# IQ modulation and phasors: which first?

Do we use IQ modulation because quadrature signals are a good physical implementation of phasors - or is it the other way around?

To elaborate a little:I'm having a hard time getting a clear explanation of we do things this way in radios. I can see that IQ signals and phasor representations go very well together. I understand that a pair of quadrature signals are needed to represent a rotating complex exponential as real values, and that modelling signals as complex exponentials is useful because when you multiply them, the phases add. It's all very elegant, but it doesn't fully explain why we do it this way. Why not make do with real functions for modelling, and use simpler mixers in the real world? I see 2 competing explanations:

1. We use complex exponentials because they're the simplest way to model a pair of quadrature signals from the real world. This raises the question of why we like quadrature signals in the real world in the first place.

2. We like quadrature signals in the real world primarily because that allows us to model things using complex exponentials.

Unfortunately no matter how hard I search and no matter how much I read, no source will commit to one of these explanations as the main reason for the whole thing. People tend to drop hints that it's a little bit of both, which is very unsatisfying. I'm not saying that it can't be partly both, but would one of these be enough on it's own? And if so, which one?

I know there is a real-world advantage to IQ modulation, in that it doesn't produce duplicate sidebands, so it's more efficient in terms of bandwidth. Is that it? Is that the fundamental motivation behind the whole thing?

• We use quadrature signals because they allow us to carry two independent data streams in the same signal (or or one data stream at double the rate). Feb 28, 2021 at 15:13
• so it's more efficient in terms of bandwidth. Is that it? Yes, a "single" (not quadrature) signal, the frequency can be positive and negative, it is impossible to tell the difference. With a quadrature signal we can tell the difference from the phase between the I and Q signals. That way the usable bandwith doubles. Feb 28, 2021 at 15:19
• Orthogonality is the word ham.stackexchange.com/a/18108/15476 Feb 28, 2021 at 15:59
• @Bimpelrekkie this confuses me a bit: "a "single" (not quadrature) signal, the frequency can be positive and negative, it is impossible to tell the difference". This amounts to saying that a single signal contains information that we can't recover. Then how did it get in there? If I put information into a signal by varying a single voltage as a function of time, then surely I can recover 100% of that, without quadrature signals? Feb 28, 2021 at 16:29
• "single signal contains information that we can't recover. Then how did it get in there" -> no it contains information encoded in phase that you choose not to recover. That is the difference between coherent and incoherent detection. Add a coherent demodulator and you can recover that information. Feb 28, 2021 at 16:39

The fundamental motivation is: "Arbitrary AM modulation and adding two sine waves together is easy, arbitrary phase modulation is hard."

Here's the details. When we send information on a carrier wave with frequency $$\f\$$ (radio, optical, etc.), we send this:

$$S(t) = A(t)\sin(2\pi f t + \phi(t))$$

Where we modulate $$\A(t)\$$ for amplitude modulation and modulate $$\\phi(t)\$$ for FM modulation (or phase modulation).

So right there we already have the phasor notation: $$S(t) = A(t)\sin(2\pi f t+\phi(t)) = \Im \left\{A(t)e^{i(2\pi f t+\phi(t))} \right\}\\ = \Im\left\{\hat{R}(t)e^{i 2\pi f t}\right\}$$ where $$\\Im\$$ is the imaginary part of the expression, and $$\\hat{R}(t)\$$ is the complex phasor in magnitude and angle of the signal, as a function of time.

So why do we use IQ? Because it's hard to make circuits that can control things with phase. But it is easy to make circuits that add two amplitude modulated signals together. So rather than thinking "phase + magnitude" we can think "sine + cosine":

So let's look at that: $$S(t) = A(t)\sin(2\pi f t+\phi(t))\\ = A(t)\cos(\phi(t))\sin(2\pi f t)+A(t)\sin(2\pi f t +\pi/2)\cos(\phi(t))\\ = A(t)\cos(\phi(t))\sin(2\pi f t)+A(t)\cos(\phi(t))\cos(2\pi f t)\\ = I(t)\sin(2\pi f t) + Q(t) \cos(2\pi f t)$$

Where $$\I(t)\$$ and $$\Q(t)\$$ are simply AM modulation on top of a sine and cosine wave (formally, people write I(t) to include the carrier too, but here let's just think of it as the baseband AM modulation of the carrier). That's very easy to make with circuits, and opens up all these arbitrary modulation techniques. So you gave two examples: you can do high efficiency single-sideband communications (which you can do in analog as well) - but you can also do any type of crazy modulation scheme that would be impractical to implement using analog circuits and AM and FM alone. For example, QPSK https://en.wikipedia.org/wiki/Phase-shift_keying#Quadrature_phase-shift_keying_(QPSK) can now be extended easily to any arbitrary number of symbols throughout the IQ plane just as easily as doing 2 - and you can place them at optimal distances away from each other for minimum bit error rate.

