I/Q Data for Dummies
whiteboard.ping.se
whiteboard.ping.se
A crucial aspect that many of the I/Q explanations miss is, that the I/Q representation (which is really phasor notation) is a feature of modulation onto a carrier wave, not baseband.
So the example starts of with drawing the sine wave and say we don't know the full wave, because we don't know if frequencies are positive or negative and because it's hard to determine the power (not even sure what he/she means there). That is wrong and directly contradicts Nyquist theory, if we sample a signal at twice it's maximum frequency we can fully reconstruct it. The reason the explanation goes wrong is because they bring in negative frequencies. Negative frequencies do not exist!
This is where the phasor/IQ representation comes in. If we modulate a signal onto a carrier wave, in other words we have a wave at some frequency and we modulate that wave in amplitude/phase/frequency, we generate frequencies higher and lower than that carrier frequency. The modulation can have components that correspond to the sine or cosine of that carrier wave. Importantly these components are orthogonal, so can carry independent data. That's where we use the phasor representation (IQ-modulation), and remove the carrier wave and represent this "artificial" baseband signal using complex numbers or sine or cosine components, or positive/negative frequencies (all these are equivalent). But it is important to remember that this is just a representation of a signal modulated onto a carrier as if there was no carrier.
That's where this explanation goes wrong.
The real reason we use I/Q sampling is because we want to frequency-shift a signal.
Why do we want to frequency-shift a signal? In radio frequency applications the signal of interest almost always has a much lower bandwidth than its highest frequency. In other words, the signal has a small bandwidth (say 40 MHz) centered around a high center-frequency (say 2.4 GHz). If we want to digitize the signal, then one way would be to use a very high sample-rate ADC (e.g. a 2.4 GHz ADC). But these are very expensive, and a much better way of digitizing the signal is to use a mixer (a frequency shifter) to shift the signal to be centered around 0 Hz and then use a relatively low sample-rate ADC (e.g. a 40 MHz ADC).
The way frequency shifting is done is by multipling the signal by a sine signal, which can be done in hardware. But this introduces a distortion to the signal because multiplying by a sine is not actually a frequency shift. It just so happens that this distortion is cancelled out by adding another copy of the signal multiplied with another sine delayed by 90°. But this addition needs to be complex (due to the relationship between sine functions and true frequency shifts), so what we do is sample the two distorted signals and do this complex addition with the digitial signals.
So the reason we have complex samples is because that's the best way we've found to do frequency shifting using real-only sine waves (this explains why we don't use complex numbers in audio signal processing; there's no need to do frequency shifting!). This tutorial goes into the details and is the best explanation I've seen on quadrature sampling (another term for I/Q sampling): https://www.dsprelated.com/showarticle/192.php
I think engineers (myself included) tend to get confused because using complex numbers makes the math simpler, and so they think that's the real reason we use them. All the talk about ambiguous frequencies or negative frequencies or needing to know the phase of a sample is true, but all of those problems could be solved without complex numbers simply by sampling twice as fast and then doing some math (again, audio DSP does just fine without quadrature sampling), so it's not a "real" reason to do this strange kind of sampling.
Edit, this is how the Airspy sdr works, to avoid iq imbalance like you get in the direct conversion receivers in most sdrs.
Second edit for terminology. Mixing is multiplying by a frquency to shift frequency. Baseband means you shifted the center of the frquencies you care about to zero, so half of the frequency content is negative. Negative frequencies are what drive that mean imaginary number into the whole thing.
It just so happens that audio DSP algorithms happen to almost never care about those exact kinds of processing, due to the way our ears and brains work. And thus, IQ data is not used in audio. But it's not because it's baseband. It's because our ears don't care about phase relationships (which is one thing you can more easily preserve in the IQ domain) and because frequency shifts like downconversion are not useful in music since they destroy the harmonic relationships in the sound.
100,200,400Hz is a consonant tone, while 110,210,410Hz is a dissonant mess
AM doesn't have this problem because it has symmetric sidebands and a carrier (so a tuning offset does not result in audio frequency shift), but SSB does. If you listen to an SSB transmission without your tuning being perfect, it sounds horrible. Voice sounds distorted, and music is hideous. I'm having trouble finding an example of the latter, probably because nobody dares put music through SSB :-) (but you can do this easily enough in gnuradio by upconverting a song with a 10Hz offset, for example)
In the frequency domain, you could think of AM demodulation as computing the width (and phase!) between the carrier images on both sidebands. It doesn't matter if the signal is a bit off to the side, because the width will be the same. You have a mirror image which gives an absolute reference.
In the IQ domain, you look at the magnitude of the vectors, not their angle, so you don't care about the frequency.
