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February 3, 2026 04:37
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| import numpy as np | |
| N = 8 | |
| h = np.array([1+1j, 0.5-0.4j, 0.2 + 0.3j]) | |
| x = np.array([1, 3, 2, 4, 7, -3, 4, 5]) | |
| assert(len(x) == N) | |
| # Convolution | |
| y = np.convolve(x, h) | |
| print(y) | |
| # Convolution matrix | |
| H = np.zeros((N + len(h) - 1, N), dtype='complex') | |
| H[0, 0] = h[0] | |
| H[1, 0:2] = [h[1], h[0]] | |
| H[2, 0:3] = h[::-1] | |
| H[3, 1:4] = h[::-1] | |
| H[4, 2:5] = h[::-1] | |
| H[5, 3:6] = h[::-1] | |
| H[6, 4:7] = h[::-1] | |
| H[7, 5:8] = h[::-1] | |
| H[8, 6:8] = [h[2], h[1]] | |
| H[9, 7:8] = [h[2]] | |
| assert(np.max(np.abs(H @ x - np.convolve(h, x)) < 1e-8)) | |
| # Construct the circulant matrix | |
| H_circ = np.zeros((N, N), dtype='complex') | |
| H_circ[0, 0] = h[0] | |
| H_circ[0, -1] = h[1] | |
| H_circ[0, -2] = h[2] | |
| for i in range(1, N): | |
| H_circ[i,:] = np.roll(H_circ[0,:], i) | |
| # Construct Fourier matrix | |
| FI = 1 / np.sqrt(N) * np.exp(1j * 2 * np.pi * np.outer(np.arange(N), np.arange(N)) / N) | |
| F = FI.conj().T | |
| # The circulant matrix is diagonalized by the DFT. That is: | |
| # H_circ = IDFT matrix * L * DFT Matrix | |
| # L = DFT Matrix * H_circ * IDFT Matrix | |
| L = F @ H_circ @ FI | |
| assert (np.max(np.abs(np.diag(L) == np.fft.fft(h, N))) < 1e-8) |
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