Unfortunately this is somewhat misleading and even wrong at times.
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.
Exactly. I wrote this explanation, but you beat me to it, so I'll just post it here:
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.
> Exactly. I wrote this explanation, but you beat me to it, so I'll just post it here:
> 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.
The reason why audio processing (not sampling) does fine without I/Q data is because our ears are almost completely insensitive to the phase relationships between different frequency components, and because additive frequency shifts are not musically useful. That is what is very hard to deal with without representing signals as I/Q. The audio world just doesn't care. Radio does. This is why most textbook audio equalizers (including those used in professional DAWs) have nonlinear phase by default (minimum-phase) unless you opt for a FIR or FFT based mode. That would never fly in radio.
That's not really the reason. It's important that the concept of phase is not something absolute, it only makes sense in relation to something. Absolute phase could be defined from the beginning of the universe which is nonsensical .
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.
We do not hear phase differences in the relative phase of different audio frequency components. Try it for yourself. Run a song through an allpass filter. It'll sound the same. In fact, speaker systems of all kinds do crazy things to the phase of signals, and nobody cares (what we care about is frequency and transient response).
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).
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.
I just tried this and you're right, complete random phase across the the whole frequency band does not noticeably change things. Funny that we learned that differently. Thanks I learnt something about audio today :).
Audio processing isn't shifted downto baseband or shifted at all, so there is no need for IQ. Its all real. If instead of a direct mix down to baseband, you tell the sdr to mix the minimum frequency in the signal you care about down to just above zero, you can work without i and q. For instance, if you mix an am radio freq down to audio frequency, its all real and you can hear it and represent it as an array of real values.
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.
You're making the mistake of assuming that the only purpose of IQ data is to represent negative frequencies after downconversion. This is not true. The IQ representation is extremely useful for certain kinds of processing, even if you're working in baseband. There are plenty of reasons to take a real baseband signal, run it through a Hilbert transform to get a Q, and process it as IQ data.
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.
I wouldn't have used the term real and baseband together, but I think I understand what you mean. I've been frustrated when people describe a modulation real when they could have deacrbed it more elegantly complex. With modern floating point registers being so large the phase loss is less important, but sometimes the representation just makes more sense symmetrical around zero (DC). Could you explain what you mean by harmonic relationships in sound? Does that imply AM will destroy some quality of the music even if you used a 22khz wide band?
I mean if you add 10Hz to all frequency components in audio, what used to be harmonics (rational multiples of the fundamental frequency) stop being harmonics and it sounds like a dissonant mess. There is no reason to ever frequency-shift music/audio by an offset (i.e. the same thing modulation does in radio, or multiplying by a carrier in the IQ time domain). The only frequency shifting we do in audio is by multiplying the frequencies (that's resampling in the time domain), which is a different story.
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)
Also, thank you for taking the time to educate me. I didn't take any signal processing classes in my EE degree, so I learned everything on the job and have gaps. How does autotune not sound horrible, if they do it right it is indistinguishable, or so I have read.
Autotune works by resampling and doing time stretching (not sure if in the time or frequency domain, depends on the technology; there are many variant ways of doing this) in order to decouple pitch and duration to make adjustments, so it doesn't break harmonic relationships.
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.
I've heard what you are talking about in SSB, I didn't know what it was. I don't quite understand the AM thing, is it that a tuning offset would grab the image of the other sideband to correct stuff?
AM reception basically uses envelope tracking, so you don't really care about the carrier frequency. It's really just "how much power am I receiving". The tuning ends up defining the window of spectrum you average power over.
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.
Yes, our ears hear power not amplitude so phase isn't so important... except maybe for strange mixing products and reflections off of walls? OK the Audiophiles can hear some of that, but generally it's true if you're listening through headphones.
We do hear relative phase relationships at any given frequency between both ears. So if you phase shift one side of a stereo signal and not the other, then yes, that is very audible.
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.
Yes, directional hearing is quite sensitive to phase, but there are often multi-reflections inside the outer ear that allow some people to hear phase discontinuities in mono.
Anyone can hear phase discontinuities because any phase discontinuity is just a burst of high frequency content.
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.
I'd add that the other reason for using I/Q in heterodyned demodulation is for SSB (Single Side Band) to reduce out of band interferers. Otherwise both Fmix +/- Ffilt will get through. Usually, transmitters are also SSB to reduce power (unless they are baseband direct modulation).
This may be true for RF applications, but it isn't universally true. Three-phase power is also modeled this way and both positive and negative frequencies definitely exist.
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).
I think the issues you mention are mostly just a matter of differing perspectives. The author starts from the perspective that the underlying signals are all fundamentally complex, and a real signal (the projection of the 'true' signal onto the real axis, or upconverted complex baseband) is transmitted for physical convenience (i.e. possible to build). It sounds like you start from the perspective that the underlying signals are real, and the "artificial" analytic signal is only introduced as a mathematical convenience ("negative frequencies do not exist!").
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.
>The reason the explanation goes wrong is because they bring in negative frequencies. Negative frequencies do not exist!
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.
Coincidentally I was planning to write a blog post about what I/Q are, inspired by this very terrible explanation. I recently started playing around with SDR and went to look up what I & Q are, found this page (it is one of the very few explanations) and got completely confused. My main objections:
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?
Yeah, it doesn't even explain the image frequency problem in traditional superheterodyne receivers, which I'm sure is pretty fundamental to why I/Q exists. (Loosely speaking, a superheterodyne receiver transforms the input frequency to some fixed intermediate frequency by shifting everything by an adjustable local oscillator frequency. However, there are two different input frequencies that produce the same frequency on the output: one above the local oscillator by the same amount as that intermediate frequency, and one below it by the same amount. Those are basically the positive and negative frequencies the article talks about I/Q data being able to distinguish.)
Sure, if you can do that, why not? I was also searching for a good explanation on I/Q data when I found this. I posted it since I found the page interesting, but also, I did not dive into the details. I'm more interested in the conceptual explanation, and not the hands-on details.
If you could add another (potentially better explanation), that could be beneficial for future learners (not necessarily limited to the HN community).
I think the author does a good job of looking at some of the practical aspects of IQ data but I see the question of "Why IQ" quite differently.
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.
I/Q is really amazing and it's an offshoot of the Euler's formula.
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.
There are some ham radio folks who put together really great videos for building an intuition on all sorts of electrical engineering / signal processing topics that require minimal math.
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:
The thing is this is all really not complicated, but unfortunately the way we (at least I) are being taught complex numbers is terrible. They essentially fall out of thin air without much explanation and sure the concept as a mathematical construct is easy enough to understand, but that doesn't really explain the why and the elegance of what you can do with them.
I really only understood this when I started doing communications (and that was after doing a PhD in physics mind you).
Yeah, you can actually simulate a good chunk of this with just trig, I actually simulated a mixer[1] in python and was able to work out how I/Q got mixed into the baseband, it was a ton of fun.
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).
Good news, you don't really have to do the math to get an intuition of what's going on... which will likely get you through it.
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.
There is math, sure, but you could treat it as a target to work towards. You really just need calculus for DSP work to start with. Remember that it's very easy to use Python or Matlab to check your answers
My only concrete advise is to avoid the Proakis book
>> I/Q Data is a signal representation much more precise
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.
I believe used notation (complex exponential) is confusing for most people. In physical implementation physical waveform will be computed I+Q (writing I+jQ seems to exclude this). I found this very helpful: https://www.youtube.com/watch?v=h_7d-m1ehoY
I think this is one of the best visualizations of I/Q signals I've ever seen. The idea of negative audio frequencies takes some getting used to, but works out very well in some DSP systems.
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.
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.