Transmit Quadrature Error Correction (Tx-QEC)

In a direct-conversion architecture like the QuadRF, Transmit Quadrature Error occurs due to mismatches between the In-phase (I) and Quadrature (Q) signal paths. These mismatches generate unwanted image frequencies where positive baseband frequency mirror into negative frequencies and vice-versa. The primary sources of these errors are:

Gain Imbalance: Differences in the conversion gain of the baseband filters and the data converters (ADCs/DACs) between the I and Q channels.
Phase Imbalance: The local oscillator (LO) in the quadrature mixer failing to maintain an exact 90-degree phase difference between the I and Q paths.
Delay Imbalance: Small variations in propagation delay between the baseband filters and data converters on the I and Q paths.

If all these variables are estimated, they can be pre-compensated in the transmitter using DSP in the FPGA.

Model of the Loopback Channel

In order to estimate the impairments (gain, phase, and delay imbalances between I and Q), we write a mathematical model of the Tx I/Q -> Rx I/Q loopback channel on the QuadRF.

If transmitting a test tone, what is received at $ y(t) $ (or I-channel) is:

$$ \begin{align} y(t) &= \cos(\omega_{RF}(t-D_{RX}) + \theta) \cdot G \cdot \cos(\omega_{RF}(t-D_{RF}-D_{RX})) \cdot \cos(\omega_{bb}(t - D_{TXI} - D_{RF} - D_{RX})) \\ &- \cos(\omega_{RF}(t-D_{RX}) + \theta) \cdot G \cdot \sin(\omega_{RF}(t-D_{RF}-D_{RX})+\theta_{TX}) \cdot g_{TX} \cdot \sin(\omega_{bb}(t - D_{TXQ} - D_{RF} - D_{RX})) \end{align} $$

Applying trigonometric sum/product identities on the left cosines, and removing the RF sum terms filtered by the baseband LPF:

$$ \begin{align} y(t) &= \frac{G}{2}\cos(\theta + \omega_{RF}D_{RF}) \cdot \cos(\omega_{bb}(t - D_{TXI} - D_{RF} - D_{RX})) \\ &+ \frac{G}{2}\sin(\theta + \omega_{RF}D_{RF}-\theta_{TX}) \cdot g_{TX} \cdot \sin(\omega_{bb}(t - D_{TXQ} - D_{RF} - D_{RX})) \end{align} $$

Now, if we correlate $ y(t) $ with the cosine that was transmitted, and use trig identities on the right (eliminating terms that integrate to zero):

$$ \sum y(t)\cdot\cos(\omega_{bb}t) = \frac{G}{2}\cos(\theta + \omega_{RF}D_{RF}) \cdot \frac{\cos(\omega_{bb}(-D_{TXI} - D_{RF} - D_{RX}))}{2} \\ + \frac{G}{2}\sin(\theta + \omega_{RF}D_{RF}-\theta_{TX}) \cdot g_{TX} \cdot \frac{\sin(\omega_{bb}(-D_{TXQ} - D_{RF} - D_{RX}))}{2} $$

Similarly, if we correlate $ y(t) $ with the sine transmitted:

$$ \sum y(t)\cdot\sin(\omega_{bb}t) = \frac{G}{2}\cos(\theta + \omega_{RF}D_{RF}) \cdot \frac{-\sin(\omega_{bb}(-D_{TXI} - D_{RF} - D_{RX}))}{2} \\ + \frac{G}{2}\sin(\theta + \omega_{RF}D_{RF}-\theta_{TX}) \cdot g_{TX} \cdot \frac{\cos(\omega_{bb}(-D_{TXQ} - D_{RF} - D_{RX}))}{2} $$

Repeating for $ z(t) $ (or Q-channel):

$$ \begin{align} z(t) &= -\sin(\omega_{RF}(t-D_{RX}) + \theta) \cdot G \cdot \cos(\omega_{RF}(t-D_{RF}-D_{RX})) \cdot \cos(\omega_{bb}(t - D_{TXI} - D_{RF} - D_{RX})) \\ &- \sin(\omega_{RF}(t-D_{RX}) + \theta) \cdot G \cdot \sin(\omega_{RF}(t-D_{RF}-D_{RX})+\theta_{TX}) \cdot g_{TX} \cdot \sin(\omega_{bb}(t - D_{TXQ} - D_{RF} - D_{RX})) \end{align} $$

