RF Design

Convergence (DPD)

Q: How many samples does a DPD adaptation typically take to converge?

It depends on the algorithm and the conditioning of the input data. An LMS-based adaptation of a memory polynomial with 30 to 60 coefficients usually needs tens of thousands of samples (several full adaptation blocks of 30,000 to 100,000 samples) and 10 to 50 update iterations to reach a stable ACPR floor. An RLS or least-squares indirect-learning estimator converges in 1 to 5 iterations because each iteration solves the full normal equations rather than taking a small gradient step, but each iteration is far more expensive: recursive RLS costs order N-squared per update via the matrix-inversion lemma, while a batch block least-squares solve costs order N-cubed per block, versus order N for LMS. In a typical 5G base station transmitter sampling the feedback path at 491.52 MSa/s, an LMS DPD might lock in 5 to 20 ms while an RLS engine locks in under 1 ms, at the cost of more FPGA multipliers.

Q: Why does my DPD diverge or oscillate instead of converging?

Divergence almost always traces to a step size that is too large for the input power level, or to a feedback path that is not properly aligned and gain-normalized. The LMS stability bound is mu less than 2 divided by the largest eigenvalue of the input autocorrelation matrix; because signal power scales that eigenvalue, a step size that is stable at back-off can blow up at full power. Normalized LMS, which divides the step by the instantaneous input energy, fixes most of this. Other causes are coefficient leakage being too low (numerical drift), time misalignment between the reference and feedback samples (even one sample of skew corrupts the estimate), and a feedback receiver that is itself clipping or compressing, which feeds a corrupted error signal back into the loop.

Q: What ACPR floor should I expect after a DPD converges?

For a well-conditioned GaN Doherty amplifier driven by a 100 MHz 5G NR signal, a converged memory-polynomial DPD typically reaches an adjacent-channel power ratio of -48 to -55 dBc, an improvement of 15 to 25 dB over the un-linearized amplifier. The residual floor is set by effects the model cannot capture: thermal and trapping memory with time constants longer than the polynomial spans, feedback receiver noise and quantization, and PA gain drift between adaptation blocks. Adding generalized memory polynomial cross terms or a Volterra kernel can push the floor another 3 to 6 dB lower, but only if the feedback signal-to-noise ratio supports it, since the estimator cannot drive the error below the noise of the observation path.

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In digital predistortion, the term describes how an adaptive estimator iteratively refines its predistorter coefficients until the residual error between the desired and observed power amplifier output settles to a stable steady-state floor. Each update block compares the captured feedback waveform against the reference, computes a coefficient correction, and applies it; the loop has converged when successive updates stop reducing the error and the adjacent-channel power ratio flattens. The speed and stability of this settling are governed by the chosen adaptation rule (LMS or RLS), the step size relative to the input autocorrelation eigenvalues, and the conditioning of the model basis. A poorly damped loop either crawls toward the floor over many blocks or diverges entirely when the step size violates its stability bound.

Category: RF Design

Typical lock time: 1 to 20 ms

Converged ACPR: -48 to -55 dBc

How an Adaptive Predistorter Settles to a Stable Solution

Digital predistortion inverts the nonlinear transfer characteristic of an RF power amplifier by pre-shaping the baseband signal so the cascade of predistorter and amplifier behaves linearly. Because the amplifier characteristic drifts with temperature, bias, supply voltage, and signal statistics, the predistorter coefficients cannot be fixed at the factory; they are estimated continuously by an adaptive algorithm. Convergence is the transient phase during which that estimator moves from an arbitrary or stale coefficient set toward the values that minimize the modeling error. Practically, an engineer watches the residual ACPR or the normalized mean-square error fall block by block and declares convergence when it stops improving within a defined tolerance, often a fraction of a dB.

The dominant architecture in modern transmitters is indirect learning. A postdistorter is trained on the captured amplifier output to learn the inverse, and its coefficients are then copied into the forward predistorter. This sidesteps the need to differentiate through the amplifier, but it makes convergence sensitive to feedback-path quality: time alignment between the reference and feedback samples must be within a fraction of a sample, gain and phase must be normalized, and the observation receiver must not itself compress. Any of these errors injects a bias that the loop converges toward, producing a stable but wrong solution that leaves the amplifier under-linearized.

