# SIGNAL CHAIN BASICS #70: Digital correction of analog quadrature modulator imbalances

Wide bandwidth radio transmitters commonly use analog quadrature modulators (AQMs) to convert complex (I + j*Q) baseband signals to radio frequencies (RF). An AQM consists of a local oscillator (LO) input, a phase splitter to generate two LO’s 90 degrees out of phase, two mixers that each mix a baseband signal to RF, and a summer to combine the two signals (**Figure 1**).

**Figure 1: Analog quadrature modulator system block diagram.**

For an ideal AQM with perfectly matched I and Q paths, a complex tone of frequency ωBB for the baseband signal (**Equations 1 and 2**)

**Click on image to enlarge.**

results in a single tone at the RF output at either ωBB -ω RF or ωBB + ωRF depending on the sign of Baseband Q (**Equation 3**):

However, there is always some imperfection from an ideal device. Three possible errors are:

1. Baseband DC offset

2. Gain mismatch between I and Q branch

3. LO phase error

represented mathematically in **Figure 2**.

**Click on image to enlarge.**

**Figure 2: Mathematical representation of an AQM with offset, gain and phase errors.**

DC offset mixes with the LO to product LO feedthrough, a tone at ωLO. The DC offset from the I and Q branches add in quadrature, so that the LO feedthrough amplitude is **Equation 4**:

LO feedthough can be corrected by adding an opposite offset in the baseband signals. Many dual high-speed DACs or integrated transmitter solutions like TI’s AFE7071 include either digital circuits to generate the baseband offset for correction. A simple method for finding the optimum DC offset values for the I and Q baseband signals is to monitor the LO feedthrough amplitude and iteratively vary first the I DC offset, and then the Q DC offset to find the minimum LO feedthrough (Figure 3). During the pass 1, the Q DC offset is held constant and the I DC offset is swept until the minimum LO feedthrough is found. During the pass 2, the I DC offset value is held at the minimum and the Q DC offset is swept until the minimum LO feedthrough is found. Ideally only one pass for each I and Q are needed, but errors in measuring the LO feedthrough minimum during the first two passes usually means three or four passes are needed.

**Figure 3: Local oscillator feedthough correction process.**

Gain and phase errors result in incomplete cancellation of the unwanted mixing product – the residual magnitude is known as sideband suppression (SBS). Starting with a gain error G in the baseband Q input and a phase error *φ* (in radians) in the LO for the I branch mixer (**Figure 2**), the amplitudes of the upper and lower sidebands are **Equations 5 and 6**:

In this case the lower sideband is dominant and the sideband suppression is the ratio (**Equation 7**):

or in dBc (**Equation 8**):

**Figure 4** shows the sideband suppression expressed in dBc vs. phase and amplitude error.

**Click on image to enlarge.**

**Figure 4: Sideband suppression (dBc) vs. phase and amplitude error.**

In a similar method as described above for LO feedthrough, Sideband suppression (SBS) can be corrected by changing the gain and phase of the baseband signal to cancel the gain and phase error of the AQM. High-speed interpolating digital-to-analog converters (DACs) such as the TI’s DAC34SH84 contain digital circuits to generate the DC offset, gain and phase change in the baseband signal, allowing easy correction of the AQM imperfections.

Although both LO feedthrough and SBS can be corrected perfectly at any one set of conditions, the optimum correction values vary with supply voltage, temperature, RF and baseband frequency, LO power, and so on. Often the calibration is performed only once during manufacturing, with the values then stored and programmed at system startup. Variation in LO feedthrough and SBS with temperature, voltage and LO power after a one-time calibration is usually indicated in the datasheet plots for the AQM (see Figures 33-44 in TI’s TRF3705 datasheet [1]). LO feedthrough and SBS usually can be maintained better than –50 dBc, which is 10-15 dB better than uncalibrated values.

**References**

* Download the TRF3705 datasheet: www.ti.com/trf3705-ca.

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