$$ z(t) = -\frac{G}{2}\sin(\theta + \omega_{RF}D_{RF}) \cdot \cos(\omega_{bb}(t - D_{TXI} - D_{RF} - D_{RX})) \\ + \frac{G}{2}\cos(\theta + \omega_{RF}D_{RF}-\theta_{TX}) \cdot g_{TX} \cdot \sin(\omega_{bb}(t - D_{TXQ} - D_{RF} - D_{RX})) $$

$$ \sum z(t)\cdot\cos(\omega_{bb}t) = -\frac{G}{2}\sin(\theta + \omega_{RF}D_{RF}) \cdot \frac{\cos(\omega_{bb}(-D_{TXI} - D_{RF} - D_{RX}))}{2} \\ + \frac{G}{2}\cos(\theta + \omega_{RF}D_{RF}-\theta_{TX}) \cdot g_{TX} \cdot \frac{\sin(\omega_{bb}(-D_{TXQ} - D_{RF} - D_{RX}))}{2} $$

$$ \sum z(t)\cdot\sin(\omega_{bb}t) = -\frac{G}{2}\sin(\theta + \omega_{RF}D_{RF}) \cdot \frac{-\sin(\omega_{bb}(-D_{TXI} - D_{RF} - D_{RX}))}{2} \\ + \frac{G}{2}\cos(\theta + \omega_{RF}D_{RF}-\theta_{TX}) \cdot g_{TX} \cdot \frac{\cos(\omega_{bb}(-D_{TXQ} - D_{RF} - D_{RX}))}{2} $$

Positive/Negative Frequencies

If we send a negative frequency tone in addition to the positive tone (both $ \pm\omega_{bb} $) to get $ y^+, z^+ $ and $ y^-, z^- $, we can sum correlation results to cancel the sine terms.

Note: Divide the sum by the number of samples $ N $. Ideally, use an $ N $ that gives an integer number of $ \omega_{bb} $ cycles so the sine/cosine terms are orthogonal.

Four constants are calculated ($ A, B, C, D $):

Let $ T_I = \cos(\omega_{bb}(-D_{TXI} - D_{RF} - D_{RX})) $

Let $ T_Q = \cos(\omega_{bb}(-D_{TXQ} - D_{RF} - D_{RX})) $

$$ B = \sum y^+(t)\cos(\omega_{bb}t) + \sum y^-(t)\cos(\omega_{bb}t) = \frac{G}{2}\cos(\theta + \omega_{RF}D_{RF}) \cdot T_I $$ $$ D = \sum y^+(t)\sin(\omega_{bb}t) + \sum y^-(t)(-\sin(\omega_{bb}t)) = \frac{G}{2}\sin(\theta + \omega_{RF}D_{RF}-\theta_{TX}) \cdot g_{TX} \cdot T_Q $$ $$ A = \sum z^+(t)\cos(\omega_{bb}t) + \sum z^-(t)\cos(\omega_{bb}t) = -\frac{G}{2}\sin(\theta + \omega_{RF}D_{RF}) \cdot T_I $$ $$ C = \sum z^+(t)\sin(\omega_{bb}t) + \sum z^-(t)(-\sin(\omega_{bb}t)) = \frac{G}{2}\cos(\theta + \omega_{RF}D_{RF}-\theta_{TX}) \cdot g_{TX} \cdot T_Q $$

Then, from the difference of correlations, another four constants ($ A', B', C', D' $):

Let $ T_{I'} = -\sin(\omega_{bb}(-D_{TXI} - D_{RF} - D_{RX})) $

Let $ T_{Q'} = \sin(\omega_{bb}(-D_{TXQ} - D_{RF} - D_{RX})) $

$$ B' = \sum y^+(t)\cos(\omega_{bb}t) - \sum y^-(t)\cos(\omega_{bb}t) = \frac{G}{2}\sin(\theta + \omega_{RF}D_{RF}-\theta_{TX}) \cdot g_{TX} \cdot T_{Q'} $$ $$ D' = \sum y^+(t)\sin(\omega_{bb}t) - \sum y^-(t)(-\sin(\omega_{bb}t)) = \frac{G}{2}\cos(\theta + \omega_{RF}D_{RF}) \cdot T_{I'} $$ $$ A' = \sum z^+(t)\cos(\omega_{bb}t) - \sum z^-(t)\cos(\omega_{bb}t) = \frac{G}{2}\cos(\theta + \omega_{RF}D_{RF}-\theta_{TX}) \cdot g_{TX} \cdot T_{Q'} $$ $$ C' = \sum z^+(t)\sin(\omega_{bb}t) - \sum z^-(t)(-\sin(\omega_{bb}t)) = -\frac{G}{2}\sin(\theta + \omega_{RF}D_{RF}) \cdot T_{I'} $$

