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Consider the Discrete Hilbert Transform, which takes a sequence $u[n]$ defined on the interval $-M\leq n\leq M$ and returns the discrete convolution

$$\tilde u[n]=\sum_{i=-M}^Mu[i]h[n-i]$$

where the impulse response function is given by

$$h[j]=\frac{2\sin^2(\pi j/2)}{\pi j}$$

(zero for all even $j$ including $j=0$, and non-zero for odd $j$).

The wikipedia article above mentions an improved version of the discrete impulse response function:

$$h_N[j]=\sum_{m=-\infty}^\infty h[n-m N]$$

which exhibits a milder frequency cut-off, such that the finite $M$ impulse response is closer to the ideal infinite impulse response $M\to\infty$.

Now my question is:

Are there any other useful variations of $h[j]$ that would improve the discrete Hilbert transform spectrum and accuracy?

Or perhaps there are some entirely different ways to improve the finite impulse response of a discrete Hilbert transform? Thanks for any suggestion!

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