# Fundamentals of I/Q Signals  When talking about frequency mixers, “I/Q” has nothing to do with an intelligence quotient. As Robert explains, the “I” stands for “in phase” and the “Q” is for “in quadrature.” In this article, he introduces you to the fundamentals of I/Q signal representation and architecture.

In the article, “Do You Speak I/O?,” Lacoste writes:

In 2012, I ended an article about frequency mixers (“Let’s Play with RF Frequency Mixers,” Circuit Cellar 263) by saying that I had only scratched the surface of the subject. In fact, I didn’t cover the important topic of so-called “I/Q” mixers in that article. If you’re wondering what “I/Q” means, let me explain.

When talking about I/Q mixers, “I” stands for “in phase” and “Q” stands for “in quadrature.” You will find these two letters in most papers on signal processing or modern radio frequency (RF) systems architectures. Unfortunately, even some of the most experienced design engineers aren’t particularly familiar with these concepts. Why? Probably because they are usually presented in mathematical terms, such as complex numbers, the Euler theorem, complex Fourier transform, and so on.
This month, my aim is to explain the fundamentals of I/Q signal representation and architecture without math. So, as usual, take a seat, breathe normally, and follow me. I’ll stay away from complex mathematics, except for a few concepts you probably learned in high school.

A frequency mixer is a frequency translation device that you can use either to move up (up-convert) or down (down-convert) any part of the RF spectrum. For the moment, let’s focus on down-converters.

Basically, a mixer is a voltage multiplier. It multiplies two voltages: the RF signal that you want to down-convert and a sine signal coming from a local oscillator (LO). The output is usually nicknamed “intermediate frequency” (IF). The magic lies behind a simple trigonometric formula. The product of two sine signals of frequencies F1 and F2 is the sum of two other sine signals. These two signals have respective frequencies F1 – F2 and F1 + F2. Figure 1 clearly illustrates what’s going on. Refer to my previous article if it’s unclear. Figure 1: A mixer works by multiplying two sine signals of frequencies FRF and FLO. Thanks to the well-known trigonometric formula reminded here, the product is the sum of two other sine signals, respectively, at frequencies FRF + FLO and FRF – FLO.

Of course, in real life, a mixer is a little more complex. But this description is sufficient enough for what I want to explain in this article. You must select the appropriate the LO and IF frequencies in order to have enough frequency separation between the two frequency terms present on the output. This enables you to remove the unwanted one with a frequency filter (high-pass or low-pass depending on the application).

Now let’s move on to modulated signals. Assume that the input RF signal is not a simple sine wave but a modulated signal that occupies a total bandwidth of BW in hertz. The band of interest is then from FRF – BW/2 to FRF + BW/2, where FRF is the central frequency of the RF signal. For example, if it is a IEEE802.11g (i.e., Wi-Fi) signal on channel 6, then you will have FRF = 2.437 GHz (the center frequency of Wi-Fi’s channel 6) and BW = 20 MHz (the modulation width of 802.11g). So, in that case, the occupied bandwidth is 2.437 GHZ ±10 MHz.

Suppose you want to translate the Wi-Fi signal to a low IF in order to digitize it. Assume that you want FIF = 50 MHz. As a mixer is operated in its linear region, it is theoretically transparent to the modulation. Therefore, you could simply mix the RF signal with a local oscillator set at a frequency of FLO = 2,437 + 50 = 2,487 MHz. The mixer’s output will include two copies of the modulated spectrum, one centered around FLO – FRF = 50 MHz and one centered around FLO + FRF = 4,924 MHz (see Figure 2). A low-pass filter will easily remove the second one. Figure 2: A classical mixer, when used as a
down-converter, generates mainly two
copies of the input spectrum, of which
one must be eliminated thanks to a
frequency-selective filter.

The signal’s occupied bandwidth is not modified by the mixer: the intermediate frequency signal will still occupies ±10 MHz around the intermediate frequency. Just a caution: You can see in Figure 2 that the spectrum of the modulated signal can be mirrored. This is due to the fact that the local oscillator frequency was set above the RF frequency. In that case, if the frequency of the RF signal increases, it comes closer to the LO frequency, and therefore the IF frequency is lower (as FIF = FLO – FRF). This shouldn’t be an issue as long as you’re aware of it.

Such an architecture is called a “low IF” design, as the RF signal is moved directly to a quite low frequency in comparison to its bandwidth. Here the occupied bandwidth of the intermediate frequency will be 50 MHz ±10 MHz (i.e., from 40 to 60 MHz).

Now imagine that you have a spectrum analyzer on the IF output and a hand on the frequency-setting knob on the local oscillator. What happens when you gently turn the knob and reduce the LO frequency? Refer to Figure 2 once again. If FLO comes closer to FRF then the generated FIF will be closer to 0 Hz. Theoretically, everything should stay fine until the local oscillator frequency is close to 10 MHz. At that point, the IF modulated signal will occupy the frequency bandwidth of 10 MHz ±10 MHz (i.e., from exactly 0 Hz up to 20 MHz).

What happens if you continue to reduce the LO frequency? Part of the IF spectrum will be lower than 0 Hz. Unfortunately, negative frequencies don’t exist, so this spectrum will be folded back on the positive frequency side and will jeopardize the useful signal. Continue and set FLO exactly at the same frequency as FRF. Then the signal will be theoretically centered at 0 Hz. You will get an occupied bandwidth from DC to BW/2—that is, full of garbage, as the two parts of the spectrum will be folded on each other (see Figure 3). Figure 3: With a standard mixer issues arise when the LO frequency is too close to the RF frequency. The output signal then cross the 0-Hz boundary and its spectrum is folded back from DC to half the bandwidth, jeopardizing its content.

You might think that this is bringing us nowhere, but there is even a name for such an RF architecture with the local oscillator exactly centered at the RF carrier frequency: zero-IF designs. So, is there a trick to avoid this spectrum fallback problem? You bet so. The answer is to use a so-called quadrature demodulator or I/Q mixer. You can analyze the concept in a few different ways. If you prefer math, read Richard Lyons’s excellent essay, “Quadrature Signals: Complex, But Not Complicated.” In what follows, I provide a more illustrative explanation.

The complete article appears in Circuit Cellar 293 (December 2014).