np.linalg.cholesky(R) when $\mathbf{R}$ is complex Hermitian, but forgetting to use Lt.conj().T instead of Lt.T in the Kronecker model.

Correction:

For complex correlation matrices, use $\mathbf{H} = \mathbf{L}_r \mathbf{H}_w \mathbf{L}_t^H$ with Hermitian transpose. For real exponential correlation, .T suffices.

MIMO

Multiple-Input Multiple-Output: a system using multiple antennas at both transmitter and receiver for spatial multiplexing or diversity.

Kronecker Model

A MIMO channel model that separates transmit and receive spatial correlations: $\mathbf{H} = \mathbf{R}_r^{1/2}\mathbf{H}_w\mathbf{R}_t^{T/2}$ .

MIMO Channel Generation