The choice of basis also shapes the convergence behavior. A simple memory polynomial with 30 to 60 coefficients is well conditioned and converges quickly, while a generalized memory polynomial or full Volterra series with hundreds of terms can model deeper memory effects but presents an ill-conditioned autocorrelation matrix that slows gradient methods and demands regularization. This is the classic trade between asymptotic accuracy, the final ACPR floor, and how many iterations and how much feedback signal-to-noise are required to get there.

Step Size, Stability, and Settling Time

For least-mean-squares adaptation the single most important knob is the step size mu. The loop is stable only when mu stays below twice the reciprocal of the largest eigenvalue of the input autocorrelation matrix; since that eigenvalue scales with signal power, a step size that is safe at back-off can drive the loop unstable at full output. Normalized LMS removes this dependence by dividing the step by the instantaneous input energy, which is why production DPD almost always uses NLMS or recursive least squares rather than plain LMS.

LMS Versus Recursive Least Squares

LMS takes one cheap gradient step per sample and needs many iterations, with convergence time tied to the eigenvalue spread of the data. RLS solves the weighted normal equations recursively, converging in a handful of iterations independent of eigenvalue spread, at a cost that grows with the square of the coefficient count per update because the matrix-inversion lemma avoids an explicit inverse. The block-least-squares estimator used in many indirect-learning DPDs is the batch counterpart: it inverts the regressor Gram matrix once per block at order N-cubed cost and reaches the floor in 1 to 5 blocks.

LMS coefficient update:
w_n+1 = w_n + μ · x_n^* · e_n

Error signal:
e_n = d_n − w_n^T x_n

LMS stability bound:
0 < μ < 2 / λ_max (λ_max = largest eigenvalue of R_xx)

Normalized LMS (power-independent):
w_n+1 = w_n + (μ / (ε + ‖x_n‖²)) · x_n^* · e_n

LS coefficient solve (indirect learning, per block):
w = (X^HX + δI)⁻¹ X^Hd

Where w = coefficient vector, x_n = regressor (basis) vector, d_n = desired sample, e_n = error, μ = step size, ε = small constant, δ = ridge regularization, X = regressor matrix. Example: NLMS memory polynomial, 45 coefficients, μ ≈ 0.3, converges in ≈ 30 blocks to about -52 dBc ACPR.

Adaptation Method Comparison

Method	Iterations to converge	Cost per update	Eigenvalue-spread sensitivity	Typical lock time	Best use
LMS	Thousands of samples	O(N)	High (slows badly)	10 to 50 ms	Low-gate-count, slow drift
Normalized LMS	Hundreds to thousands	O(N)	Moderate	5 to 20 ms	Power-varying signals
RLS	1 to 5 iterations	O(N²) per update	Low (insensitive)	< 1 ms	Fast lock, high accuracy
Block LS (indirect)	1 to 5 blocks	O(N³) per block	Low	1 to 5 ms	5G base stations
Levenberg-Marquardt	3 to 10 iterations	O(N³) per iter	Low	Offline / lab cal	Stubborn nonlinear models

Common Questions

Frequently Asked Questions

How many samples does a DPD adaptation typically take to converge?

It depends on the algorithm. An LMS adaptation of a 30 to 60 coefficient memory polynomial usually needs tens of thousands of samples and 10 to 50 update iterations to reach a stable ACPR floor. An RLS or block least-squares estimator converges in 1 to 5 iterations because each solves the full normal equations: recursive RLS costs O(N²) per update via the matrix-inversion lemma, while a batch block-LS solve costs O(N³) per block. In a 5G transmitter sampling feedback at 491.52 MSa/s, NLMS might lock in 5 to 20 ms while an RLS engine locks in under 1 ms.

Why does my DPD diverge or oscillate instead of converging?

Almost always a step size that is too large for the input power, or a misaligned feedback path. The LMS bound is μ < 2/λ_max, and since signal power scales λ_max, a step stable at back-off blows up at full power. Normalized LMS fixes most of this. Other causes: coefficient leakage too low (numerical drift), even one sample of reference-to-feedback time skew, or a feedback receiver that is clipping and feeding back a corrupted error.

What ACPR floor should I expect after a DPD converges?

For a well-conditioned GaN Doherty amplifier on a 100 MHz 5G NR signal, a converged memory-polynomial DPD typically reaches -48 to -55 dBc, a 15 to 25 dB improvement over the un-linearized PA. The floor is set by uncaptured effects: long thermal and trapping memory, feedback noise and quantization, and gain drift between blocks. Generalized memory polynomial or Volterra cross-terms can buy another 3 to 6 dB, but only if feedback signal-to-noise supports it.

Related Terms

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