Exercises

ex-sp-ch23-01

Easy

Implement ZF detection for a 2x2 MIMO system with BPSK and verify that it correctly recovers the transmitted symbols in a noise-free case.

Solution
Implementation
H = np.array([[1, 0.5], [0.3, 1]], dtype=complex)
x = np.array([1, -1], dtype=complex)
y = H @ x
x_hat = np.linalg.pinv(H) @ y
print(f"Recovered: {np.sign(np.real(x_hat))}")

ex-sp-ch23-02

Easy

Compute the SVD of a 4x4 MIMO channel and verify that U\mathbf{U} and V\mathbf{V} are unitary.

Solution
Implementation
H = (np.random.randn(4,4) + 1j*np.random.randn(4,4)) / np.sqrt(2)
U, S, Vh = np.linalg.svd(H)
print(f"U unitary: {np.allclose(U @ U.conj().T, np.eye(4))}")
print(f"V unitary: {np.allclose(Vh @ Vh.conj().T, np.eye(4))}")

ex-sp-ch23-03

Easy

Implement ML detection for a 2x2 MIMO system with BPSK (MNt=4M^{N_t} = 4 candidates).

Solution
Implementation
from itertools import product
const = np.array([1, -1], dtype=complex)
best_x, best_d = None, np.inf
for x_cand in product(const, repeat=2):
    x_vec = np.array(x_cand)
    d = np.sum(np.abs(y - H @ x_vec)**2)
    if d < best_d: best_d, best_x = d, x_vec

ex-sp-ch23-04

Easy

Show that for a diagonal channel H=diag(h1,h2)\mathbf{H} = \text{diag}(h_1, h_2), ZF detection is equivalent to per-element division.

Solution
Proof

For diagonal H\mathbf{H}, Hβˆ’1=diag(1/h1,1/h2)\mathbf{H}^{-1} = \text{diag}(1/h_1, 1/h_2), so ZF reduces to x^k=yk/hk\hat{x}_k = y_k / h_k.

ex-sp-ch23-05

Easy

Compute the condition number of 1000 random 4x4 MIMO channels and plot the histogram.

Solution
Implementation
conds = [np.linalg.cond((np.random.randn(4,4)+1j*np.random.randn(4,4))/np.sqrt(2)) for _ in range(1000)]

ex-sp-ch23-06

Medium

Simulate 2x2 MIMO BER for ZF, MMSE, and ML detectors with QPSK over 10000 channel realizations. Plot BER vs Eb/N0E_b/N_0.

Solution
Implementation
# See code supplement ch23/python/mimo_detection.py

ex-sp-ch23-07

Medium

Implement V-BLAST with ordered SIC for a 4x4 MIMO system. Compare BER with ZF and MMSE.

Solution
Implementation
# See code supplement ch23/python/mimo_detection.py

ex-sp-ch23-08

Medium

Implement SVD precoding with water-filling for a 4x4 MIMO channel. Compare capacity with equal power allocation.

Solution
Implementation
# See code supplement ch23/python/mimo_precoding.py

ex-sp-ch23-09

Medium

Implement ZF precoding for a 4x2 multi-user MIMO downlink (4 BS antennas, 2 single-antenna users). Show zero inter-user interference.

Solution
Implementation
H = (np.random.randn(2,4) + 1j*np.random.randn(2,4)) / np.sqrt(2)
W = H.conj().T @ np.linalg.inv(H @ H.conj().T)
W /= np.linalg.norm(W, axis=0, keepdims=True)
# H @ W should be diagonal
print(np.abs(H @ W).round(3))

ex-sp-ch23-10

Medium

Simulate massive MIMO with M=32,64,128M = 32, 64, 128 antennas and K=8K = 8 users. Show that conjugate BF BER improves with MM and approaches ZF.

Solution
Implementation
# See code supplement ch23/python/massive_mimo.py

ex-sp-ch23-11

Hard

Implement a sphere decoder for 4x4 MIMO with QPSK. Compare its BER with ML and ZF, and measure the average number of nodes visited.

Solution
Approach

Use QR decomposition and tree search with Schnorr-Euchner enumeration.

ex-sp-ch23-12

Hard

Simulate the effect of imperfect CSI on ZF precoding. Model channel estimation error as H^=H+ΟƒeE\hat{\mathbf{H}} = \mathbf{H} + \sigma_e \mathbf{E} and show the BER floor.

Solution
Approach

The residual interference from imperfect CSI creates a BER floor that depends on Οƒe2\sigma_e^2.

ex-sp-ch23-13

Hard

Implement regularized ZF (RZF) precoding for multi-user MIMO: W=HH(HHH+Ξ±I)βˆ’1\mathbf{W} = \mathbf{H}^H(\mathbf{H}\mathbf{H}^H + \alpha\mathbf{I})^{-1}. Find the optimal Ξ±\alpha that maximizes sum rate.

Solution
Approach

Sweep Ξ±\alpha from 0 to K/SNRK/\text{SNR} and compute sum rate for each.

ex-sp-ch23-14

Hard

Simulate pilot contamination in massive MIMO: two cells with M=64M = 64 antennas each, K=4K = 4 users per cell, sharing the same pilot sequences. Show the SINR floor.

Solution
Approach

The estimated channel includes contributions from both cells' users, creating permanent inter-cell interference.

ex-sp-ch23-15

Challenge

Implement MIMO-OFDM with per-subcarrier ZF detection for a 2x2 system over a frequency-selective channel. Compare with single-antenna OFDM.

Solution
Approach

At each subcarrier kk, the 2x2 channel H[k]\mathbf{H}[k] is obtained from the FFT of each antenna pair's channel impulse response.

ex-sp-ch23-16

Challenge

Implement the diversity-multiplexing trade-off (DMT) simulation: show that a 2x2 MIMO system can achieve diversity order 4 (Alamouti) or multiplexing gain 2 (V-BLAST), but not both simultaneously.

Solution
Approach

Implement Alamouti STBC for full diversity and V-BLAST for full multiplexing. Plot BER slope and rate to visualize the trade-off.

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