Exercises

ex-sp-ch21-01

Easy

Compute the free-space path loss at 3.5 GHz for distances of 10, 100, and 1000 meters.

Solution
Implementation
import numpy as np
fc = 3.5e9
for d in [10, 100, 1000]:
    pl = 20*np.log10(4*np.pi*d*fc/3e8)
    print(f"d={d}m: PL={pl:.1f} dB")

ex-sp-ch21-02

Easy

Generate 100000 Rayleigh fading samples and verify that the mean power is 1.0 and the CDF matches the theoretical Rayleigh CDF.

Solution
Implementation
rng = np.random.default_rng(42)
h = (rng.standard_normal(100000) + 1j*rng.standard_normal(100000)) / np.sqrt(2)
print(f"Mean |h|^2 = {np.mean(np.abs(h)**2):.4f}")

ex-sp-ch21-03

Easy

Generate Ricean fading samples for K=0,5,10K = 0, 5, 10 dB and plot the envelope distributions.

Solution
Implementation
for K_dB in [0, 5, 10]:
    K = 10**(K_dB/10)
    h = np.sqrt(K/(K+1)) + np.sqrt(1/(K+1)) * (rng.standard_normal(N) + 1j*rng.standard_normal(N)) / np.sqrt(2)

ex-sp-ch21-04

Easy

Compute the maximum Doppler frequency for a mobile at 120 km/h at carrier frequencies of 900 MHz, 2.1 GHz, and 28 GHz.

Solution
Implementation
v = 120 / 3.6  # m/s
for fc in [900e6, 2.1e9, 28e9]:
    fd = v * fc / 3e8
    Tc = 0.423 / fd
    print(f"fc={fc/1e9:.1f} GHz: fd={fd:.1f} Hz, Tc={Tc*1e3:.2f} ms")

ex-sp-ch21-05

Easy

Build a 4×44 \times 4 exponential correlation matrix with ρ=0.8\rho = 0.8 and verify it is positive definite using np.linalg.eigvalsh.

Solution
Implementation
rho = 0.8
R = rho ** np.abs(np.arange(4)[:,None] - np.arange(4)[None,:])
eigvals = np.linalg.eigvalsh(R)
print(f"Eigenvalues: {eigvals}")
print(f"All positive: {np.all(eigvals > 0)}")

ex-sp-ch21-06

Medium

Simulate BPSK BER over Rayleigh fading for Eb/N0=0E_b/N_0 = 0-3030 dB and compare with the closed-form Pˉb=12(1−γˉ/(1+γˉ))\bar{P}_b = \frac{1}{2}(1 - \sqrt{\bar\gamma/(1+\bar\gamma)}).

Solution
Implementation
# See code supplement ch21/python/fading_ber.py

ex-sp-ch21-07

Medium

Generate time-correlated Rayleigh fading using the spectral shaping method (filter white noise with Jakes spectrum). Verify the autocorrelation matches J0(2πfdΔt)J_0(2\pi f_d \Delta t).

Solution
Implementation
# See code supplement ch21/python/jakes_model.py

ex-sp-ch21-08

Medium

Generate correlated MIMO channels using the Kronecker model with 4×44 \times 4 configuration and ρ=0.5\rho = 0.5. Compute the ergodic capacity at SNR = 10 dB.

Solution
Implementation
Nr, Nt = 4, 4
snr = 10  # linear
N_ch = 1000
caps = []
for _ in range(N_ch):
    H = kronecker_mimo(Nr, Nt, Rr, Rt)
    C = np.log2(np.real(np.linalg.det(np.eye(Nr) + snr/Nt * H @ H.conj().T)))
    caps.append(C)
print(f"Ergodic capacity: {np.mean(caps):.2f} bps/Hz")

ex-sp-ch21-09

Medium

Implement the 3GPP UMa path loss model (LOS and NLOS) and plot coverage probability vs distance for a 43 dBm transmitter at 3.5 GHz.

Solution
Implementation
# See code supplement ch21/python/path_loss_models.py

ex-sp-ch21-10

Medium

Generate a spatially correlated shadow fading map using the FFT-based method with decorrelation distance 50 m. Plot the resulting heatmap.

Solution
Implementation
# See code supplement ch21/python/shadow_fading.py

ex-sp-ch21-11

Hard

Implement the Weichselberger (virtual channel) model as an alternative to Kronecker. Compare capacity distributions for the two models.

Solution
Approach

The Weichselberger model uses H = Ur @ (Omega .* Hw) @ Ut.H where Ur, Ut are eigenvectors of Rr, Rt and Omega is a coupling matrix.

ex-sp-ch21-12

Hard

Estimate the Ricean KK-factor from 10000 channel samples using both the moment method and the MLE method. Compare accuracy.

Solution
Approach

The moment method uses K=E[∣h∣2]2−Var[∣h∣2]/(E[∣h∣2]−…)K = \sqrt{E[|h|^2]^2 - \text{Var}[|h|^2]}/(E[|h|^2] - \sqrt{\ldots}). MLE fits the Rice distribution via scipy.stats.rice.fit.

ex-sp-ch21-13

Hard

Simulate a wideband channel with 3GPP TDL-A model. Compute the frequency response and verify the coherence bandwidth matches 1/(5τrms)1/(5\tau_{\text{rms}}).

Solution
Implementation
# Compute frequency response via FFT of the impulse response

ex-sp-ch21-14

Hard

Implement 2-branch MRC diversity over Rayleigh fading. Verify that the BER slope changes from 1/SNR1/\text{SNR} to 1/SNR21/\text{SNR}^2.

Solution
Approach

Generate two independent fading coefficients h1,h2h_1, h_2. MRC combining: y=h1∗r1+h2∗r2y = h_1^* r_1 + h_2^* r_2, then decide based on Re(y)\text{Re}(y).

ex-sp-ch21-15

Challenge

Implement a complete 3GPP CDL-A (Clustered Delay Line) channel model with proper angle-of-arrival generation, per-cluster Doppler shifts, and antenna array response.

Solution
Approach

See 3GPP TR 38.901 Section 7.5 for the full CDL generation procedure.

ex-sp-ch21-16

Challenge

Build a geometry-based stochastic channel model (GBSM) for a vehicular scenario with scatterers placed around the TX and RX. Generate time-varying channels and compare statistics with the Clarke-Jakes model.

Solution
Approach

Place scatterers on rings around TX/RX, compute path delays and Doppler shifts from geometry, then sum contributions coherently.

Chapter SummaryReferences & Further Reading