Exercises

ex-sp-ch24-01

Easy

Compute the steering vector for an 8-element ULA with d=λ/2d = \lambda/2 at θ=30°\theta = 30°.

Solution
Implementation
N, d_lambda = 8, 0.5
theta = np.radians(30)
a = np.exp(1j * 2 * np.pi * d_lambda * np.arange(N) * np.sin(theta))

ex-sp-ch24-02

Easy

Compute the HPBW for ULAs with N=4,8,16,32N = 4, 8, 16, 32 elements at d=λ/2d = \lambda/2.

Solution
Implementation
for N in [4, 8, 16, 32]:
    print(f"N={N}: HPBW ≈ {102/N:.1f}°")

ex-sp-ch24-03

Easy

Show that two ULA steering vectors at different angles are orthogonal when d=λ/2d = \lambda/2 and the angles correspond to DFT frequencies.

Solution
Implementation
N = 8
a1 = steering_vector(N, 0.5, np.arcsin(0))
a2 = steering_vector(N, 0.5, np.arcsin(2/N))
print(f"Inner product: {np.abs(a1.conj() @ a2):.6f}")  # ≈ 0

ex-sp-ch24-04

Easy

Compute the array gain in dB for arrays with N=4,16,64,256N = 4, 16, 64, 256 elements.

Solution
Calculation
for N in [4, 16, 64, 256]:
    print(f"N={N}: Gain = {10*np.log10(N):.1f} dB")

ex-sp-ch24-05

Easy

Generate a DFT codebook for a 16-element ULA and plot all 16 beam patterns.

Solution
Implementation
codebook = dft_codebook(16, 0.5)

ex-sp-ch24-06

Medium

Simulate two sources at 20°20° and 25°25° with SNR = 20 dB and 100 snapshots. Compare Bartlett, Capon, and MUSIC DOA estimates.

Solution
Implementation
# See code supplement ch24/python/beamforming.py

ex-sp-ch24-07

Medium

Implement Capon beamforming with diagonal loading. Show how loading improves robustness when L<2NL < 2N.

Solution
Approach

Add δI\delta\mathbf{I} to R^\hat{\mathbf{R}} before inversion.

ex-sp-ch24-08

Medium

Implement the ESPRIT algorithm for DOA estimation and compare with MUSIC.

Solution
Approach

Use the shift invariance property of the ULA to estimate DOAs.

ex-sp-ch24-09

Medium

Simulate hybrid beamforming with 64 antennas and compare spectral efficiency with NRF=2,4,8N_{\text{RF}} = 2, 4, 8 RF chains.

Solution
Implementation
# See code supplement ch24/python/hybrid_beamforming.py

ex-sp-ch24-10

Hard

Implement the MDL (Minimum Description Length) criterion to automatically estimate the number of sources for MUSIC.

Solution
Approach

MDL selects KK that minimizes a penalized log-likelihood.

ex-sp-ch24-11

Hard

Implement a 2D MUSIC algorithm for a UPA and estimate azimuth and elevation of multiple sources.

Solution
Approach

Extend MUSIC to scan over (θ,ϕ)(\theta, \phi) using UPA steering vectors.

ex-sp-ch24-12

Hard

Simulate null-steering beamforming: maintain unit gain at a desired direction while placing nulls at specified interferer directions.

Solution
Approach

Solve CHw=f\mathbf{C}^H\mathbf{w} = \mathbf{f} where C\mathbf{C} contains steering vectors and f\mathbf{f} specifies gain/null constraints.

ex-sp-ch24-13

Hard

Implement wideband beamforming using a tapped delay-line beamformer for a frequency-selective scenario.

Solution
Approach

Apply frequency-dependent weights to compensate for the frequency-dependent steering.

ex-sp-ch24-14

Challenge

Implement the alternating minimization algorithm for hybrid precoder design that minimizes ∥Fopt−FRFFBB∥F\|\mathbf{F}_{\text{opt}} - \mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\|_F.

Solution
Approach

Alternate between optimizing FRF\mathbf{F}_{\text{RF}} and FBB\mathbf{F}_{\text{BB}}.

ex-sp-ch24-15

Challenge

Build a complete 5G NR beam management simulation: beam sweeping, beam measurement, beam selection, and beam tracking with mobility.

Solution
Approach

Implement SSB beam sweeping, CSI-RS measurement, and beam failure recovery.

Chapter SummaryReferences & Further Reading