Thermal Convolution

The junction temperature T_j(t) is computed as the convolution of the instantaneous power dissipation P_diss(t) with the device's thermal impulse response h_θ(t). This is expressed as:

• T_j(t) = T_amb + ∫ P_diss(τ) · h_θ(t - τ) dτ
• The impulse response h_θ(t) encapsulates the complete thermal dynamics of the die, attach, package, and heatsink
• This operation captures the thermal lag where temperature changes lag behind power envelope variations
• Unlike simple RC filtering, convolution accounts for the distributed nature of heat flow across multiple material layers

The thermal impulse response h_θ(t) is the fundamental kernel that characterizes a device's electro-thermal behavior. Extraction methods include:

• Pulsed measurement: Applying a short power pulse and recording the transient temperature decay to derive the impulse response via deconvolution
• Foster network fitting: Fitting a sum of exponentials to measured heating curves, where each RC pair contributes a time constant
• Finite Element Analysis (FEA): Simulating the 3D heat equation across the device geometry to compute the theoretical impulse response
• The extracted response typically spans microseconds to seconds, reflecting multiple thermal time constants from junction to ambient

In digital predistortion systems, thermal convolution is implemented as a discrete-time Finite Impulse Response (FIR) filter:

• T_j[n] = Σ_{k=0}^{M} h_θ[k] · P_diss[n-k] where M is the thermal memory length
• The power dissipation waveform P_diss[n] is derived from the squared magnitude of the baseband signal envelope
• Filter length M must span the slowest thermal time constant (often milliseconds), requiring hundreds to thousands of taps at typical sampling rates
• Efficient implementation uses multi-rate processing where thermal convolution runs at a decimated rate, exploiting the low bandwidth of thermal dynamics

Thermal convolution operates alongside faster electrical memory effects, creating a multi-rate nonlinear system:

• Electrical memory (ns to µs): Caused by bias network impedances, trapping, and matching network dispersion
• Thermal memory (µs to ms): Governed by self-heating and thermal impedance
• The combined effect produces thermal-induced spectral asymmetry in the output spectrum that cannot be corrected by memoryless or short-memory predistorters
• Advanced DPD architectures cascade thermal convolution with Volterra or memory polynomial models to jointly compensate both timescales
• The separation of timescales allows decoupled identification of electrical and thermal model parameters

The power dissipation P_diss(t) driving thermal convolution is a nonlinear function of the signal envelope:

• For Class-AB amplifiers, instantaneous efficiency varies with signal amplitude, making P_diss(t) a nonlinear mapping of the envelope
• Low-frequency envelope components (within the thermal bandwidth) cause envelope frequency heating — dynamic temperature swings that track modulation patterns
• Modern wideband signals (e.g., 5G NR with 100 MHz bandwidth) contain envelope frequencies from DC to tens of MHz, fully exciting the thermal response
• Accurate thermal convolution requires a power dissipation model that accounts for both DC bias power and signal-dependent efficiency variations

Integrating thermal convolution into a DPD system creates a thermal-aware predistorter:

• A real-time thermal estimator runs the convolution of the transmitted envelope with the stored thermal impulse response
• The estimated junction temperature T_j[n] is used to index temperature-dependent DPD coefficients or to adjust the predistortion function
• Implementation options include:
  • Temperature-compensated LUTs: Multiple LUT banks selected by temperature state
  • Parameter interpolation: DPD coefficients smoothly interpolated between temperature-calibrated endpoints
  • Augmented basis functions: Thermal state variables added directly to the predistorter basis set
• This approach maintains linearity across wide temperature swings, critical for outdoor base station equipment

The fundamental transfer function that defines the dynamic relationship between power dissipation and junction temperature rise. Represented as a complex frequency-dependent quantity, it serves as the kernel in the thermal convolution integral. Extracted from transient thermal measurements or finite element simulations, thermal impedance captures the distributed RC time constants of the die, attach, and package layers.

The characteristic time governing how quickly a device responds to changes in power dissipation. Multiple time constants—ranging from microseconds for near-junction effects to milliseconds for package-level heating—define the memory span of thermal convolution. These constants are extracted from the thermal impedance curve and directly determine the length of the finite impulse response filter used in discrete-time convolution implementations.

A behavioral representation of thermal impedance using a series of parallel RC stages. Each stage contributes an exponential decay term to the thermal impulse response, making the convolution computationally efficient. While the Foster network lacks direct physical correspondence to material layers, it provides an excellent curve-fit to measured heating transients and is widely used in compact electro-thermal simulators.

A physically derived thermal model where each RC stage corresponds to a distinct material layer (die, solder, baseplate). Capacitors are connected to thermal ground, representing heat storage in each layer. The Cauer network's impulse response, when convolved with the power waveform, yields junction temperature with direct physical interpretability, making it preferred for finite element correlation and package design optimization.

An augmented behavioral model that extends standard memory polynomials with low-frequency thermal terms. By convolving the instantaneous power envelope with a thermal impulse response, it generates a temperature regressor that captures slow gain and phase drift. This structure separates short-term electrical memory from long-term thermal memory, enabling more accurate predistortion with fewer coefficients.

The dynamic temperature fluctuation driven by the low-frequency components of the modulated signal envelope. Since thermal time constants (microseconds to milliseconds) overlap with the envelope bandwidth of modern wideband signals, the junction temperature tracks the instantaneous average power. Thermal convolution captures this effect by filtering the squared envelope magnitude through the device's thermal impedance.