Understanding IQ Vectors: Signal Representation in Complex Form

In signal processing, the IQ vector is a powerful way to represent a signal using complex numbers, separating it into its In-phase (I) and Quadrature (Q) components. Think of it as a way to describe a signal using both its magnitude and phase.

I and Q Components

• I (In-phase): Represents the real part of the signal.
• Q (Quadrature): Represents the imaginary part of the signal.

After data mapping (modulation schemes like BPSK, QPSK, and QAM), binary data is transformed into a complex signal, which we call the IQ vector signal.

Visualizing IQ Data

Let’s consider a single sample of I/Q data. Suppose we have:

• I = 0.70
• Q = 0.42

Here’s a visual representation of this IQ vector:

Image: IQ data vector

From this rectangular plot, we can derive several key characteristics:

• Amplitude of the I (real) signal: 0.7
• Magnitude (A): Using the Pythagorean theorem, $A = \sqrt{0.7^2 + 0.42^2} = \sqrt{0.49 + 0.1764} = \sqrt{0.6664} \approx 0.816$
• Phase Angle (φ): Using trigonometry, $\phi = \tan^{-1}(0.42 / 0.7) \approx 30.96$ degrees. This is the angle the vector makes with the I-axis.

Importantly:

• $I = A \cos(\phi)$
• $Q = A \sin(\phi)$

Different Representations of IQ Data

We can represent the IQ data vector in several forms:

• Rectangular (Cartesian) Form: As a complex number $I + Qi$ or as a vector $[I, Q]$.
• Polar Form: Using amplitude and phase: $A\cos(\phi)$ and $A\sin(\phi)$, where $I = \text{Amplitude} \cdot \cos(\text{angle})$ and $Q = \text{Amplitude} \cdot \sin(\text{angle})$.
• Euler’s Form: $Ae^{i\phi}$

IQ Vector Signal Generation (VSG)

Advanced RF Vector Signal Generators (VSGs) are used to generate IQ data. The output of the physical layer is typically in the form of an IQ data vector. The specifics of the physical layer differ based on IEEE/3GPP standards such as WLAN, WiMAX, and GSM. To generate the complex IQ data vector, we need to specify parameters like:

• Signal Bandwidth
• Modulation Type
• Forward Error Correction (FEC) technique
• MAC Payload

IQ Vector Signal Analysis (VSA)

Vector Signal Analyzers (VSAs) are used to analyze IQ data signals. A popular option is the Agilent 89600 VSA. To analyze the IQ signal, you need to provide the VSA tool with:

• Bandwidth of the IQ Signal
• Sampling Rate at which the IQ data was acquired