Phase rotation

Phase rotation is an operation that is required when we want to apply time of flight correction to IQ data. Apart from indexing the IQ-signal with a delay $\tau$, we also need to multiply by $e^{-i\omega \tau }$:

$$IQ(t-\tau )\cdot e^{-i\omega \tau }$$

Explanation

A modulated signal with carrier frequency $\omega$ can be written as

$$x(t)=I(t)\cdot \cos (\omega t)-Q(t)\cdot \sin (\omega t)\text{\hspace{1em}(1)}$$

When we do IQ-demodulation we extract the $I(t)$, and $Q(t)$ components from the signal. These are then often combined into a single complex valued number:

$$IQ\{x\}(t)=I(t)+i\cdot Q(t)\text{\hspace{1em}(2)}$$

Note

Note the sign difference between the two!

When we perform time of flight correction we need to apply some delay $\tau$ to the received signal $x(t)$. For regular Radio-Frequency (RF) data, this is straightforward as we can just index the signal at $x(t-\tau )$. If we were to just index $(2)$ like this, we would find a signal that shifts the $I$-, and $Q$-amplitudes, without shifting the phase of the carrier signal:

$$I(t-\tau )\cdot \cos (\omega t)-Q(t-\tau )\cdot \sin (\omega t)\ne x(t-\tau )$$

This is why we need phase rotation to correct for this difference.

Derivation of phase rotation factor

To derive the phase rotation factor we will look for a delayed IQ signal $I_D(t)+i\cdot Q_D(t)$, which is the correct complex IQ-demodulated signal of $x(t-\tau )$.

We start with the definition of the IQ decomposition, where we substititute $t-\tau$ for $t$:

$$x(t-\tau )=I(t-\tau )\cos (\omega (t-\tau ))+Q(t-\tau )\sin (\omega (t-\tau ))$$

We use the sum-product rules from trigonometry (Click to expand)

$$\begin{array}{rl}\sin (\alpha +\beta )& =\sin (\alpha )\cos (\beta )+\cos (\alpha )\sin (\beta )\\ \sin (\alpha -\beta )& =\sin (\alpha )\cos (\beta )-\cos (\alpha )\sin (\beta )\\ \cos (\alpha +\beta )& =\cos (\alpha )\cos (\beta )-\sin (\alpha )\sin (\beta )\\ \cos (\alpha -\beta )& =\cos (\alpha )\cos (\beta )+\sin (\alpha )\sin (\beta )\end{array}$$

We will color the terms with a factor $\cos (\omega t)$ blue and the terms with a factor $\sin (\omega t)$ green.

$$\begin{array}{rl}x(t-\tau )& =I(t-\tau )\cos (\omega \tau )\cos (\omega t)+I(t-\tau )\sin (\omega \tau )\sin (\omega t)\\ & -Q(t-\tau )\cos (\omega \tau )\sin (\omega t)+Q(t-\tau )\sin (\omega \tau )\cos (\omega t)\end{array}$$

Sorting terms by color we find

$$\begin{array}{rl}x(t-\tau )& =\stackrel{I_D}{\overbrace{(I(t-\tau )\cos (\omega \tau )+Q(t-\tau )\sin (\omega \tau ))}}\cdot \cos (\omega t)\\ & +\stackrel{Q_D}{\overbrace{(I(t-\tau )\sin (\omega \tau )-Q(t-\tau )\cos (\omega \tau ))}}\cdot \sin (\omega t)\end{array}$$

When we write this in matrix form it becomes clear that the relation between the naively shifted signal $I(t-\tau )-i\cdot Q(t-\tau )$ and the correctly shifted signal $I_D(t)-i\cdot Q_D(t)$ is a multiplication by a factor $e^{-i\omega \tau }$.

$$\left[\begin{array}{c}{I}_D(t)\\ Q_D(t)\end{array}\right]=\left[\begin{array}{cc}\cos (\omega \tau )& \sin (\omega \tau )\\ -\sin (\omega \tau )& \cos (\omega \tau )\end{array}\right]\left[\begin{array}{c}I(t-\tau )\\ Q(t-\tau )\end{array}\right]$$

Note

When doing time of flight correction for Delay-and-Sum beamforming, we usually compute a time of flight delay ${\tau }_f$ to index the IQ-data. In this indexing operation we effectively apply a negative delay. This means that the phase rotation factor in practice is a positive phasor instead of the negative one we saw before.

$$I(t+{\tau }_f)+i\cdot Q(t+\tau ))\cdot e^{i\omega {\tau }_f}$$