Before this:Complex signals & I/QThe Fourier transform
The analytic signal & Hilbert transform
Key takeaways A real signal’s spectrum is always mirror-symmetric — every positive frequency has a redundant negative-frequency twin. The Hilbert transform phase-shifts the signal by 90° to make the quadrature part; pairing it with the original gives the analytic signal, a complex signal whose negative-frequency half is cancelled. That is how a one-wire real signal becomes clean I/Q.
The I/Q lesson took two-number samples for granted. This one answers where the second number comes from when a signal starts life as a single real stream. It builds on complex signals & I/Q and the Fourier transform.
The mirror in every real spectrum
Take the Fourier transform of any purely real signal and you always find the same thing: the spectrum is symmetric about zero. Whatever sits at +5 kHz has an identical mirror image at −5 kHz. That twin carries no extra information — it is a mathematical consequence of the signal being real — but it is what makes a lone real stream unable to say which side of a frequency a component is on.
The Hilbert transform: a 90° phase shifter
The tool that removes the mirror is the Hilbert transform. Think of it as a filter that leaves every frequency’s amplitude untouched but shifts its phase by exactly 90° — a quarter cycle. Feed a cosine in and a sine comes out; that is precisely the quadrature relationship between I and Q.
Building the analytic signal
Now assemble the complex signal:
I (in-phase) = the original real signal
Q (quadrature)= its Hilbert transform (90 deg shifted)
analytic signal = I + jQ
Because of how the 90°-shifted copy adds and subtracts across the spectrum, the two mirror halves cancel on the negative side and reinforce on the positive side. What remains is a complex signal with a one-sided spectrum — the analytic signal. Its magnitude at each instant is the signal’s envelope and its angle is the instantaneous phase, the very quantities demodulation reads.
Where this happens in practice
Most SDRs never run a literal Hilbert transform on a real stream — their hardware produces I/Q directly by mixing the antenna signal against two oscillators 90° apart, a quadrature downconversion that achieves the same one-sided result in analog. The Hilbert view still matters: it is the theory that explains why two channels 90° apart capture everything a real signal holds, and it is exactly how you’d convert a real recording (say, an audio file) into I/Q in software. The result feeds the same complex baseband pipeline either way.
Quick check: what does forming the analytic signal accomplish?
Recap
- A real signal’s spectrum has a redundant negative-frequency mirror.
- The Hilbert transform shifts every frequency by 90° to form the quadrature part.
- Pairing original + Hilbert gives the analytic signal, whose mirror half cancels.
- SDR hardware reaches the same one-sided I/Q by quadrature downconversion.
Next up: the operation at the heart of every filter — convolution and the impulse response.
Frequently asked questions
What does the Hilbert transform actually do?
The Hilbert transform shifts every frequency component of a real signal by exactly 90 degrees. Used as the imaginary (quadrature) part alongside the original real signal, it produces the analytic signal — a complex signal whose spectrum has no negative-frequency half, so it can be represented and processed as clean I/Q.
Why do we want to get rid of negative frequencies?
A real signal’s spectrum is always mirror-symmetric: every positive frequency has an identical negative-frequency twin carrying no new information. That mirror is what makes a real signal ambiguous about which side of a carrier it sits on. Removing it with the analytic signal leaves one clean copy, which is exactly what lets an SDR tell a signal above centre from one below.