In SSB you only have one sideband, and often no carrier at all, so there is no reference. You need to nail the frequency to get a proper signal out. And even then AIUI your phases will be random, though that doesn't matter for audio.
Audio time stretching (or equivalently, changing the pitch without changing time) is not a clearly defined process with a mathematical description (unlike plain resampling or modulation) but rather a semi-heuristic process that takes into account psychoacoustics. But yes, in practice, for small adjustments of a monophonic sample like a voice, modern algorithms sound really good.
You are right nobody can hear phase, but nobody can see phase either again because you need to relate (interfere) to something. However it does make a difference if we think about the superposition (interference) of different audio frequency components. We would definitely here some of those phase differences.
That said iq does not make sense in audio processing because it's baseband. There is no carrier wave.
The same is not true for radio. There, corrupting the phase relationships corrupts the data (for many systems).
Phase is relative, but our ears don't care about relative phase either (at least as long as you don't stick nonlinear filters after, then it starts mattering, but usually in audio things are fairly decorrelated anyway so it only matters in quite specific cases).
Here is an example: https://twitter.com/marcan42/status/1282685645731672064
Demo: https://twitter.com/zwegner/status/1282859889447116809 (interestingly, you can hear the change in Twitter's low quality encode, but it goes away at higher qualities, so it seems their crappy AAC encoder does care about relative phase :-))
It's counterintuitive how little our ears care about phase across frequency bands. This is not true for other kinds of signals.
However, because its virtually unknown to audio folks theres perceptible nodes everywhere, if you can hear them.
But nodes and mixing products are independent of overall phase across the power spectrum, in a linear system. So if you apply the same phase change to both left and right, the distribution of nodes in the room won't change. The only time these inter-frequency phase relationships start to matter is when you introduce nonlinearities, like distortion.
But typical reflections off of surfaces are largely linear as far as I know, and any linear operation will not introduce any power spectrum changes that are phase dependent. As far as I know, the the ear canal can be largely modeled as a linear system (to within the thresholds of hearability).
The only way to hear phase is to introduce a nonlinearity. That then generates harmonics (or sometimes even lower frequencies), and their power spectrum depends on the specific phase relationships of the incoming signal.
A physical example of a nonlinearity would be a vibrating surface that hits another surface at a certain excursion. Depending on the relative phases of the excitation signal, you can have different peak excursion, and therefore clearly get a different result if one phase set makes it reach the other surface and another one doesn't.
> The real reason we use I/Q sampling is because we want to frequency-shift a signal.
> Why do we want to frequency-shift a signal? In radio frequency applications the signal of interest almost always has a much lower bandwidth than its highest frequency. In other words, the signal has a small bandwidth (say 40 MHz) centered around a high center-frequency (say 2.4 GHz). If we want to digitize the signal, then one way would be to use a very high sample-rate ADC (e.g. a 2.4 GHz ADC). But these are very expensive, and a much better way of digitizing the signal is to use a mixer (a frequency shifter) to shift the signal to be centered around 0 Hz and then use a relatively low sample-rate ADC (e.g. a 40 MHz ADC).
> The way frequency shifting is done is by multipling the signal by a sine signal, which can be done in hardware. But this introduces a distortion to the signal because multiplying by a sine is not actually a frequency shift. It just so happens that this distortion is cancelled out by adding another copy of the signal multiplied with another sine delayed by 90°. But this addition needs to be complex (due to the relationship between sine functions and true frequency shifts), so what we do is sample the two distorted signals and do this complex addition with the digitial signals.
I'm not sure I understand you correctly, but I would not say you distort the signal when you multiply with a sine wave. Essentially you create to frequency components the sum and difference frequencies (f1+f2, f1-f2), now if f1 is your modulated signal (so some f1+fmod, where fmod is a band and can be positive and negative) and you want to convert to baseband you would select f2 so that it's at the carrier (f1=f2) then you generate a baseband signal at 0 carrier frequency and a signal at 2xf1 which is usually outside your detector bandwidth so not detected. However this process only gives you half of the frequencies of your fmod, to get the other half you need to multiply with cosine(f2) which essentially gives you the component that was at 2xf1 now at baseband. So to handle that more elegantly in math you add the two components up as real and imaginary components, essentially that enables you to drop the cos/sin(f1) terms from your equations.
I agree that that's a place where the explanation goes wrong, but negative frequencies do exist, they cancel out the imaginary part of the positive frequency. A real-valued signal has equal amplitude at f and -f, if you're talking about Fourier transforms. Admittedly, Fourier transforms depart slightly from the more elementary definition of frequency as one divided by the period, but that's a linguistic snag, not a conceptual one.
IMO either perspective is fine - neither is "wrong" or "right."
