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Given the spectrum of analog signal $x_a(t)$ which is imaginary and band-limited, find the lowest sampling frequency to be able to reconstruct $x_a(t)$ from samples $x[n]$.

My attempt: Bandwidth of the signal is $\Delta \Omega = \Omega_2 - \Omega_1$ Such the sampling frequency should be $\Omega_T = 2\Delta \Omega$

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  • $\begingroup$ I'd suggest drawing the spectrum of the sampled signal, just to make sure that there is no aliasing in the band of interest and that the proposed filter can indeed reconstruct the original signal. $\endgroup$ – MBaz Dec 2 '17 at 16:40
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Your understanding is not correct for complex sampling:

For the specific case of a complex bandpass (analytic) signal whose spectrum is by definition zero for the negative frequencies such as:

$$ H(\Omega) = \begin{cases} 0 &, \text{ by def. for } \Omega < 0 \\ A_\Omega &, \text{ nonzero for } \Omega_1 < \Omega < \Omega_2 \\ \end{cases}$$

then the necessary minimum sampling rate which would avoid aliasing (spectral overlap) is given by $$ \Omega_s > ( \Omega_2 - \Omega_1 ) $$ Note that if the signal were real with a symmetric bandwidth then the minimum sampling rate would be twice that of the complex case. Also then the allowed range of valid sampling rates would be found differently.

The reconstrcution filter that would yield the original complex signal $x(t)$ back will be a complex bandpass filter, not a real one. You can obtain the complex bandpass filter $H_+(\Omega)$ from the real bandpass filter $H_r(\Omega)$ using a Hilbert transformer such as:

$$ h_+(t) = h_r(t) + j \mathcal{H} \{h_r(t) \} $$

Note that for a given sampling rate you will be getting $F_s$ complex samples per second which is equivalent to $2 \times F_s$ real samples per second. Hence the apparent advantage of complex sampling is actually not realized , as the total number of samples per second will be the same in both cases.

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  • $\begingroup$ Thank you for the answer, do you have any links to something I can read about the spectrum of a complex bandpass signal? $\endgroup$ – DSP son Dec 2 '17 at 20:02
  • $\begingroup$ I'm glad I could help. Internet has links for everything: online classes, tutorials, books. Unfortunately, thus, I haven't bookmarked one such. However you shall search for communication theory books for the utilization of complex bandpass (analytic) signals in modulation theory. Look for I/Q sampling as well. By the way, you can select the answer and/or upvote it, if you find it useful and/or correct. That's how the questions are closed when answered properly... $\endgroup$ – Fat32 Dec 2 '17 at 20:12
  • $\begingroup$ Thank you again, I'm looking at interesting information as we speak. One thing, if it is not too much, how would the ideal reconstruction filter look? Would it be onesided from $\Omega_1$ to $\Omega_2$ ? $\endgroup$ – DSP son Dec 2 '17 at 21:07
  • $\begingroup$ @DSPson Yes the ideal reconstruction filter will be one sided bandpass. That's why I called it a complex filter. This way it will reject all other images including the one at the negative frequencies $-\Omega_2 < \Omega < -\Omega_1$ which would not be rejected by a real filter (whose spectrum is conjugate symmetric (has even magnitude) )... $\endgroup$ – Fat32 Dec 2 '17 at 21:12
  • $\begingroup$ @DSPson Thank you too. Please don't hesitate to come back for more questions, as you would encounter during your research... $\endgroup$ – Fat32 Dec 2 '17 at 21:25

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