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February 4, 2026 10:00
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| import numpy as np | |
| import matplotlib.pyplot as plt | |
| T = 1 | |
| delta_f = 1 / T | |
| M = 8 | |
| N = 8 | |
| OVERSAMPLING = 128 | |
| t = np.arange(-10 * N * M * T, 10 * N * T * M, 1 / OVERSAMPLING) | |
| def alpha_kl(k, l, t, M, N, T): | |
| s = np.zeros_like(t, dtype='complex') | |
| delta_f = 1 / T | |
| for n in range(N): | |
| EXPPARAM = (t - l * T / M - n * T) | |
| s_ = np.exp(1j * 2 * np.pi * n * k / N) * \ | |
| np.exp(1j * np.pi * delta_f * M * EXPPARAM) * \ | |
| np.sinc(delta_f * M * EXPPARAM) * M * delta_f | |
| s = s + s_ | |
| return np.sqrt(T / M / N) * s | |
| import numpy as np | |
| sample_vector = alpha_kl(0, 0, t, M, N, T) | |
| vector_len = len(sample_vector) | |
| # 2. Initialize a 3D array of zeros | |
| # Shape: (M rows, N columns, vector_len depth) | |
| alpha_array = np.zeros((M, N, vector_len), dtype='complex') | |
| # 3. Populate the array using nested loops | |
| for k in range(M): | |
| for l in range(N): | |
| alpha_array[k, l, :] = alpha_kl(k, l, t, M, N, T) | |
| # Send symbols on these | |
| symbols = np.array([1, 1j, 1+1j, -1j]) | |
| waveform = np.zeros_like(alpha_array[0][0], dtype='complex') | |
| waveform += symbols[0] * alpha_array[0][0] | |
| waveform += symbols[1] * alpha_array[0][1] | |
| waveform += symbols[2] * alpha_array[1][0] | |
| waveform += symbols[3] * alpha_array[1][1] | |
| # Recover the symbols | |
| symbols_recovered = np.zeros_like(symbols, dtype='complex') | |
| symbols_recovered[0] = np.trapezoid(waveform * np.conj(alpha_array[0][0]), t) | |
| symbols_recovered[1] = np.trapezoid(waveform * np.conj(alpha_array[0][1]), t) | |
| symbols_recovered[2] = np.trapezoid(waveform * np.conj(alpha_array[1][0]), t) | |
| symbols_recovered[3] = np.trapezoid(waveform * np.conj(alpha_array[1][1]), t) | |
| print(symbols) | |
| print(symbols_recovered) | |
| # Let's delay and frequency offset one of them | |
| waveform = symbols[0] * alpha_array[0][0] | |
| waveform_delay = np.roll(waveform, OVERSAMPLING // M * 2) | |
| print(np.trapezoid(waveform_delay * np.conj(alpha_array[0][2]), t)) | |
| waveform_delay_doppler = np.roll(waveform * np.exp(1j * 2 * np.pi * 2 / T / N * t), OVERSAMPLING // M * 2) | |
| print(np.trapezoid(waveform_delay_doppler * np.conj(alpha_array[2][2]), t)) | |
| triplets = [ | |
| # (gain, normalized time delay, normalized doppler shift) | |
| # normalized means between 0 and M - 1 and 0 and N - 1 | |
| (1.0, 2, 2), | |
| (0.0, 2, 3), | |
| ] | |
| actual_signal = np.zeros_like(t) * 0.0j | |
| for i in triplets: | |
| gain, time_delay, doppler = i | |
| actual_signal += np.roll(gain * waveform * np.exp(1j * 2 * np.pi * doppler / T / N * t), OVERSAMPLING // N * time_delay) | |
| dd_domain_rx = np.zeros((M, N), dtype='complex') | |
| for k in range(M): | |
| for l in range(N): | |
| dd_domain_rx[k][l] = np.trapezoid(actual_signal * np.conj(alpha_array[k][l]), t) | |
| # plt.imshow(np.abs(dd_domain_rx)) | |
| # plt.show() | |
| # We want to do something like Doppler focusing. We will just keep shifting the signal by T / M multiples and just add them all up. | |
| assert(np.max(np.abs((np.roll(actual_signal, -32) * np.exp(-1j * 2 * np.pi * 2 / T / N * t)) - waveform)) < 1e-8) | |
| test_nu = 1 / N / T | |
| PHI = np.zeros_like(actual_signal) * 0.0j | |
| for p in range(M): | |
| PHI += np.roll(actual_signal, -p * OVERSAMPLING // M) * np.exp(-1j * 2 * np.pi * test_nu * (t + p * T)) | |
| plt.plot(t, np.abs(PHI)) | |
| plt.show() |
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