# What is IQ in the context of SDRs?

What is IQ, in the context of software-defined radios (SDRs)?

• Comments are not for extended discussion; this conversation has been moved to chat. – Mike Waters Apr 18 '20 at 20:57

"IQ" refers to the practice of having two mixers with their oscillator phase 90 degrees apart, and then the digitization and processing of those two streams of data.

One practical problem is in a superheterodyne receiver, the LO must be selected and the filters designed for the best image response.

The problem can be understood intuitively like this: say there is a spinning disk, with a mark on its diameter. The disk is spinning so quickly it's a blur, but you can use a strobeoscope to observe it. You know approximately how fast the disk is spinning, and you set the strobeoscope to this frequency. It will then appear the disk is spinning slowly, with the speed corresponding to the difference in speed between the strobeoscope and the disk. Depending on the direction of spin, you can tell if the disk is spinning faster or slower than the strobescope.

If you separate the horizontal and vertical positions of the mark into functions, you will find they are the trigonometric functions $$\sin()$$ and $$\cos()$$, which are the same function but 90 degrees apart:

The problem is a single RF mixer operates like a stroboscope that only gives you the position of the mark in one dimension. With just one dimension you can tell the difference in speed, but you can't tell if that difference is positive or negative.

The mathematics of how this works is elegantly expressed in Euler's formula:

$$e^{ix} = \cos x + i \sin x$$

It turns out this is really useful. The reason a frequency mixer has sum and difference frequency products is because

$$\cos(x) = \cos(-x)$$

But if we allow the calculations to have complex values we can write signals not like this:

$$s(t) = \cos(\omega t)$$

$$s(t) = e^{i \omega t}$$

and because

$$e^{ix} \ne e^{-ix}$$

we no longer have that pesky issue of dealing with images at every mixing stage.

In analog hardware this is implemented by having two mixers. They each get the same RF signal, but their local oscillators are 90 degrees out of phase. This means one mixer (call it $$I$$) produces the $$\cos$$ component in Euler's formula, and the other mixer ($$Q$$) produces the $$\sin$$ component. After these signals have been digitized we can treat each pair of samples and the real and imaginary part of a complex number, and then go on processing them as if it's a single, complex-valued signal.

It turns out complex numbers show up in all kinds of DSP math. For example, the discrete Fourier transform (DFT) can be thought of as calculating the correlation of the input with a $$\sin$$ and a $$\cos$$ function at each bin. That means with Euler's formula we can consider the DFT to operate on complex numbers.

Also, many demodulation algorithms will involve shifting things up and down in frequency, and in the digital domain, just like the analog domain, if we do this only with real numbers then we have to be concerned with rejecting image frequencies at every step.

These are just a few examples. The point is generally IQ processing simplifies both the analog and digital implementations of an SDR.

• Great answer. The part about complex numbers and real numbers in the last paragraph kinda comes out of nowhere -- might help to have a sentence to introduce that, something like You can see that we can use complex numbers to model the two components, with real numbers in one dimension and imaginary numbers in the other. – Caleb Apr 18 '20 at 16:54
• @Caleb thanks for the feedback, tried to make it a little better. – Phil Frost - W8II Apr 19 '20 at 21:09

To just answer the literal question:

I stands for Inphase and Q stands for quadrature.

These are the two baseband signals you get when you mix the RF signal with a cosine of the carrier frequency, and with a 90° shifted version of that cosine, respectively (and properly low-pass filter afterwards).

We call the first the inphase component, because it is literally what you get in phase with the cosine.

Mathematically, mixing-and-filtering is an inner product on the vector space of $$L^2$$ signals (it's effectively the integral of the point-wise multiplication of two functions – the RF signal and the local oscillator). That is: it's a projection, very much in the geometric sense! That explains the name of the quadrature component, as it's orthogonal to the inphase component.

Being orthogonal also means, just like in geometry: no matter which length you have orthogonal to the other vector, the projection on that vector is going to be 0.

That is very important – it means that I and Q, respectively, are two signals that are, together, equivalent to the RF signal in what they contain in signal "content", but they are independent; no matter what you do to the I part, it doesn't change the Q part.

That gives us a way of understanding any RF signal around any carrier frequency by describing the equivalent baseband signal in terms of I and Q signals, which individually have half the bandwidth of the RF signal.

That is the strength of it – no matter to which carrier frequency your transmitter mixes its signal, the equivalent IQ baseband doesn't change (only the LO frequency).

And the same applies to all the nice linear channel models we have: You could describe what happens to your signal between your transmitter and your receiver by describing what happens at the carrier frequency – or you could do the same directly in baseband, and ignore the fact that you have a mixer to go up from baseband to RF, then some amplifier, an antenna, the air, another antenna, an LNA, a mixer in between and just describe all this as a channel on the baseband, mathematically (and SDR, being software, is very good at math).

Now, to do things mathematically with the baseband, it has proven elegant to associate complex numbers with the I and Q components: The complex baseband signal at any time $$t$$ is just $$I(t) + jQ(t)$$, with $$j=\sqrt{-1}$$, the imaginary unit.

A Fourier series can represent a real-world signal as a weighted sum of harmonically-related sine and cosine waves. Sine and cosine waves are out of phase by one-quarter of a cycle and so are said to be in quadrature to each other. Thus, the weighted sums are known as the In-phase (I) and Quadrature (Q) components.

