## Introduction

I was reading EETimes this month when I saw an interesting article on analog quadrature modulators (AQMs). I have not looked at these devices in a while and I noticed that some of my early issues with AQMs may not be a problem anymore. My issues had to do with the sensitivity of AQMs to small errors — DC offsets and small phase errors. Today’s versions of these circuits incorporate self-calibration capabilities that eliminate my previous concerns. The article is a good one and worth reading closely.

As I usually do, I wrote up a Mathcad worksheet as I read the article so that I could duplicate their analysis and make sure that I understand the material. This post is based on this worksheet. My design data package (i.e. all data associated with an electronics design) typically contains a number of these worksheets that describe component operation and design details.

This analysis effort is not just an academic one for me — I actually have a home project where one these devices will be very useful.

## Background

When I think of quadrature modulators, I usually think of the Weaver single-sideband (SSB) modulator shown in Figure 1, which is intended to produce the lower SSB signal. All of the discussion to follow will assume that the lower sideband is desired and the upper sideband is not. The argument can easily be flipped for an upper SSB system.

Figure 1: SSB (Lower) Generation Using the Weaver Modulator.

I list here some of the key points about the Weaver modulator:

• The left-hand side of Figure 2 simply converts the input signal into its in-phase and quadrature components.

I will not be spending any further time on this section. The Wikipedia does have a discussion of the concept.

• The right-hand side actually generates the single sideband version of the input signal.

This portion of the circuit is the focus of my discussion here.

• The right-hand side will generate the lower sideband or upper sideband depending on whether you have a sum or difference function at the output, respectively.

You can easily see how the upper sideband can be generated by changing the plus sign on the right-hand side of Equation 1 to a minus.

I always liked the visual symmetry of the Weaver modulator. Its operation is easy to understand because it represents the physical realization of a trigonometric formula, which I show in Equation 1. Its operation is well described using the product-to-sum trigonometric formulas.

 Eq. 1 $\displaystyle \cos \left( {{\omega }_{C}}\cdot t \right)\cdot \cos \left( {{\omega }_{B}}\cdot t \right)+\sin \left( {{\omega }_{C}}\cdot t \right)\cdot \sin \left( {{\omega }_{B}}\cdot t \right)=$ $\displaystyle \frac{1}{2}\cdot \cos \left( \left( {{\omega }_{C}}-{{\omega }_{B}} \right)\cdot t \right)+\cos \left( \left( {{\omega }_{C}}+{{\omega }_{B}} \right)\cdot t \right)$ $\displaystyle +\frac{1}{2}\cdot \cos \left( \left( {{\omega }_{C}}-{{\omega }_{B}} \right)\cdot t \right)-\cos \left( \left( {{\omega }_{C}}+{{\omega }_{B}} \right)\cdot t \right)$ $\displaystyle =\cos \left( \left( {{\omega }_{C}}-{{\omega }_{B}} \right)\cdot t \right)$

where

• ωC is angular frequency of the carrier.
• ωB is the angular frequency of the baseband signal.

The Weaver modulator is often used today in digital realizations of SSB modulators. However, 30+ years ago (i.e. back in my day), EE professors warned their students to beware of analog implementations of this circuit because it is sensitive to the various DC and phase errors that occur in analog systems. If you look at Equation 1, their warning makes sense. Because analog electronics always has small DC offsets, the carrier will not be perfectly cancelled out — an effect usually referred to as carrier leakage. Similarly, small phase errors will mean that the upper sideband will not be completely cancelled out — sort of a sideband leakage. These errors used to take an enormous amount of effort to remove. Things are different today. We can put circuitry on the modulator chips that can introduce compensating DC and phase errors that will nearly eliminate the upper sideband leakage.

To model these errors, I will focus on the modulator portion of Figure 1, which I expand for easier viewing in Figure 2.

## Analysis

### DC Offset

I go through modeling the effect of DC offsets in Figure 3, which shows how we can:

1. Introduce DC offsets into the in-phase (DCi) and quadrature inputs (DCq).
2. Simplify the resulting expression to show that the modulator produces a desired term at ωC – ωB and an undesired term at ωC.
3. Show the magnitude of the undesired term is $\sqrt{DC_{i}^{2}+DC_{q}^{2}}$.

Figure 3: Modeling the Effect of a DC Offset on Quadrature Modulation.

Figure 4 illustrates the process of reducing the effect of the DC offsets by introducing compensating offsets using step-wise refinement:

1. [red line] Introduce DC shifts into the in-phase path and find the offset that minimizes the carrier leakage.
2. [blue line] Applying the in-phase DC shift determined above, introduce DC shifts into the quadrature-phase path and find the offset that minimizes the carrier leakage.
3. [pink line] Applying both the in-phase and quadrature DC shifts determined above, again find the new in-phase offset that minimizes the carrier leakage. This update should be small.
4. [green line] Applying both the updated in-phase and quadrature DC shifts determined above, again find the new quadrature-phase offset that minimizes the carrier leakage.

The example in the article and in my Figure 4 iterated four times. In theory, you will iterate until you get the error down to the level your application requires.

Figure 4: Step by Step Removal of the Carrier Leakage.

In the Figure 4 example, you will see that I add a small random component to the measured carrier leakage. The original article mentioned that noise caused the minimization algorithm to take multiple passes. The addition of a small amount of noise in the leakage model confirms that statement.

### Phase Errors

The effect of phase errors in the system are modeled by the Sideband Suppression ratio (SBS), which I show in Equation 2.

 Eq. 2 $\displaystyle SBS({{G}_{LO}},\phi )=10\cdot \log \left( \frac{1+{{\left( 1+{{G}_{LO}} \right)}^{2}}+2\cdot \left( 1+{{G}_{LO}} \right)\cdot \cos \left( \phi \right)}{1+{{\left( 1+{{G}_{LO}} \right)}^{2}}-2\cdot \left( 1+{{G}_{LO}} \right)\cdot \cos \left( \phi \right)} \right)$

where

• GLO the gain error (i.e. difference from 1) in the local oscillator relative to the input path.
• φ the phase shift of the local oscillator relative to the phase shifts present in the input path.

Figure 5 shows how we can go about deriving Equation 3.

Figure 5: Derivation of Sideband Suppression Ratio Equation.

We can now plot Equation 2 versus phase and gain errors using a contour plot (Figure 6).

Figure 6: Sideband Suppression Contour Plot Versus Phase Error and Amplitude Error.

## Conclusion

This post is a discussion of a magazine article that was about a circuit that I intend to use shortly in a home project. It was interesting to see how a circuit architecture that used to have serious problems has now evolved in a very simple way to minimize these problems.