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If a passband signal is real or complex, can we find its equivalent baseband signal?

I know that an equivalent baseband representation exists for a real passband signal. Is this also true for a complex passband signal?

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No, there is no standard way to represent a complex passband signal in baseband. However, this is not often a problem, at least in communications, because most passband signals of interest are real (see counter-examples in the comments).

In general, the spectrum of a complex passband signal $s(t)$ has no symmetries. This is an example of the magnitude spectrum of a complex passband signal with bandwdth $W$:

enter image description here

To obtain a complex baseband representation, you may downconvert the right-hand side of the spectrum and lose the left-hand side, or upconvert the left-hand side and lose the right-hand side. Neither is useful, because half of the signal is lost in the process.

One possible approach would be to obtain two complex baseband signals, one for the left-hand spectrum and one for the right-hand spectrum.

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    $\begingroup$ I'd disagree – the baseband transform is simply multiplication with a complex sinusoid, so equivalent baseband is really just the same signal mutliplied with $e^{j2\pi f_c t}$, but I'd agree, why I'd call it "equivalent" when it's just a shifted version is questionable. $\endgroup$ Sep 3, 2021 at 12:42
  • $\begingroup$ @MarcusMüller To me, the problem is this: the low-pass equivalent of a real passband signal can be used to reproduce the original signal -- there is nothing in the passband signal that is not captured in the low-pass representation. This is not true of a complex passband signal, because either the positive or negative frequencies are lost when the signal is shifted. For a real passband signal this is not a problem, because its spectrum is symmetric. $\endgroup$
    – MBaz
    Sep 3, 2021 at 12:50
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    $\begingroup$ I agree with MBaz that they are only equivalent for real passband signals. But I don't agree that all passband signals of interest are real. There are situations where we do use passband signals of interest that are complex: one example is when we split the analog signal in quadrature and sample what now represents a complex IF signal with two ADC's. (this is done in cases when the additional ADC and quadrature splitter is cheaper/simpler than the higher performance anti-alias filter that would otherwise be required). $\endgroup$ Sep 3, 2021 at 15:11
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    $\begingroup$ @DanBoschen Good point; I'll edit my answer. $\endgroup$
    – MBaz
    Sep 3, 2021 at 15:14
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    $\begingroup$ Thanks a lot for spending your valuable time to help me understand the concept. $\endgroup$ Sep 6, 2021 at 12:34
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Yes. This is commonly done in IQ SDRs which use a (often Tayloe IQ mixer) sampling frequency offset in an earlier wider bandwidth intermediate frequency or IF stage IQ data stream from the signal frequency of interest (to help reduce the effects of first stage IQ mixer IQ imbalance or offset on the signal of interest).

To convert the spectrum at some offset from the center of an IF IQ data stream to baseband, one simply complex multiplies the IQ data by a complex exponential at the frequency of the offset, and then low pass filters the new baseband IQ data to remove the other circularly rotated frequencies outside the new baseband bandwidth of interest, then optionally resampling to a new lower IQ data rate.

This can be done in multiple stages for multi-down-conversion super-het style IQ signal paths (in order to demodulate multiple bands or sub-band signals simultaneously, thru multiple signal sinks, etc)

An IQ SDR transmitter can do the opposite, IQ modulate up from narrow-band baseband to an offset frequency in the complex IQ IF data stream, to be later upconverted again to the strictly real transmitter output, possibly increasing the sample rate of the data along the way.

In the waterfall for complex spectrum, one simply sees a shift in the signal of interest (or rotation of the waterfall) from center baseband to some other spot, or vice-versa.

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  • $\begingroup$ mcHF QRP transceivers and RTL-SDR software commonly use this kind of offset in the initial IQ data stream. $\endgroup$
    – hotpaw2
    Sep 4, 2021 at 10:19
  • $\begingroup$ But this is converting a real passband signal to a complex baseband signal (with a small carrier offset), isn't it? The OP is asking about converting a complex passband signal itself. I believe Mbaz is correct that there is no "standard way" for a generic complex passband signal as he wrote, although I can't think of a good example where I have used a complex passband signals other than single sided ones. $\endgroup$ Sep 4, 2021 at 12:13
  • $\begingroup$ If there’s a carrier offset larger than the signal bandwidth in a much wider bandwidth IQ stream, then isn’t the passband signal complex? And the equivalent baseband can also be also complex. The OP didn’t specify that it has to be strictly real. $\endgroup$
    – hotpaw2
    Sep 4, 2021 at 16:01
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    $\begingroup$ Not necessarily- The carrier offset could be such that the positive and negative frequencies move in opposite sign (a real carrier offset). The equivalent baseband signal is typically complex, unless we are dealing with modulations that are strictly real themselves (AM, BPSK, etc). $\endgroup$ Sep 5, 2021 at 1:36
  • $\begingroup$ Thanks a lot for spending your valuable time to help me understand the concept. $\endgroup$ Sep 6, 2021 at 12:35
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There’s yet another possible answer if the positive and negative frequencies in a complex signal have non overlapping spectrum. Just take the real part.

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