You are on the right track, so here is a hint:
Since these are all binary functions with time step of one (and I'm too lazy to type integrals), I'm just going to treat it like a sum with the index being the end-time. So you get
$$y(4) = h(1)\cdot x(4) + h(2)\cdot x(3) + h(3)\cdot x(2) + h(4)\cdot x(1) $$
Given that both $|h| \le 1$ and $|x| \le 1$ it should ...
I´m not sure if this really applies to your problem since it may be another issue, but I can tell from my experience with acoustic impulse response measurements (only there you want to estimate the response from the input signal, but the deconvolution should be the same):
In case your transfer function E is somehow bandlimited and has zero (or very low) ...
I think you are using the wrong tool for the job. Semitones are spaced logarithmically but FIR filters have linear frequency resolution. If you want to reliably distinguish between the low E and the low F on the bass guitar you need a frequency resolution of better than 2 Hz which requires 10s of thousands of taps (at 48 kHz sample rate).
That's why most ...
You really do want to filter using $x_a$.
Suppose we design a low-pass FIR filter with the following response.
Let's then use them to filter white noise.
Notice that the impulse response that has been fftshifted does not yield a good filter output.
The reason is that the filter command uses an FFT length that is $N+M-1$ in length rather than $M$ ($M$ is ...
Everything that you can apply an FFT to is discrete in both time and frequency which means it's also periodic in both domains with the FFT length $N$.
Symmetry is clearly defined as $x[-n] = x[n]$ For signals that are periodic with N that extends to $x[-n+2kN] = x[n+2mN]$ where $m$ and $n$ ae integers.
Would it be correct to say that both xa and xb are ...
I didn't know the convolution theorem for the DHT before, but it's pretty clear that if it exists, it must be about circular convolution, just like for the DFT.
You're comparing that with acyclic convolution, so the results differing is no surprise.