Supply-Dependent Gain Compression

As the instantaneous drain voltage (V_DD) decreases during envelope tracking, the power amplifier's transistor enters a region of reduced gain, causing supply-dependent gain compression. Unlike fixed-supply AM-AM distortion, this compression point shifts dynamically with the envelope signal. The gain variation is not purely a function of input power but a joint function of instantaneous input power and instantaneous supply voltage, requiring a two-dimensional characterization surface rather than a simple one-dimensional compression curve.

The gain compression behavior is fully characterized by iso-gain contours plotted on the PA's characteristic plane (input power vs. supply voltage). These contours reveal the nonlinear mapping between the operating point and the resulting gain. Key observations from iso-gain analysis:

• Constant gain cannot be maintained at low V_DD even with reduced input power
• The compression threshold shifts leftward as supply voltage decreases
• Non-uniform contour spacing indicates nonlinear gain sensitivity to supply variation
• Contour density reveals regions of high distortion sensitivity where small voltage errors cause large gain errors

Supply-dependent gain compression introduces coupled amplitude and phase distortions that fixed-supply DPD cannot correct. As drain voltage modulates:

• AM-AM distortion: Output amplitude deviates from the expected linear relationship due to gain compression at lower supply voltages
• AM-PM distortion: The transistor's input and output capacitances are voltage-dependent, causing supply-modulated phase shift that varies with both input power and instantaneous V_DD
• Cross-term distortion: The interaction between RF envelope dynamics and supply modulation creates intermodulation products not present in static operation This coupling necessitates dual-input behavioral models that treat RF input and supply voltage as independent variables.

The most severe gain compression occurs when the instantaneous drain voltage approaches the transistor's knee voltage (V_knee). In this region:

• The transistor transitions from saturation to linear (triode) operation
• Gain drops precipitously, often by several dB per volt of supply reduction
• The output waveform experiences hard clipping rather than soft compression
• Harmonic generation increases dramatically, worsening spectral regrowth ET-DPD systems must apply aggressive gain expansion in this region to linearize the output, but this demands precise knowledge of the instantaneous compression characteristic and risks overdriving the PA if misaligned.

Supply-dependent gain compression is not memoryless. Thermal and trapping memory effects cause the compression characteristic to depend on recent signal history:

• Self-heating: High-power operation raises junction temperature, altering the gain-compression relationship for subsequent low-voltage periods
• Charge trapping: In GaN and GaAs devices, trapped charge modifies the electric field distribution, creating hysteresis in the iso-gain contours
• Supply modulator memory: The modulator's finite bandwidth and output impedance introduce dynamic supply voltage errors that interact with the PA's compression behavior These effects require Volterra-based or recurrent neural network models with memory terms that capture the time-dependent nature of the compression.

The shaping function that maps instantaneous envelope magnitude to supply voltage directly determines which regions of the gain compression surface are traversed. Design trade-offs include:

• Aggressive shaping (deep voltage tracking): Maximizes efficiency but forces operation deep into the compression region, requiring strong DPD correction that may degrade EVM
• Conservative shaping (voltage headroom): Maintains linearity but sacrifices efficiency gains
• Iso-gain shaping: Follows iso-gain contours to maintain constant gain without DPD, but limits efficiency improvement The optimal shaping function is a co-design parameter jointly optimized with the DPD model to balance efficiency and linearization capability.

Constant gain curves plotted on a power amplifier's characteristic plane, typically mapping instantaneous input power against instantaneous drain voltage. These contours are the primary visualization tool for characterizing supply-dependent gain compression. A perfectly linear ET PA would exhibit perfectly horizontal, evenly spaced contours. In reality, the contours bunch together at low voltages and high powers, revealing the compression region. System architects use these plots to design shaping functions that map the RF envelope to a supply voltage trajectory that avoids the most severe compression zones, balancing efficiency against linearity.

A modeling framework that treats the power amplifier as a system with two independent inputs: the complex baseband RF signal and the dynamic supply voltage. This is essential for capturing supply-dependent gain compression because a single-input model cannot distinguish between gain variations caused by RF amplitude and those caused by supply modulation. Common implementations include:

• Augmented Volterra series with cross-terms between RF and supply kernels
• Polynomial models with supply-dependent coefficients
• Neural networks with a two-dimensional input vector These models form the basis for extracting ET-aware DPD coefficients.

The amplitude-to-amplitude nonlinearity that arises specifically from dynamic supply modulation. As the drain voltage changes to track the envelope, the PA's gain compression point shifts dynamically. A signal peak occurring when the supply voltage is low will experience severe gain compression, while the same peak at a higher supply voltage may remain linear. This creates a supply-dependent gain function that cannot be corrected by a conventional static AM/AM predistorter. The DPD must implement a multi-dimensional correction that accounts for both the instantaneous input power and the instantaneous supply voltage.

A memoryless predistortion structure indexed by two variables: instantaneous input power (or magnitude) and instantaneous supply voltage. Each table entry stores a complex gain correction value. This structure directly addresses supply-dependent gain compression by applying different predistortion coefficients depending on the operating point on the iso-gain contours. Key implementation considerations:

• Table dimensions: Power bins × Voltage bins
• Interpolation: Bilinear interpolation between adjacent entries to avoid quantization artifacts
• Adaptation: Both dimensions must be updated during online training to track thermal drift and aging

The precise time-synchronization of the RF signal path and the envelope tracking supply voltage path at the transistor drain. A timing mismatch as small as a few nanoseconds causes the supply voltage to be applied to the wrong RF sample, resulting in severe supply-dependent distortion that appears as gain compression or expansion on the wrong signal features. This misalignment creates a distinct signature in the AM/AM scatter plot, where the cloud of points broadens asymmetrically. DPD alone cannot fully compensate for a gross delay mismatch; the alignment must be corrected in the analog or digital domain before linearization.

An extension of the classical Volterra series that incorporates dynamic supply voltage terms to model the interaction between RF input history and supply voltage history. The model includes cross-kernels that capture how past supply voltages affect current gain (supply memory effects) and how RF and supply histories jointly produce nonlinear distortion. A simplified representation:

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y(n) = Σ h_rf(k) x(n-k) + Σ h_supply(l) v(n-l) + Σ Σ h_cross(k,l) x(n-k) v(n-l) + ...

This structure is critical for wideband ET systems where both RF and supply memory effects contribute to supply-dependent gain compression.