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Discrete Fourier Transform Vs. Inverse Discrete Fourier Transform (Division by Number of Samples) [closed]

I've noticed that there are two possible notations for both DFT and IDFT. The First One DFT: $$ y_i = \sum_{j = 0}^{n - 1} x_j \cdot \omega_n^{i \cdot j}, i = 0, \dots, n - 1 $$ IDFT: $$ x_i = ...
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Use Butterworth and Chebychev filters

I need to calculate frequency response, phase response and apply to signals the Butterworth, Chebychev1 and Chebychev2 band-pass filters. I'm developing in C++ with Qt, and I'm looking for algorithms ...