Editor’s Note: This is a two-part series where DI authors Damian and Phoenix Bonicatto explore IQ signal representation and negative frequencies to ease the understanding and development of SDRs.

Part 1 explains the commonly used SDR IQ signal representation and negative frequencies without the complexity of math.

Part 2 (to be published) presents a device that allows you to play with and display live SDR signal spectrums with negative frequencies.

Introduction

Software-defined radio (SDR) firmware makes extensive use of I/Q representation of the received and transmitted signal. This representation can simplify and add ease to the manipulation of incoming signal. I/Q data also allows us to work with negative frequencies. My goal here is to explain the I/Q representation and negative frequencies without the complexity usually invoked by obscure terms and non-intuitive mathematics. Also, I will present a device that you can build to allow you to play with and display live spectrums with negative frequencies. So, let’s get started.

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I/Q and quadrature concepts

What is I/Q data? “I” is short for in-phase and “Q” is short for quadrature. It’s the first set of SDR terms that sound mysterious and tends to put people off—let’s just call them I and Q. Simply, if you have a waveform, like you see on an oscilloscope, you can break it into two sinusoidal components—one based on a sine, and another based on a cosine. This is done by using the trig “angle sum identity”. The I and Q are the amplitudes of these components, so our signal is now represented as:

Where: “A” is the original signal amplitude and:

We have just created the in-phase signal, I*cos(ωt), and the quadrature signal, Q*sin(ωt). Just to add to the confusion, when we deal with the in-phase and quadrature signals together it is referred to as “quadrature signaling” …sigh.

[Note: In SDR projects IQ data (or I/Q data) is generally referring to the digital data pairs at each sample interval.]

Now, you may wonder why we would go through the trouble to convert the data to this quadrature form and what this form of the signal is good for. In receivers, for example, just using the incoming signal and mixing it with another frequency and extracting the data has worked since the early days of radio. To answer this, let’s look at a couple of examples.

Example of the benefits of quadrature form

First, when doing simple mixing of an incoming signal you get, as an output, two signals—the sum of the incoming signal and the mix frequency, and the difference of these two frequencies. The following equation demonstrates this by use for the trig product identity:

To continue in your receiver, you typically need to filter one of these out, usually the higher frequency. (The unwanted resultant frequency is often called the image frequency, which is removed by an image filter.) In a digital receiver this filter can take some valuable resources (cycles and memory). Using the I/Q form above, a mix can be created that removes either just the sum or just the difference without filtering.

You can see how this works in Figure 1. First, define the mix signal in an I/Q format:

Mix Signal I part = cos(ω_mt)
Mix Signal Q part = sin(ω_mt)

Figure 1 Quadrature (complex-to-complex) mix returning the lower frequency.

(There is more to this, but this mix architecture is the basic idea of this technique.)

You can see that only the lower frequency is output from the mixer. If you want the higher frequency and to remove the lower frequency, just change where the minus sign is in the final additions as shown in Figure 2.

Figure 2 Quadrature mix returning the higher frequency.

This quadrature, or complex-to-complex, mixing is a very powerful technique in SDR designs.

Next, let’s look at how I/Q data can allow us to play with negative frequencies.

When you perform a classical (non-quadrature) mix, any result that you get cannot go below a frequency of zero. The result will be two new frequencies: the sum of the input frequencies and the absolute value of the difference. This absolute value means the output frequencies cannot go negative. In a quadrature mixer the frequency is not constrained with an absolute value function, and you can get negative frequencies.

Let’s think about what this means if you are sweeping one of the inputs. In the classical mixer as the two input frequencies approach each other, the difference frequency will approach 0 Hz and then start to go back up in frequency. In a quadrature mixer the difference frequency will go right through 0 Hz and continue getting more and more negative.

One implication of this is that, in a sampled system you’re working on, bandwidth is the sample rate divided by 2. When using a quadrature representation, you have a working bandwidth that is twice as large. This is especially handy when you have a system where you want to deal with a large range of frequencies at a time. You can move any of the frequencies to baseband; the higher frequencies will stay in their relative position in the positive frequencies; and the lower frequencies will stay in their relative positions in the negative frequencies. You can slide up and down, by mixing, without image filters or corrupting the spectrum with images. Another very powerful technique in SDR designs.

A tool for exploring IQ data

This positive and negative spectrum is very interesting but unfortunately the basic FFT on your oscilloscope probably won’t display them. It typically only displays positive frequencies. Vector network analyzers (VNAs) can display negative frequency but not all labs have one. You can play around in tools like MATLAB, but I usually like something a little closer to actual hardware and more real-time to get a better feel for the concept. A signal generator and a scope always help me. But I already said a scope does not display negative frequency. Well, the tool presented in Part 2 will allow us to play with I/Q data, negative frequencies, and mixing.

Editor’s Note: An Arduino-Nano-based device will be presented in Part 2 that can generate IQ samples based upon user frequency, amplitude, and phase settings. This generated data will then display the spectrum showing both positive and negative frequencies. Stay tuned for more!

Damian Bonicatto is a consulting engineer with decades of experience in embedded hardware, firmware, and system design. He holds over 30 patents.

Phoenix Bonicatto is a freelance writer.

Related Content

• Exploring software-defined radio (without the annoying RF) – Part 1
• Exploring software-defined radio (without the annoying RF)—Part 2
• SDR Basics Part 3: Transmitters
• The virtual reality of 5G – Part 2 (measurements)
• Ultra-wideband I/Q demodulator improves receiver performance

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