# Analog Complex IQ Mixers

Excerpts from “Analog Complex IQ Mixers – Just Say No”, David Rowe, 2018

Complex IQ analog mixers are exactly that: complex. You think you are simply shifting a signal in frequency, but real world mixers also insert unwanted carrier and image signals. This article explains why, and discusses further complications from real-world imperfections.

A common communication task, shown in Figure 1, is to shift a complex signal $e^{j \omega_{m} t}$ by multiplication (mixing) with a complex carrier $e^{j \omega_{c} t}$. The real output analog signal is a sinusoidal signal with frequency $w_{m} + w_{c}$ rads/s.

Figure 1: Ideal Complex Mixer

A complex mixer is often implemented using the design illustrated in Figure 2.

Figure 2: Implementation of Ideal Complex Mixer

The real and imaginary part of the modulating signal is multiplied by the real and imaginary part of the complex carrier signal.

In mathematical form, the real part is:

\begin{align}
\Re\{e^{j \omega_{m} t} e^{j \omega_{c} t}\} & = \Re\{[\cos(\omega_{m}t) + j\sin(\omega_{m}t)][\cos(\omega_{c}t) + j\sin(\omega_{c}t)]\} \\
& = \cos(\omega_{m}t)\cos(\omega_{c}t)-\sin(\omega_{m}t)\sin(\omega_{c}t) \\
& = \frac{1}{2}[\cos((\omega_{c}+\omega_{m})t)+\cos((\omega_{c}-\omega_{m})t)] \\
& + \frac{1}{2}[\cos((\omega_{c}+\omega_{m})t)-\cos((\omega_{c}-\omega_{m})t)] \\
& = \cos((\omega_{c}+\omega_{m})t)
\end{align}

Similarly, the imaginary part is:
\begin{align}
\Im\{e^{j \omega_{m} t} e^{j \omega_{c} t}\} & = \Im\{[\cos(\omega_{m}t) + j\sin(\omega_{m}t)][\cos(\omega_{c}t) + j\sin(\omega_{c}t)]\} \\
& = \cos(\omega_{m}t)\sin(\omega_{c}t)+\sin(\omega_{m}t)\cos(\omega_{c}t) \\
& = \frac{1}{2}[\sin((\omega_{c}+\omega_{m})t)+\sin((\omega_{c}-\omega_{m})t)] \\
& + \frac{1}{2}[\sin((\omega_{c}+\omega_{m})t)-\sin((\omega_{c}-\omega_{m})t)] \\
& = \sin((\omega_{c}+\omega_{m})t)
\end{align}

Note how the $\cos((\omega_{c} – \omega_{m})t)$ and $\sin((\omega_{c} – \omega_{m})t)$ terms cancel out in the real and imaginary parts, respectively. These are the “images”, that is, $\omega_{m}$ below the carrier frequency $\omega_{c}$.

However, real world analog IQ mixers have imperfections. Figure 3 models a real world mixer.

Figure 3: Model of Real World Complex Mixer

The extra components introduce the following various effects:

• There is typically a gain imbalance $a$ between the real and imaginary arms. An $a=0.9$ would represent an error of 0.1 or -20dB gain imbalance.
• Real world analog mixers do not have perfect isolation between the carrier and output ports, so a little carrier will leak through. This is modelled as $d_{i}$ and $d_{q}$, as each mixer will have slightly different carrier leakage.
• Finally, the phase shift between the real and imaginary parts of the carrier will never be exactly 90 degrees. This phase error is modelled by $\alpha$.

These imperfections lead to an unwanted image signal at $cos((\omega_{c}-\omega_{m})t)$, and carrier at $cos(\omega_{c}t)$, both of which are typically suppressed by a few 10’s of dB relative to the desired signal.

People will suggest all sorts of clever schemes to improve balance and suppress the carrier and image, but they generally lead to a good deal of frustration and extra work. Avoid these schemes, and analog complex mixers if you possibly can.

Fortunately sample rates are steadily increasing, making direct sampling of HF, VHF and soon microwave frequencies possible. This means complex IQ mixers can be implemented in DSP, where they belong.

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