# Restore real signal from its complex representation if sampled under Nyquist frequency

Assume we have:

• real signal: $$s(t)$$
• its analytic representation: $$s^+(t) = s(t) + j*H(t)$$, where $$H(t)$$ - Hilbert transform of $$s(t)$$

Spectrum of $$s^+(t)$$ has only positive part, so may be sampled with frequency $$f_p = F_n/2$$ (without aliasing); where $$F_n$$ - Nyquist frequency for real signal. Its for example LTE20 case which IQ representation is sampled with $$f_p=30.72MHz$$.

The question is: how to restore real signal from analytic representation if sampled under $$F_n$$?

(Typical literature response is that just imaginary part can be discarded, but they silently assumes that $$f_p >= F_n$$.)

You simply can't.

The analytic signal contains the exact same amount of information as the original real-valued signal, and if either is undersampled, you get the loss of information that undersampling gets you.

That's it. Unless you've got any secondary information about the signal that you forgot to mention, there's nothing you can do.

• Does it mean that 3GPP made big mistake specifying LTE IQ data signal as undersampled? It means that our world is in big danger! :) – Andrew123 Feb 6 at 10:19
• no, it doesn't mean that (LTE is not undersampled, never heard of that). It probably means you're misinterpreting some document, probably. – Marcus Müller Feb 6 at 11:03
• Are you perhaps mentally shifting LTE to baseband, noticing the highest frequency is at 30.72/2 MHz, and then confusing the complex baseband with a real signal? – Marcus Müller Feb 6 at 11:31

"You simply can't..." - it isn't true. Let me explain by pictures...

Base band signal with bandwith B:

S(f) sampled with Fp > 2*B:

S(f) sampled with Fp < 2*B:

Analytic representation of S(f) (complex signal, has no band mirror on negative frequencies):

Analytic signal sampled with B < Fp < 2*B:

As you can see information isn't lost. One possibility to restore original signal from its "undersampled" complex representation would be resampling Sp' to Fp > 2*B and then manually restore lost mirrored negative part of spectrum. But the question is if there is more robust method?

• Can you write code that recovers the real part as you describe? I'm afraid you'll find it impossible: real and analytic are one-to-one, like x and x * 2. – OverLordGoldDragon Feb 17 at 0:03
• Andrew – ah I think you and I might have applied different understandings of "if sampled under $F_N$"! You're of course right, if you sample both real and imaginary part at a rate of $B$ in your scenario, reconstruction is easy. – Marcus Müller Feb 20 at 18:38