For more than a century, the purpose of radio has been to move information. We do this by adding information to a radio signal at the transmitter, and detecting that information at the receiver. You call this modulation and demodulation.

Modulation simply impresses information onto a carrier wave by adjusting its amplitude, frequency or phase. Complex signals let you manipulate these changes with algebra using a two dimensional vector. This vector can be described by its Magnitude and Phase. It can also be described by two points in the complex plane, representing its real and imaginary components. We call these points I and Q, or in-phase and quadrature, by convention.

Quadrature simply means that we describe the same signal in two components that are 90º different in phase. The amazing discovery was that by creating a complex quadrature signal, we end up with a pair of numbers (I and Q, or tips of a vector) that tell us everything we need to know about the signal and its information.

To quote Gerry Youngblood back in 2002, “Give Me I and Q and I Can Demodulate Anything”.

Using I/Q emerged as an efficient way to move information to and from baseband. Baseband is where the information really lives. For example, audio is the baseband for your local FM station or your mobile phone. Complex baseband representation is essential for communications, including telephone, radio, cellular, wireless networks, satellites and fiber optics. It’s all about information which is independent of the carrier signal. So, remember that quadrature is all about getting signals to baseband where they become much simpler to work with.

At baseband, I/Q is the information. When you add the baseband to an oscillator you get the radio signal plus information. When you subtract the oscillator from the radio signal you get the baseband information. So, in DSP, all you really need to know is your I/Q pair of numbers and your sampling rate. All the rest is just math. And complex numbers make that math more workable. They also provide the glue between algebra, trigonometry and exponents.

Let’s take a deep breath and nail this I/Q thing once and for all. To do digital signal processing, we need analytic functions that have derivatives. This provides the ability to evaluate the rate of change anywhere. An analytic signal is a positive value sinusoid expressed by a complex number. Quadrature transformation generates analytic signals.

You convert a real signal into an analytic signal by adding an imaginary component. The imaginary component is simply the real component shifted by 90º. In math, you do this by using a Hilbert Transform. In DSP, you use a quadrature mixer. Both of these methods are shown above.

What we call I and Q are simply the magnitudes of the in-phase and quadrature parts. I and Q signals are numbers which represent changes to both the magnitude and phase. Together, they are an analytic pair that perform magic.

What magic? With I and Q, you can use DSP to create any type of modulation, as described in this hands on video. Conversely, I and Q allow you to detect any form of modulation or signal, as explained by by Rick Lyons in his tutorial.

Among the benefits of using complex numbers and SDR receiver quadrature signals are:

• Symmetry between the time and frequency domains, and between analysis and synthesis
• Removal of negative frequencies so you end up with a true bandpass signal
• Simplification of calculations down to vectors rather than trigonometry
• Properly implemented, the elimination of aliases and images, making filtering easier
• Because the negative frequencies are removed, you can sample at Fs rather than the Nyquist rate.

To learn more, check out I/Q Data for Dummies and DSP Related.

## Coming Up Next

This completes our look at two of the fundamental mysteries: complex numbers and quadrature signals. Next, we move on to receiver performance – less math, I promise.

If you want to learn a lot more about digital signal processing, I can recommend two books that opened this stuff up for me around ten years ago.

The first is The Scientist and Engineer’s Guide to Digital Signal Processing by Steven Smith. It’s great for beginners and available free. The second book is Understanding Digital Processing by Richard Lyons. I would describe this book as intermediate level and full of great tips and tricks.

These two books, together with some QEX articles and the signal processing library in Intel’s Integrated Performance Primitives, were all I needed to write my first software defined receiver.

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