Oversimplified Window Method

Sample ideal Hilbert-transform kernel $h(t)=1/\pi t$ to get

$$\begin{eqnarray*} {\hat h}_i(n) &\isdef & \frac{1}{\pi nT} \qquad \qquad \hbox{(\emph{Sampled} Hilbert transform kernel)} \\ [5pt] {\hat h}_w(n) &\isdef & w(n){\hat h}_i(n) \quad\;\; \hbox{(\emph{Windowed} ideal impulse response)} \\ [5pt] {\hat H}_w(\omega) &=& (W\ast {\hat H})(n) \quad \hbox{(\emph{Smoothed} ideal frequency response)} \end{eqnarray*}$$

Design Parameters:

fs = 22050;     % sampling rate (Hz)
T = 1/fs;       % sampling period (sec)
M = 257;        % FIR filter length = window length
N = 8*(M-1);    % for interpolated spectral displays
beta = 8;       % beta for Kaiser window

Filter Design:

% Choose our sampled time axis to avoid time zero
% to avoid a division by zero in hr below:
n = [-N/2+0.5:N/2-0.5];    % Time axis (avoid t=0)
hi = T ./ (pi*n*T);        % Sampled Hilbert kernel
hr = sin(pi*n) ./ (pi*n);  % 1/2 sample delay filter
h = (hr + j*hi)/2;         % Sampled ideal final filter
plot(f,fftshift(max(-100,20*log10(abs(fft(h)))))); grid;
...