“Why” and “How” of I and Q

Why “I” and “Q”?

“I” and “Q” processing is a prominent concept in discussions of Digital Signal Processing (DSP).  Why?  “I” and “Q”, as such don’t exist in nature.  “I” and “Q” are mathematical creations for DSP.  I’ll explain why and offer you a chance to create “I” and “Q” using GNU Radio in a modeling mode.

Digital Signal Processing, the heart of SDR, is entirely mathematically determined.  A key mathematical function of DSP is Euhler’s Complex Exponential.  The mathematics of Euhler’s Complex Exponential make it possible to analyze and understand “sinusoids”, that is all signals with all modulations.  Terms and Abbreviations   Conversely, these mathematics can be used to mathematically filter, mix, amplify, or what have you, any sinusoid.  In other words, Euhlers Complex Exponential is used to perform any and all signal processing functions for DSP.  An essential construct for this math function is that it process two signals, reliably 90 degrees out of phase relative to one another.

The stable 90 degree phase relationship of the two signals allows the analysis of signals and modulations.  Hence the “I” and “Q”.  In the end, “I” samples and “Q” samples designate two digital streams: one signal sampled “in phase” and the other signal sampled in “quadrature phase” or 90 degrees out of phase relative to “I”.  The two sampled signals are at the same frequency, but not sampled with the same phase.  Hence “I” and “Q”.  No “I”, no “Q”, no DSP.  For state of the art SDR, “I” and “Q” signals are synthesized immediately following signal digitization and become the digital input for DSP.

How to synthesize I and Q?

Synthesizing digital “I” and “Q” data for DSP is typically accomplished using a pair of Multipliers configured as a Quadrature Multiplier.  One Multiplier multiplies the sample stream by the “I” component of the down conversion oscillator.  The other Multiplier multiplies the “Q” component of the down conversion oscillator.  This process is simpler than you might imagine.  The following GNU Radio flow graph illustrates this process: Quadrature Multiplier demo

NOTE: The “I” and “Q” concepts are shared in the analog and digital realms.  The somewhat confusing relm specific descriptive terminology deserves some comment.  The preceding “I” and “Q” discussion uses terminology specific to the digital realm, the processing of digital data streams from sampled signals, in other words DSP.   In the case of analog “I” and “Q” based signal processing, the terminology differs but the concept is the same.  The analog system input is a “signal”, not a “data stream”.  Analog “I” and “Q” signals are created in a Quadrature Mixer. The digital system input is a “data stream” not a “signal”.   Digital “I” and “Q” data are created in a Quadrature Multiplier.  A further example of the complexity of language verses the realities of implementation is first generation SDR.  These early SDR implementations, e.g. SDR-1000, Softrock, were implemented using Quadrature Mixers to produce analog “I” and “Q” signal outputs prior to digitization and subsequent digital signal processing.  Second generation SDR, so called “advanced SDR”, employ Quadrature Multipliers to digitally generated “I” and “Q” data streams following digitization.

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