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The Discrete Fourier Transform (DFT, what the FFT algorithm computes) has the origin in the top-left corner. It relates a time-domain signal sampled at n = 0..N-1, and a frequency-domain signal sampled at k = 0..N-1. k is assumed periodic, such that k = N is the same as k = N. The same periodicity can be assumed for n, which is why the convolution computed ...

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The OP clarified in comments under the question that the intention is to compensate for group delay distortion introduced in the hardware. The typical approach to optimally compensate for this (in a least squared sense) with processing alone is to use the Wiener-Hopf equations to determine the coefficients of an equalizer that can be implemented with an FIR ...

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That's actually rather simple. A little bit of trial and error will do the trick. The standard specifies upper and lower limit for the transfer function of the filter. So you can just try a normal Butterworth bandpass and see what happens. The picture below shows a 3rd octave design at 1kHz and a sample rate of 48 kHz. A 2nd order bandpass isn't quite a good ...

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