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[EDIT: note that the note you refer to compute the discrete-time Fourier transform, via a continuous argument in frequency. And not a DFT. You are apparently computing a 3-point DFT] What I usually call the size-2 binomial filter is $\beta_1=\frac{1}{2}[1\;1]$, the 2-point moving average, whose Fourier representation is well-known, or easy to compute. $$... 3 I don't really understand the thing about cosines (as in: how is that helpful?) – a DFT is really just a mapping from a complex vector with N elements to a complex vector with N elements; and your calculation seems to be wrong, and I'm not sure where, but doing two of the three elements of the DFT manually might actually be enough to clear things up. \... 2 It is a little late, but i'm also working on convolution reverb at the moment. If it is still of interest, you can use my code. Simply call the function convolution_reverb and pass the paths to the two audio files (audio and impulse response, both need to be .wav files), as well as the name for the result file to be created. import numpy as np from wave ... 1 The input-output relation of such a system can be written as$$y(t)=\int_{-\infty}^{\infty}\delta(t/2-\tau)x(\tau)d\tau\tag{1}$$Note that the system is linear but time-varying, and such systems can generally be described by a two-dimensional impulse response h(t,\tau):$$y(t)=\int_{-\infty}^{\infty}h(t,\tau)x(\tau)d\tau\tag{2} In the given example ...
With the help of the two answers above, I think I finally understood what the paper was all about. Let $Z_n = \sum_{i=1}^n X_i$ where $X_i \sim \text{Bernoulli}(p)$. The sum of $n$ Bernoulli random variables can be found by convolution in the time domain. Then $Z_n$ is a binomial variable. In the document, the authors mistakenly wrote \$B_2 = \frac{1}{4}\...