LUT Granularity

Granularity is fundamentally determined by the step size between adjacent table entries. A smaller step size yields higher resolution, allowing the LUT to capture sharp gain compression transitions. However, this directly increases the total number of entries, consuming more BRAM or distributed memory in an FPGA. The quantization of the input envelope into discrete address bits introduces a fundamental quantization error floor that cannot be overcome by adaptation algorithms alone.

Optimal granularity is rarely uniform. Non-uniform LUTs allocate higher entry density in regions where the power amplifier exhibits rapid nonlinearity changes—typically near the 1 dB compression point and saturation. Sparse spacing is used in the linear region to save memory. This companding approach maximizes correction accuracy per stored coefficient, often implemented via a mu-law or A-law companding function on the indexing magnitude.

The effective granularity can be enhanced beyond the physical table size through interpolation. Linear, quadratic, or cubic interpolation between adjacent entries smooths the predistortion function, reducing spectral regrowth caused by discontinuous gain transitions. This allows a coarser physical table to approximate the performance of a much finer one, trading multiplier logic for memory blocks.

Granularity interacts multiplicatively with memory depth. A multi-dimensional LUT that compensates for memory effects requires entries for each combination of current envelope and delayed taps. A table with 256 amplitude bins and 4 memory taps of depth 4 each creates an address space of 256^5 entries. LUT partitioning and pruning are essential to prevent granularity from causing an unmanageable memory explosion.

The granularity directly maps to the address bus width of the memory block. A 10-bit address provides 1024 entries, defining the maximum resolution. In high-speed FPGA implementations, the critical path through the address calculation logic often limits the achievable granularity. Designers must balance the ADC/DAC resolution of the feedback path against the LUT granularity, as a table finer than the converter's effective number of bits provides no practical benefit.

Finer granularity slows LMS convergence. With more coefficients to adapt, the algorithm requires more iterations to settle, and the risk of misadjustment noise increases. Coarser tables average the error over a wider amplitude range, converging faster but leaving residual distortion. Multi-rate adaptation schemes often adapt a coarse table quickly for initial lock, then progressively refine a finer table.

The distortion introduced by representing a continuous predistortion function with a finite number of discrete amplitude levels. Coarse granularity directly increases quantization noise, as the true correction value is rounded to the nearest stored coefficient. This error manifests as elevated adjacent channel leakage ratio (ACLR) and incomplete cancellation of amplifier nonlinearity. The relationship is governed by the number of bits used per LUT entry and the total number of entries across the input dynamic range.

A mathematical technique for estimating predistortion values between discrete table entries to reduce the effective granularity without increasing memory size. Common methods include:

• Linear interpolation: Simple, low-latency, but introduces slope discontinuities
• Polynomial interpolation: Higher accuracy using quadratic or cubic fits between adjacent entries
• Spline interpolation: Smooths transitions across multiple segments Proper interpolation allows coarser physical granularity while maintaining fine effective resolution, trading computational complexity for memory savings.

A look-up table architecture with variable spacing between entries, allocating higher density in regions where the power amplifier exhibits rapid gain compression. This contrasts with uniform granularity, where entries are equally spaced across the input range. Non-uniform spacing optimizes correction accuracy by concentrating entries where the AM-AM and AM-PM characteristics change most steeply—typically near the amplifier's 1 dB compression point and saturation region—while using coarser spacing in linear operating zones.

The residual nonlinearity resulting from approximating the predistortion function between stored table entries. This error is inversely proportional to LUT granularity: finer spacing reduces the distance between known points, lowering interpolation error. The error magnitude depends on:

• The curvature of the amplifier's inverse transfer function
• The interpolation method used (linear vs. higher-order)
• The local derivative of gain compression Minimizing this error drives the selection of minimum acceptable entry density.

Techniques for reducing the total number of stored coefficients while preserving effective predistortion accuracy. Compression methods interact directly with granularity decisions:

• Vector quantization: Clusters similar correction values to share entries
• Polynomial segmentation: Stores polynomial coefficients instead of raw gain values for each entry
• Huffman coding: Applies lossless compression to coefficient storage These approaches allow designers to implement finer logical granularity within constrained hardware memory budgets.

The technique of dividing a large multi-dimensional look-up table into smaller sub-tables to reduce memory requirements while preserving predistortion accuracy. When memory depth is incorporated for amplifier memory effects, the total address space grows exponentially. Partitioning decomposes this space into independent one-dimensional tables with independent granularity per dimension, allowing fine spacing on the instantaneous envelope axis while using coarser granularity on historical sample dimensions where sensitivity is lower.