Angle Sum and Difference Identities

($ G $ is positive, because phase is fully absorbed into other terms)

$$ G^2/4 = A^2 + (D')^2 + B^2 + (C')^2 $$ $$ \text{atan2}(-A-D', B - C') = (\theta + \omega_{RF}D_{RF}) + \omega_{bb}(-D_{TXI} - D_{RF} - D_{RX}) $$ $$ \text{atan2}(-A+D', B + C') = (\theta + \omega_{RF}D_{RF}) - \omega_{bb}(-D_{TXI} - D_{RF} - D_{RX}) $$

$$ (G \cdot g_{TX})^2/4 = (-D)^2 + (-A')^2 + C^2 + (B')^2 $$ $$ \text{atan2}(D+A', C - B') = (\theta + \omega_{RF}D_{RF} - \theta_{TX}) + \omega_{bb}(-D_{TXQ} - D_{RF} - D_{RX}) $$ $$ \text{atan2}(D-A', C + B') = (\theta + \omega_{RF}D_{RF} - \theta_{TX}) - \omega_{bb}(-D_{TXQ} - D_{RF} - D_{RX}) $$

Which can be easily rearranged to extract the following:

Loop gain: $ |G| = 2\sqrt{A^2 + (D')^2 + B^2 + (C')^2} $
Tx Gain QE: $ |g_{TX}| = \frac{2\sqrt{(-D)^2 + (-A')^2 + C^2 + (B')^2}}{G} $
Tx Phase QE: $ \theta_{TX} = \frac{\text{atan2}(-A-D', B - C') + \text{atan2}(-A+D', B + C')}{2} - \frac{\text{atan2}(D+A', C - B') + \text{atan2}(D-A', C + B')}{2} $
RF Mixer Tx-Rx Phase: $ \theta + \omega_{RF}D_{RF} $
Tx BB mismatch delay: $ \omega_{bb} \cdot (D_{TXI} - D_{TXQ}) $
Total loopback delay: $ \omega_{bb} \cdot (D_{RF} + D_{RX} + (D_{TXI} + D_{TXQ})/2) $

✓ Calculations confirmed in MATLAB and physical testing on the QuadRF.

For determining delays involving $ \omega_{bb} $ unambiguously, Tx-Rx correlation should be time-aligned enough such that phase doesn't alias within a period of $ \omega_{bb} $.

Online Tracking Calculation

(After performing near-loopback time alignment)

As long as the total Tx, RF, Rx delay is small enough relative to $ |\omega_{bb}| $ that the cosine coefficient $ T_I $ isn't too small (assuming $ T_I $ and $ T_Q $ aren't too small):

$$ \theta_{TX} = \text{atan2}(-A, B) - \text{atan2}(-D, -C) $$

Or using conjugate multiply:

$$ \begin{align} \theta_{TX} &= \text{angle}\left[ (B-jA) \cdot \text{conj}(C+jD) \right] \\ &= \text{atan2}(-BD-AC, BC - AD) \end{align} $$

Given a small difference angle, using the approximation $ \text{atan2}(Y,X) \approx Y/X $ (which cancels $ g_{TX} $):

$$ \theta_{TX} = \frac{AC+BD}{AD-BC} $$

We can also calculate (assuming $ T_I $ isn't too small):

$$ \theta + \omega_{RF}D_{RF} = \text{atan2}(-A, B) $$

If $ T_I = T_Q $ (pretty much true for low freq tones):

$$ g_{TX} = \sqrt{\frac{C^2+D^2}{A^2+B^2}} $$ $$ G \cdot T_I = 2\sqrt{A^2+B^2} $$

Implementation Notes

Calculate each of these times cosine or sine baseband, integrated.

Sending pos/neg tones: Note that the correlation start trigger should be consistently the same relative to the baseband tone so the delay variables don’t change.
The number of correlation samples should ideally be such that $ \omega_{bb} $ completes an integer number of cycles.
Sending a different baseband frequency: Resolves delay phase ambiguity.
Offsetting RF PLL: Correlating with different baseband frequencies introduces a new phase offset. Note that $ D_{RX} $ and $ D_{RF} $ are unresolvable without an RF PLL offset.