For example, single sideband modulation (SSB) comprises a Q component weighted by the Hilbert transform of the weight of the I component. Digital signal processors (DSP) "do the math" to calculate the weights needed to produced the desired results.

If you're looking for just a basic definition, I/Q data is the result of a special form of sampling. Unlike "normal" sampling which takes a single measurement of a signal at each division of the sample rate, I/Q sampling gathers two measurements for each sample: both an "In phase" and a "Quadrature" component — thus the acronym.

The "quadrature" part only makes sense in a context where you are mixing the signals before sampling them. Although the ultimate signal of interest may be chosen selectively via software only, a typical SDR still gets "tuned" to a particular range of frequencies to start with. This is done by multiplying the incoming RF with an "LO" signal, similar to the first stage of an analog heterodyning receiver. The center frequency of the tuned-in range is determined by the frequency of that "LO" signal.

Now you can think of the in-phase component as the "normal" sample, and the quadrature component is equivalent to a second sample taken "90º later" in terms of the tuning frequency (i.e. delayed in time by 1/4 of the LO's period). In practice, both the I and Q "ADC inputs" would be provided simultaneously as separate signals from the mixer circuit, and the two components are taken together as a single "vector" sample at each point in time.

Capturing a two-component I/Q sample is more useful than just doubling the acquisition rate of "normal" [scalar] sampling. The vector I/Q samples are better for DSP processing because they avoid some ambiguity issues that come about in the mixer/heterodyning process. See Phil's comments below in response to an earlier version of this answer.

I found Whiteboard Web's I/Q Data for Dummies to be a great intro to the concept personally, as it works through questions like "why isn't normal sampling good enough?" at least from a processing perspective. (Especially now that other posters here have provided greater clarity on what it is trying to say… ;-)

• It does provide more information: with IQ sampling it's possible to know if a frequency component at the mixer's input is above or below the LO frequency. Not so by simply doubling the sampling rate: you end up with the same bandwidth, but the frequency ambiguity remains. – Phil Frost - W8II Apr 19 '20 at 4:21
• I see where you are hung up -- IQ sampling isn't just sampling sometimes and then sampling in between those points. It's not the phase of when you do the sampling that's significant, it's the phase of the local oscillator. Just taking more samples at a different time does not have the same effect as changing the LO phase. – Phil Frost - W8II Apr 20 '20 at 18:49
• @PhilFrost-W8II Hmm, trying to understand. (Maybe this needs a new Q&A but not even sure the question yet….) It sounds like I am wrong about what the 90º-off-phase is relative to — it's not the phase of the sampling rate, but the LO rate that matters? ~~How does this work with the "direct sampling mod" trick folks use with SDRs where there is no mixer?~~ Ah, I see that at least with the RTL-SDR when direct sampling you have to choose either I or Q which I presume then gets treated as a "normal" sample to use my own terminology above. – natevw - AF7TB Apr 20 '20 at 19:46
• It still seems off, but I'm trying to put my finger on exactly what. I think it's that you say "90º later", implying you could achieve this 90 degree phase shift by waiting in time. It's certainly true in a lot of engineering we consider phase shifts and delay as the same thing, but that doesn't work here. A delay in time, like from adding a length of transmission line, can provide a 90 degree phase shift for only one frequency. But in IQ sampling, the phase shift is 90 degrees for all frequencies, because it's achieved not by adding a delay, but by changing the LO phase. – Phil Frost - W8II Apr 20 '20 at 23:07
• If you have GNU Radio companion, try making a demo for yourself: put a complex signal source with a frequency you can vary into a complex time sink, and look at how the relative phase between the real and complex parts changes with frequency. You'll see at all frequencies, the two are 90 degrees apart, though which one leads the other flips as you cross over 0 Hz. There's no way to achieve that with a delay. – Phil Frost - W8II Apr 20 '20 at 23:10

IQ is a type of sampled data. Sampled data (a vector of numbers) allows digital processing (a bunch of arithmetic on finite size numbers), instead of using analog circuits (inductors, capacitors, etc.), to produce some signal transform (for instance: filtering or demodulation, etc.).

IQ usually describes pairs of unequally spaced or offset heterodyned samples taken at a sample rate (per pair) frequency related to the frequency bandwidth of interest, often well below the actual RF frequency of the signal of interest. Regular (non-IQ) sampling is taking equally spaced samples at a rate above (often well above) the high end of a (radio or audio) frequency band of interest.

To be considered a good IQ signal, the pair of samples need to be taken in quadrature (1/4, 3/4 time spacing), or be samples of a pair of waveforms that result from heterodyning by a quadrature mixer. This allows the IQ data to be fed to a complex FFT, where the positive and "negative" frequency halves of the FFT result will not just degenerate to complex conjugate mirrors of each other, as would be the FFT result if the FFT was fed, real component only, with equally spaced samples. Thus you get twice a much useful "stuff" out of a given length IQ FFT.

• hm, not really a type of sampled data, I and Q (for Inphase and Quadrature) just describe the two analog baseband signals after mixing down with the LO and it's 90°-shifted version. Sampling usually comes after. – Marcus Müller Apr 17 '20 at 17:05