AND you still get to do normal AM, FM, and whatever other modulation you want!

EDIT: Demodulation too! You can modulate and demodulate any arbitrary signal if you have I and Q, which is why software defined radios can pretty much do anything on the fly. You just amplitude modulate and add, or you just detect the relatively slow I(t) and Q(t) signals - which is really easy to do in a circuit - then process in software.

Do we use IQ modulation because quadrature signals are a good physical implementation of phasors - or is it the other way around?

Neither.

The world of signals is two-dimensional. Electrical, mechanical, pressure waves, gravity waves on water, I don't know why, they just are. Trying to represent them with one dimension doesn't work (unless the other dimension is always constant, like in FM). Using three is superfluous. Two is just right, that just happens to be the way the world is. Deal with it. We can note that signals transport energy, and energy needs two components, for instance something force-like paired with something displacement-like. You can have any force you like, and if the displacement is zero, then no energy is shifted, and vice versa. That is consistent with needing two dimensions, but I'm not sure it's an explanation that convinces me.

So we have to use some form of 2-D modulation to fully represent signals. IQ is one useful form of this. R-theta is another useful form, if we don't get too close to the origin. For general signals, we can't be assured of missing the origin, so IQ is the usual choice.

Phasors are two dimensional, so can be used to represent signals. We often switch between an R-theta representation and a Cartesian IQ or complex number representation, depending on which models the actual signal more conveniently.

In all cases, the signal is the fundamental thing. How we model it is our choice, and doesn't affect the signal.

• The world of signals is two-dimensional. - what does that mean? Are you saying that physical signals are inherently 2-dimentional? Feb 28, 2021 at 16:37
• @JohnB I've added a rant to my 2-D signals. Have you anything better you could suggest? I'm sure there are good explanations to be had somewhere. Feb 28, 2021 at 17:15
• thanks, I can't suggest anything better, because I don't quite know what you're trying to say. You seem to be saying that representing a signal as a one dimensional quantity, i.e. a real function of time - is inadequate. But in what way? If I encode a signal as real-valued amplitude as a function of time, do I not encode all the information? Feb 28, 2021 at 17:19
• @JohnB You've only put half the information on that it can carry. All the info you put in is there, and you can extract it all, as would happen if you were using FM for instance. Representing in one dimension is just fine, if only that dimension is carrying data. You have to choose the right bases for your axes of course, IQ will return two dimensions for FM, whereas R-theta returns only one. That's it! No matter how many dimensions you use, you can find a basis where only two eigenvalues are significant, the extra ones will be noise like. Feb 28, 2021 at 17:30
• You've only put half the information on that it can carry - I thought that this was only true if I mix my signal with a carrier, because I then I get 2 sidebbands and I'm using twice as much bandwidth as I need for the information I've got. I know that IQ helps with that - but that's a modulation issue and not inherent to signals generally. But you're saying that the original signal already has that problem? I can't get my head around that. Feb 28, 2021 at 18:02

Traditionally, we used I modulation, such as on/off keying, or amplitude modulation.

Then systems of Q modulation (adjusting the phase of the carrier - or its integral, frequency) to carry information, keeping the I term (amplitude) constant.

IQ modulation in various forms developed more recently, combining both I and Q modulation as a way to improve efficiency; both power and spectral efficiency.

SSB as an early form has probably sometimes been described and can be implemented without phasor theory, (e.g. as AM, followed by bandpass filtering one sideband) while phasor notation has also been employed in other circles (such as motor control).

so there's no need to assume one led to the other; it's likely they developed independently and came together.

• Your second paragraph sounds awfully like you're describing R-theta mod, or just theta mod, not IQ mod. Feb 28, 2021 at 17:43
• Then systems of Q modulation (adjusting the phase of the carrier - So you're saying that I represents the amplitude and Q represents the phase? I hadn't thought of it like this. I thought they both encode amplitude, and both encode phase? Feb 28, 2021 at 20:44
• I over-simplified to an extent, and Neil is correct to point out. But you can see I (Q=0) as pure AM and Q=+/-1 (I=0) as pure phase modulation. Once you get off the I and Q axes yes, both encode both.
– user16324
Feb 28, 2021 at 21:01