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hFilt = designfilt('hilbertfir','FilterOrder',n,'TransitionWidth',TW); n is the order of the filter and TW is the transition width. d = fdesign.hilbert b = firpm(n,f,fresp,w,'hilbert') n - order of the filter f - normalized frequency points for transition bandwidth w - weight of the points

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Two samples is sufficient and we can determine if the phase difference is indeed $\frac{3\pi}{2}$ as follows: Start with the hypothesis that the phase difference $\phi_1-\phi_2$ is $\frac{3\pi}{2}$, which is equivalent to $-\frac{\pi}{2}$ If and only if the phase is in such quadrature, then $s_1(t)$ and $s_2(t)$ will be the real and imaginary components of ...

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HINT: It looks like they shifted the frequency response by half the sampling frequency, i.e., $$H_{HP}\left(e^{j\omega}\right)=H_{LP}\big(e^{j(\omega-\pi)}\big)\tag{1}$$ Frequency shifting corresponds to modulation (multiplication) in the time domain. Now you just have to figure out the modulation sequence that achieves the correct frequency shift.

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