Shaping Function

A shaping function is a static, memoryless transfer function that maps the instantaneous envelope magnitude |x(n)| of the baseband signal to a target drain voltage V_dd(n) for the power amplifier. This mapping is typically implemented as a look-up table (LUT) indexed by the signal's instantaneous power or magnitude. The function is designed offline using iso-gain contour analysis of the PA's characteristic curves, ensuring that for every input envelope value, the selected supply voltage maintains the required gain while minimizing DC power consumption. Unlike adaptive algorithms, the shaping function itself does not change during operation—it is a fixed, pre-characterized curve.

The shaping function embodies the fundamental efficiency-linearity trade-off in envelope tracking. An aggressive shaping function that deeply modulates the supply voltage down to the ET efficiency knee maximizes power savings but introduces significant supply-dependent gain compression and ET-induced AM/PM distortion. Conversely, a conservative shaping function that maintains higher minimum voltages preserves linearity at the cost of reduced efficiency. The optimal shaping function is a compromise, often designed with a voltage floor (V_min) below which the supply is not allowed to drop, preventing the PA from entering its highly nonlinear compression region while still capturing the majority of efficiency gains.

Shaping functions are derived from the PA's iso-gain contours—constant-gain curves plotted on the two-dimensional plane of supply voltage (V_dd) versus input power (P_in). These contours are obtained through load-pull measurements or nonlinear vector network analyzer (NVNA) characterization under varying DC bias conditions. The shaping function is constructed by selecting a target gain value and tracing the corresponding iso-gain contour across the input power range. For each input power level, the contour specifies the minimum V_dd that maintains the desired gain, directly yielding the shaping table entries. This process ensures the PA operates at the boundary of compression for maximum efficiency.

The shaping function directly influences the complexity required of the digital predistorter (DPD). An aggressive shaping function that pushes the PA deep into compression generates strong nonlinearities with significant memory effects, requiring a high-order DPD model such as a generalized memory polynomial (GMP) or augmented Volterra series. The shaping function and DPD must be co-designed: the shaping function determines the operating trajectory across the PA's nonlinear surface, and the DPD must be trained on data that captures this specific trajectory. A poorly chosen shaping function can produce nonlinearities that exceed the correction capability of any practical DPD, resulting in residual spectral regrowth.

The shaping function's output—the target supply voltage waveform—must be physically realizable by the supply modulator. The envelope bandwidth of this waveform is typically 3-5× the RF signal bandwidth due to the nonlinear magnitude-to-voltage mapping. If the shaping function generates voltage transitions that exceed the modulator's slew rate capability, the actual supply voltage will lag behind the target, causing envelope-bandwidth mismatch distortion. Shaping functions are therefore often bandwidth-limited through filtering or designed with smooth transitions to ensure the resulting voltage waveform stays within the modulator's tracking bandwidth, trading some efficiency for realizable tracking fidelity.

For advanced ET-DPD systems, the simple 1D shaping function is extended to a 3D look-up table (3D LUT) indexed by both instantaneous input power and the current supply voltage. This structure captures the supply-dependent nonlinear behavior of the PA, where the optimal predistortion correction depends not only on the input envelope but also on the instantaneous operating voltage. The 3D LUT effectively combines the shaping function and the memoryless DPD into a single unified structure, enabling joint optimization of efficiency and linearity. Each table entry stores a complex gain correction value, and interpolation is used between indexed points for smooth operation.

Constant gain curves plotted on a power amplifier's characteristic plane—typically over supply voltage and input power—that serve as the foundational map for designing shaping functions. By selecting a trajectory along these contours, designers can maintain constant linear gain while dynamically modulating the drain voltage. The shaping function is essentially a parameterized path through this contour space, trading off efficiency against linearity at each instantaneous envelope power level.

The nonlinear variation in a power amplifier's gain as a function of its instantaneous drain voltage. As the supply modulator reduces voltage to save power, the PA's gain compresses nonlinearly, introducing AM-AM distortion. The shaping function must account for this compression characteristic: a poorly designed mapping can push the PA deep into compression at low supply voltages, generating severe distortion that even an advanced digital predistorter struggles to correct.

The operating point on a power amplifier's efficiency curve where a small reduction in output power results in a sharp drop in efficiency. The shaping function defines the lower boundary of envelope tracking operation relative to this knee. Below the knee, the efficiency gains from further supply voltage reduction are negligible, and the shaping function typically clamps the supply voltage to a minimum value to avoid unnecessary distortion without meaningful efficiency benefit.

A fundamental limitation where the required bandwidth of the dynamic supply voltage—dictated by the shaping function's output—exceeds the tracking capability of the supply modulator. The shaping function can be deliberately bandwidth-limited or smoothed to reduce high-frequency content, trading off instantaneous efficiency for modulator feasibility. This co-design constraint means the shaping function is never designed in isolation but always with the modulator's slew rate and bandwidth specifications in mind.

Unwanted phase modulation of the output RF signal caused by the dynamic variation of the power amplifier's supply voltage. As the shaping function commands the supply modulator to change the drain voltage, the PA's input capacitance and transistor parasitics shift, introducing a supply-dependent phase shift. The shaping function design must consider this AM/PM conversion characteristic, as aggressive voltage swings can generate phase distortion that complicates the digital predistorter's correction task.

A memoryless predistortion structure indexed by instantaneous input power and supply voltage to apply a complex gain correction. The shaping function provides the supply voltage index, creating a direct coupling between the envelope path and the DPD correction path. This 3D LUT compensates for the static nonlinearities of an envelope tracking PA by storing pre-characterized correction values for every combination of input magnitude and supply voltage defined by the shaping function's mapping.