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![]() ![]() ![]() The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. The DFT is therefore said to be a frequency domain representation of the original input sequence. It has the same sample-values as the original input sequence. ![]() An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. In mathematics, the discrete Fourier transform ( DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. Its similarities to the original transform, S(f), and its relative computational ease are often the motivation for computing a DFT sequence. The respective formulas are (a) the Fourier series integral and (b) the DFT summation. The spectral sequences at (a) upper right and (b) lower right are respectively computed from (a) one cycle of the periodic summation of s(t) and (b) one cycle of the periodic summation of the s(nT) sequence. Fig 2: Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner. The FFT algorithm computes one cycle of the DFT and its inverse is one cycle of the DFT inverse. The inverse DFT (top) is a periodic summation of the original samples. Right column: The DFT (bottom) computes discrete samples of the continuous DTFT. Its Fourier transform (bottom) is a periodic summation ( DTFT) of the original transform. Center-right column: Original function is discretized (multiplied by a Dirac comb) (top). The inverse transform is a sum of sinusoids called Fourier series. Fourier transform (bottom) is zero except at discrete points. Center-left column: Periodic summation of the original function (top). Left column: A continuous function (top) and its Fourier transform (bottom). Roughly 5% of the matrix has nonzero values.Fig 1: Relationship between the (continuous) Fourier transform and the discrete Fourier transform. If someone know how to do this much more efficiently it would be greatly appreciated. ![]() Test_matrix=1īackground: In my code, these two lines take 75% of the time for a 4000x4000 matrix whereas the ensuing topological sort and DP algorithm take only the rest of the quarter. I don't necessarily need to use these types if a faster option exists.įor timing purposes, the following matrix is a good representative: test_matrix = np.zeros(shape=(4096,4096),dtype=np.float32) I was wondering whether this might be doable with some sort of sparse matrix (CSR, CSC, COO etc.) from scipy.sparse but I am unfamiliar with them and have not got that working. (The lists are called preds and descs because they refer to the predecessors and descendants in a DAG when the matrix is interpreted as an adjacency matrix but this is not essential to the question.) nonzero() for v in range(matrix.shape)]Įxample input: matrix = np.array(,])Įxample output preds = ), array(), array(), array(, dtype=int64)]ĭescs =, dtype=int64), array(), array(), array()] nonzero() for v in range(matrix.shape)]ĭescs =. The following is a working piece of code: preds =. I am looking for the fastest way to obtain a list of the nonzero indices of a 2D array per row and per column. ![]()
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