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I thought about this a bit, and I wondered whether something that numerically approximates the wished-for signal might be good. So I implemented a project onto convex sets (POCS) approach to synthesizing such a signal. The idea is: Initialize with a random signal. Take the FFT. Zero out the upper coefficients. Take the inverse FFT. Zero out the upper ...

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This is not an answer, rather a rationale I am following. (I know there is the "comment" functionality, but I don't have unlocked it yet) Anyway, I would shortly describe my observation as: your answer reduces in determining whether a generic signal (not necessarily an analytic one) with finite support in time/frequency can have one-sided infinite ...

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Disclaimer: I understand the question as "is there an $x(t)$ that's $x(t<0)=0$ with $X(\omega < 0) = 0$". I present some ideas rather than proofs. Approach 1 I'll answer in terms of the inverse Fourier transform. We seek some $X(\omega)$ such that the complex sinusoids it weighs, in summation (integration), cancel to zero for $t < 0$ - but ...

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The Fourier transform of a causal signal cannot be zero in any interval $[\omega_1,\omega_2]$ with $\omega_1<\omega_2$. This follows from the Paley-Wiener condition, which states that if $A(\omega)=|X(\omega)|$ is the magnitude of the Fourier transform $X(\omega)$ of a causal signal $x(t)$, then the integral \int_{-\infty}^{\infty}\frac{\big|\ln A(\...

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The primary purpose of those filters is to pass the desired signal bandwith and reject the carrier feed-through at $\omega_h$ and the double frequency component at $2\omega$. Given that, I would recommend designing the filters with a passband and rejection requirement at those frequencies in mind with a goal of minimizing filter complexity (which means ...

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Reconstruction is possible so long as NOLA is obeyed - which is an easier criterion (on synthesis information) to meet than what you seek (analysis information). To discriminate temporal variations finer than $T$, the window's temporal width must be $\leq T$. You can use ssqueezepy's window_resolution with appropriate unit conversion (mult by $f_s$) to ...

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In general for the FT, in order for the result for the DFT or IDFT to be real, the waveform MUST be real and even (symmetric about vertical axis) or imaginary and odd (anti-symmetric about the vertical axis), or made up of the sum of such components. A simple example with a cosine and continuous-time Fourier Transform as detailed below should clear this up ...

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(while we expected it to be real). We certainly did not expect this to be real. Quite on the contrary: a real time domain signal has a spectrum that is conjugate symmetric. Be zeroing out the negative frequency we clearly broke this symmetry and that means that the analytical signal MUST be complex (other than the trivial case a $x[n] = const$)

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There are routines that will provide the Hilbert coefficients directly, but an approach I like to use given its simplicity and clarity in functionality is to transform a Half Band filter to a Hilbert as follows: Step 1: Estimate the number of taps needed from the specifications using these commonly used estimators. The one Marcus Mueller provided would be ...

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You can't implement an ideal Hilbert transformer: its impulse response is non causal and infinite in time. So you can only implement an "approximate" one and the best way to do this depends on the specific requirement of your application: what's your frequency range of interest, how much magnitude and/or phase deviation can you tolerate, are you ...

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