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Building an analog Differentiate and Cross-Multiply (DACM) demodulator using the AD633 is a classic and robust way to recover audio vibrations from an I/Q microwave radar/interferometer.
Because acoustic vibration velocity is directly proportional to the Doppler frequency φd/dt, an DACM analog demodulator extracts intelligible audio while eliminating the "null point" fading common to single-channel Doppler radars.
Mathematical Foundation
The baseband signals coming from the I/Q mixer are:
It = A * cos(φt) Qt = A * sin(φt) |
The instantaneous phase is: φt = arctan(Qt / It)
Taking the time derivative yields:
d | Q | (It * dQ/dt) - (Qt * dI/dt)
--- | arctan(---) | = ---------------------------
dt | I | I2t + Q2t
|
Phase vs. Velocity: φt is the physical surface displacement. Its derivative φd/dt is the surface velocity.
Because human ears and speakers respond to velocity/pressure waves, you do not need an integrator; φd/dt is already your audio signal.
The Denominator: (I2 + Q2): For a stationary radar pointed at a vibrating window or wall, the return amplitude remains roughly constant over audio timescales (I2 + Q2 = A2 ≈ const.) Omitting the analog divider avoids catastrophic division-by-zero noise spikes when signals fade.
(Optional) For proper amplitude compensation, calculate: D = I2 + Q2
Using three additional analog multipliers:
- Multiply: I * I
- Multiply: Q * Q
- Add their outputs using an op-amp summing amplifier. This creates: D = I2 + Q2
- Then divide the cross-product (numerator) output by D using another analog divider. Most analog multipliers can also be used as dividers.
- This gives the actual phase-rate output.
- Importantly, the denominator must not approach zero; the circuit needs a minimum signal-level threshold.
Stage A: Input Buffering and DC Offset Cancellation
The #1 failure mode in analog DACM is DC offset. Static clutter and mixer local oscillator (LO) leakage create fixed DC biases:
It = Ibias + A * cos(φt) Qt = Qbias + A * sin(φt) |
If the Ibias and Qbias are not zero, the cross-multiplication will distort and null-cancellation will fail.
Use a dual op-amp (e.g., NE5532/TL072/OPA2134/etc.) as a unity-gain buffer.
Place a multi-turn trimpot 10 kΩ between +15V and -15V with a high-value summing resistor to the inverting input) on both I and Q channels.
Tuning: With the radar pointed at your target, adjust the trimpots until the X-Y circular arc on an oscilloscope in X/Y mode until the dot is centered precisely at 0V, 0V.
Stage B: Band-Limited Differentiators
An ideal differentiator (gain ∝ freq.) amplifies high-frequency noise and will oscillate if built with an ideal op-amp. You must build a practical, band-limited differentiator:
Topology: Inverting op-amp configuration
Input Path: Series combination of Cin = 0.047 µF and Rin = 1 kΩ
Feedback Path: Parallel combination of Rf = 10 kΩ and Cf = 1000 pF
Transfer Function
Low-frequency Differentiation: f < 1 / (2 * pi * Rf * Cin) ≈340 Hz
Flat Gain / Roll-Off Above Voice Band: fstop ≈ 1 / (2 * pi * Rin * Cin) ≈3400 Hz
This limits differentiation strictly to the acoustic/speech band (300-3400 Hz) and keeps RF hiss from swamping the multipliers.
Outputs
Inverting Differentiator Produces: -dI/dt and -dQ/dt
Stage C: Cross-Multiplication with Two AD633s
The AD633 has the transfer function:
(X1 - X2)(Y1 - Y2)
W = ------------------ + Z
10V
|
You can compute the entire term (I * (dQ/dt)) - (Q * (dI/dt)) / 10V using two Analog Devices AD633 ICs without needing an external op-amp subtractor, by taking advantage of the differential inputs and the Z summing pin.
Audio Post-Processing
Bandpass / Low-Pass Filter: The output W2 should pass through a 2nd-order active low-pass Sallen-Key filter (e.g., cut-off at 3.5 kHz to 4 kHz) to eliminate multiplication feedthrough and high-frequency mixer noise.
AC Coupling: A simple high-pass coupling capacitor (1 µF film cap) removes any slight multiplier DC offset.
Audio Amplifier: Feed the signal through a 10 kΩ log volume potentiometer into an LM386, TDA2822, or a dedicated headphone amplifier.
Practical Alignment & Tips
Targeting: Microwaves at 10 GHz or 24 GHz reflect well from glass windows, thin paneling, and soda cans. Use a small horn antenna or dielectric lens on your radar module to increase spatial directivity and boost signal-to-noise ratio.
Checking Quadrature Balance: Ensure the I/Q amplifiers have matched gains. If the gain between the two channels differs by more than ~5%, your circle will turn into an ellipse, causing harmonic distortion in the audio output.
If You Experience Large Range Drift: If the target drifts slowly in range, A2 = I2 + Q2 may modulate the volume. Instead of building an analog divider (which risks high-gain oscillation near signal dropouts), feed I and Q through a dual AGC circuit (or dual matched VCAs like the THAT2180/THAT4301) before the demodulator to keep the carrier amplitude constant.
For an I/Q microwave interferometer, a conventional audio AGC cannot be used on the two channels independently. If channel I and channel Q For an I/Q microwave interferometer, a conventional audio AGC cannot be used on the two channels independently. If channel adjust their gains separately, their circular trajectory (I2 + Q2 = R2) deforms into a pulsating ellipse, corrupting the phase angle φt and introducing severe distortion into the DACM circuit.
To preserve the phase geometry, a ganged dual-AGC circuit is required:
Identical Gain: A single, shared control voltage (Vc) drives both channels simultaneously: Iout = G * Iin and Qout = G * Qin
DC-Coupled Architecture: The carrier is a low-frequency vector. The gain control elements must operate linearly down to true DC.
Subsonic Response (τ > 0.5s): The AGC loop must only respond to slow signal fading, range changes, or thermal drift (< 0.5 Hz). It must not react to audio frequencies (100-5000 Hz) allowing acoustic phase vibrations to pass through untouched.