*Low Cost Characterization of RF Transceivers through IQ Data Analysis*is a paper from the 2007 ITC and I read it.

The Authors are: E. Acar and S. Ozev

**Summary**

The authors discuss how previous work suggests measuring either high or low level parameters in order to test an RF system. However, both methods have drawbacks. The high level measurements such as Error Vector Magnitude (EVM) or Bit Error Rate (BER) typically require very long test times. The technique of measuring low-level parameters can require multiple specific test setups according to the measurement being taken, which can be costly and slow. Other efforts to reduce the need for expensive RF instrumentation have been suggested, including moving RF test circuits to test boards and utilizing DFT features.

The authors work improves on past work by taking a simple test setup typically used for Error Vector Magnitude (EVM) or Bit Error Rate (BER) measurements and developing analytical methods to determine lower level performance parameters without having to use the multiple expensive test setups typically required. This has the benefits of faster test times on less expensive equipment.

In order to calculate the lower level parameters the authors construct a mathematical model of a quadrature transmitter. The quadrature transmitter has a digital input signal, an IQ modulator and a power amplifier. This type of system is susceptible to three basic potential problems.

1. Noise

2. Gain and phase mismatches between the I and Q signals

3. Non-linearity problems

I wrote about IQ signals in another blog post.

The authors then go on to explain how they derive the mathematical model for the output stage of a quadrature transmitter, similar to Figure 1 (there should be amplifiers on the two inputs and the output – not shown). The models account for the three potential problems listed above in the form of an expression for noise, phase and gain imbalance and non-linear compression.

Figure 1. Quadrature Transmitter

The authors go on to discuss some models for noise that are modeled as Gaussian distributions. They then talk about how to obtain the constellation point of a symbol under the effect of this type of noise.

I was not familiar with the idea of a “symbol” or what a constellation diagram is. I’m gathering that a symbol is sort-of a piece of digital information. So in a digital modulation/demodulation there are discrete digital values encoded on the carrier, these are called symbols.

A constellation diagram is a plot of the complex plane of the I and Q signals where each symbol is assigned a spot on the diagram. Well, it’s not that each symbol is arbitrarily assigned a location; it’s where the symbol (that piece of encoded data) will fall on the complex (IQ) plane. The constellation diagram is then useful for visualizing the impairments on the system (impairments like: noise, non-linearity, and mismatch). Figure 2 shows a constellation diagram with for ideal symbol point and where the points are skewed as the result of impairments, where the ideal points form the square and the impairments form the parallelogram.

The authors go on to discuss some models for noise that are modeled as Gaussian distributions. They then talk about how to obtain the constellation point of a symbol under the effect of this type of noise.

I was not familiar with the idea of a “symbol” or what a constellation diagram is. I’m gathering that a symbol is sort-of a piece of digital information. So in a digital modulation/demodulation there are discrete digital values encoded on the carrier, these are called symbols.

A constellation diagram is a plot of the complex plane of the I and Q signals where each symbol is assigned a spot on the diagram. Well, it’s not that each symbol is arbitrarily assigned a location; it’s where the symbol (that piece of encoded data) will fall on the complex (IQ) plane. The constellation diagram is then useful for visualizing the impairments on the system (impairments like: noise, non-linearity, and mismatch). Figure 2 shows a constellation diagram with for ideal symbol point and where the points are skewed as the result of impairments, where the ideal points form the square and the impairments form the parallelogram.

Figure 2. Example constellation diagram with ideal and impaired symbol points

Continuing with the constellation diagram, an impairment of a gain imbalance (a higher gain of either the I or Q signal relative to the other one) would cause the constellation diagram square to become a rectangle. This is because, either the I or Q amplitude would push the points out on the I or Q axis. When we incorporate a phase imbalance we get the parallelogram shape.

The authors go into how to analyze the shape of the constellation diagram to determine the gain and phase imbalance (based on the magnitudes and angles)

The authors state the definition of the Third Order Input Intercept Point is, “the input power at which the power of the third order term is equal to the power of the fundamental term.” IIP3 is a way to measure the non-linearity of the system. If I understand, this is a measure of non-linearity because a perfectly linear system would no power at the harmonic frequencies, but when there is non-linearity you start getting the power shifted to the harmonic frequencies and the power of the fundamental frequency goes down. So, from the definition, there is some input power level where the fundamental and third-order harmonic has equal powers. I’m guessing here, but I think this means you could take two systems and compare their IIP3 points to see which will work at higher power before getting to this established point of non-linearity.

Directly testing the BER can be slow because typically systems do not have very many errors and so it takes a long input vector to get an accurate measure of the real BER. The authors propose a statistical method to calculate BER from the constellation diagram. So, basically, they run a smaller vector and look at the variation in the constellation points to estimate the number of bit errors that would occur.

The authors continue on to discuss using constellation analysis on a multi-carrier system and derive the equations for gain/phase imbalance and IIP3.

They use an OFDM modulation for this work.

The authors evaluate their methods by simulation and build a demonstration system from discrete components.

For the simulation the simple test setup is modeled in MATLAB, and then used a 2000 bit input vector. The different impairments tested were for phase and gain imbalance. The authors show that calculated error in calculating the imbalances was less than two percent for all test cases.

The details of the multi-carrier system analysis are discussed.

I learned a lot by reading this paper but it was almost too much to take on. I don’t have much background in RF and modulation; I’m trying to learn that. I didn’t really summarize all of the results as it was too much. I’ll try again and keep building up what I know.

__Gain and Phase Imbalance__Continuing with the constellation diagram, an impairment of a gain imbalance (a higher gain of either the I or Q signal relative to the other one) would cause the constellation diagram square to become a rectangle. This is because, either the I or Q amplitude would push the points out on the I or Q axis. When we incorporate a phase imbalance we get the parallelogram shape.

The authors go into how to analyze the shape of the constellation diagram to determine the gain and phase imbalance (based on the magnitudes and angles)

__Third Order Input Intercept Point (IIP3)__The authors state the definition of the Third Order Input Intercept Point is, “the input power at which the power of the third order term is equal to the power of the fundamental term.” IIP3 is a way to measure the non-linearity of the system. If I understand, this is a measure of non-linearity because a perfectly linear system would no power at the harmonic frequencies, but when there is non-linearity you start getting the power shifted to the harmonic frequencies and the power of the fundamental frequency goes down. So, from the definition, there is some input power level where the fundamental and third-order harmonic has equal powers. I’m guessing here, but I think this means you could take two systems and compare their IIP3 points to see which will work at higher power before getting to this established point of non-linearity.

__Bit Error Rate__Directly testing the BER can be slow because typically systems do not have very many errors and so it takes a long input vector to get an accurate measure of the real BER. The authors propose a statistical method to calculate BER from the constellation diagram. So, basically, they run a smaller vector and look at the variation in the constellation points to estimate the number of bit errors that would occur.

__Multi-Carrier Systems__The authors continue on to discuss using constellation analysis on a multi-carrier system and derive the equations for gain/phase imbalance and IIP3.

They use an OFDM modulation for this work.

__Results__The authors evaluate their methods by simulation and build a demonstration system from discrete components.

For the simulation the simple test setup is modeled in MATLAB, and then used a 2000 bit input vector. The different impairments tested were for phase and gain imbalance. The authors show that calculated error in calculating the imbalances was less than two percent for all test cases.

The details of the multi-carrier system analysis are discussed.

**My Takeaway**I learned a lot by reading this paper but it was almost too much to take on. I don’t have much background in RF and modulation; I’m trying to learn that. I didn’t really summarize all of the results as it was too much. I’ll try again and keep building up what I know.