> because it's hard to determine the power (not even sure what he/she means there)
I think they are referring to the fact that having the analytic representation makes envelope detection trivial.
Three-phase/three-wire electric power doesn't have three linearly independent currents or line-to-line voltages. Rather than fight the constraint that Ia+Ib+Ic == 0, we transform the system into one in which the constraint doesn't exist. The three (redundant!) 120-degree three-phase signals are transformed into a virtual two-phase reference system and we operate on the two phases that way. Sometimes we even "frequency shift" the signal into the reference frame of a physical (or virtual) rotor.
There are some interesting side-effects. In the alpha-beta (stator) frame, unbalanced three-phase load looks like the sum of a dominant positive 60 Hz and smaller negative 60Hz signal. Diode-rectified loads are dominated by the negative fifth and positive seventh harmonics in the stator reference frame, but are frequency-shifted to negative and positive sixth in the rotor's reference frame.
Three-phase four-wire (A, B, C, and Neutral) does have three linearly independent currents and voltages. But their disturbances are so different from each other that it is frequently convenient to model this system as the sum of a complex alpha-beta line-to-line current plus a virtual common-mode phase (the zero phase).
See also the Clarke and Park transforms.
1. It's actually a fairly simple concept but this page jumps into pages of formulas and irrelevancies. It is also missing motivation for why I/Q exist. No way is this "for dummies"
2. This: "I'd say the true signal is complex, and the real signal is an incomplete projection of it" is factually incorrect. The signal coming down the antenna is real.
I feel like I could explain it better in 1/4 of the text. Would anyone be interested in that?
If you could add another (potentially better explanation), that could be beneficial for future learners (not necessarily limited to the HN community).
https://pysdr.org/content/sampling.html#
first.
I really just wanted something that just explained what I and Q were and why they existed, without going into what Nyquist and Fourier transforms.
A few that I think are useful or have interesting components are:
* https://pysdr.org/content/sampling.html#quadrature-sampling
* https://visual-dsp.switchb.org/
* Understanding Digital Signal Processing by Richard Lyons
I would, as I read this article but I also think a clearer explanation could be made.
When I started with SDR and wrote some Fourier Transform code I wanted to know why if I fed it a sine wave at frequency X, I got two peaks in the transform. Experts said, "Well that is because you have a real signal only, if you add the quadrature component you'll see just one peak." Which is true, if you do that you see one peak, but WHY was my question.
The answer to that question was that there is a difference between discrete mathematics and continuous mathematics. And more specifically, when you operate using discrete mathematics, you need to understand both the slope and the magnitude of the the function you are working with.
A discrete version of sin(ωt) (where ω = 2πf) that is "real only" as some would say, has zero values in the quadrature side. That happens when the signal is the sum of two signals that are the complex conjugate of each other (their quadrature (or imaginary) parts differ only in sign). The discrete Fourier transform correctly picks up both of those signals and identifies them as +f and -f.
The best book to understand I/Q if your background is not engineering is probably this book "Digital Signal Processing in Modern Communication System" by Andreas Schwarzinger [1].
This book will teach from the basic principles of communication from the I/Q up to OFDM, basically most of the essential techniques of modern communication as mentioned in its title.
[1]https://www.amazon.com/Digital-Signal-Processing-Communicati...
I've tried to get into communication theory to work on some moderately complex SDR projects, but it seems I cant escape math for this.
For I/Q signals in particular, W2AEW has a couple great videos that explain the intuition and then demonstrate things in action on real instruments:
I really only understood this when I started doing communications (and that was after doing a PhD in physics mind you).
Once you've got the data in I/Q lots of the modulation modes just fall out really easily(I recommend reading about AM, FM and BPSK/QPSK as a starting point which then you can generalize into QAM).
My only concrete advise is to avoid the Proakis book
Try out GNU Radio... you can just use your audio I/O instead of messing with RF, etc. Build a few graphs where you use complex signals, multiply them together, etc... and you can get the hang of it, for no $$ and only a few hours time.
Is precise the right word here? It’s more complete, less information is lost (sampling is never lossless though) but there’s no difference in precision.
If i have IQ data then i can derive the data that would be preserved by any other modulation (phase, frequency, amplitude shifting). I can’t recover the original IQ samples from a sampled FM signal.
Negative frequencies are like when a 3 phase generator runs backwards... everything plugged into it also runs backwards... the sign of the frequency tells you the direction of rotation, and the frequency is the speed.
If you had 2 antennas 1/4 wave apart, and used them for I and Q, you would then be able to tell which direction the signal was coming from along the I/Q antenna axis. With 3 or more antenna, you can determine compass direction of the source.
The same is true for 2 microphones, at any frequency where the microphones are 1/4+N wavelengths apart.