You've seen Gilbert cells and log/antilog amps separately. Now let's settle a recurring design question: when you need to multiply two analog signals, which architecture wins? Both exploit BJT exponential behavior, but they have wildly different bandwidth, accuracy, and quadrant capabilities.

The Log-Antilog Approach: Take log(A), take log(B), sum them, take antilog. Mathematically: e^(ln A + ln B) = A·B. You build it with two log amps feeding a summing junction, then an antilog amp. The catch: logs are only defined for positive numbers. This is a one-quadrant multiplier — inputs must be unipolar (typically positive currents).

The Translinear (Gilbert Cell) Approach: Uses the translinear principle directly — a loop of BJT V_BE drops where ΣV_BE(clockwise) = ΣV_BE(counterclockwise) forces a current product relationship. With proper biasing and differential inputs, you get a four-quadrant multiplier — both inputs can swing positive or negative.

Bandwidth Reality Check:

• Log-antilog: Limited by the log amp's compensation cap (needed for stability across decades). Typical bandwidth: 10 kHz to 100 kHz. The bandwidth also varies with signal level — small signals are slower because the transistor's r_e is larger.
• Translinear/Gilbert: Open-loop current-mode operation. No frequency compensation needed. Bandwidth: 10 MHz to 500 MHz easily. The AD834 hits 500 MHz at ±1% accuracy.

Accuracy Reality Check: Log-antilog wins here. Over 4-5 decades of input range, a good log/antilog multiplier (AD538) achieves 0.25% total error. Gilbert cells typically deliver 0.5-2% because matching the eight transistors in the core is hard, and offset voltages directly create feedthrough errors.

Real-World Example — RF Power Measurement: You want to compute V² for true power on a 100 MHz signal. A log-antilog squarer dies above 100 kHz, so it's useless. An AD835 (Gilbert cell) handles it with ease — square the input, lowpass filter, done. Conversely, for computing log-ratios in a chemical pH sensor (slow signals, 6-decade dynamic range), the log-antilog approach is vastly more accurate.

Rule of Thumb: If your signal bandwidth exceeds 100 kHz OR you need bipolar (four-quadrant) operation, pick a Gilbert cell. If you need more than 3 decades of dynamic range with sub-percent accuracy and signals are slow, pick log-antilog. For pure squaring of AC signals (RMS detection, modulation), Gilbert wins almost always.

Quick calculation: For a Gilbert multiplier with V_offset = 1 mV at an input rated for ±1 V full scale, feedthrough error = 1 mV / 1 V = 0.1%. Trim the offset and you'll hit 0.02% — but only at DC; AC feedthrough is set by transistor mismatch, not offset.