Zadoff-Chu Sequence

Every sample in a Zadoff-Chu sequence has an identical magnitude, resulting in a peak-to-average power ratio (PAPR) of 0 dB. This constant envelope characteristic is critical for DPD training because it excites the power amplifier uniformly across its entire operating range without introducing amplitude variations that could bias the extracted model. The sequence drives the PA at a consistent power level, ensuring the behavioral model captures nonlinearity across the full compression curve rather than just at average power levels.

The cyclic autocorrelation of a Zadoff-Chu sequence is zero for all non-zero lags. Mathematically, the autocorrelation function is a Kronecker delta function — a single impulse at zero lag and zero elsewhere. This ideal autocorrelation property means the sequence is perfectly uncorrelated with shifted versions of itself, enabling precise channel estimation and distortion measurement without self-interference. In DPD applications, this allows the coefficient extraction algorithm to isolate the PA's nonlinear response from the training signal's own structure.

The Discrete Fourier Transform (DFT) of a Zadoff-Chu sequence also exhibits constant magnitude across all frequency bins. This dual-domain flatness — constant amplitude in both time and frequency — ensures uniform excitation of the power amplifier across the entire signal bandwidth. For wideband DPD applications targeting 5G NR signals with 100 MHz or more of instantaneous bandwidth, this property guarantees that all frequency components are equally represented in the training data, preventing frequency-dependent bias in the extracted predistorter coefficients.

Zadoff-Chu sequences generated from different root indices are nearly orthogonal to each other. The absolute value of the cross-correlation between two sequences with different roots equals 1/√N, where N is the sequence length. For long sequences (e.g., N=839 in LTE PRACH), this cross-correlation approaches zero. This property enables multi-user or multi-antenna DPD training scenarios where different transmit paths can be excited simultaneously with distinct Zadoff-Chu sequences without mutual interference during model extraction.

Unlike OFDM signals that exhibit PAPR values of 8-13 dB, Zadoff-Chu sequences maintain exactly 0 dB PAPR. This allows DPD engineers to characterize the power amplifier at its maximum saturated output power without the crest factor reduction processing typically required for operational waveforms. The sequence drives the PA into deep compression uniformly, revealing the full nonlinear transfer characteristic. This is particularly valuable for Doherty amplifier optimization, where accurate modeling of the carrier and peaking amplifier interaction at peak power is essential.

Zadoff-Chu sequences can be generated for any prime length N, with N-1 distinct root indices available. This flexibility allows DPD system designers to match the training sequence length to the memory depth of the power amplifier model. For example, a sequence of length N=1021 provides sufficient samples to extract a generalized memory polynomial model with memory depth M=5 and nonlinearity order K=7. The prime-length constraint ensures the zero autocorrelation property holds exactly, without the approximation errors that occur with truncated or zero-padded sequences.

The defining mathematical family to which Zadoff-Chu sequences belong. A CAZAC sequence maintains a constant envelope in the time domain, yielding a peak-to-average power ratio (PAPR) of 0 dB. This property ensures the training signal does not excite additional nonlinearities in the power amplifier under test. Simultaneously, its perfect periodic autocorrelation—a delta function at zero lag—provides ideal orthogonality for channel estimation and model extraction.

A signal processing technique often paired with Zadoff-Chu sequences in OFDM-based systems. By appending a copy of the sequence's tail to its beginning, the cyclic prefix transforms linear convolution into circular convolution. This preserves the sequence's perfect autocorrelation properties even after passing through a dispersive channel, enabling precise time-domain alignment of the feedback signal with the reference for accurate DPD coefficient estimation.

A parameter u (where u is coprime to the sequence length N) that generates a family of orthogonal Zadoff-Chu sequences from a single base formula. Different root indices produce sequences with zero cross-correlation, allowing multiple transmitters in a MIMO array to transmit training signals simultaneously without mutual interference. This is critical for concurrent multi-antenna DPD extraction in massive MIMO beamforming systems.

The mathematical domain in which Zadoff-Chu sequences are defined and applied. Represented as I/Q samples, the sequence's complex-valued nature allows it to probe both amplitude (AM-AM) and phase (AM-PM) distortion simultaneously. The constant amplitude in the complex plane ensures the training signal exercises the power amplifier across its full 360-degree phase range without amplitude variation, isolating nonlinear phase distortion for precise behavioral modeling.

The primary in-band metric used to validate the effectiveness of a Zadoff-Chu-trained DPD system. EVM measures the constellation deviation between the ideal transmitted sequence and the linearized PA output. Because the Zadoff-Chu sequence has a known, deterministic structure, any deviation in the received constellation directly quantifies residual nonlinear distortion, providing a precise, repeatable figure of merit for linearization performance.

The process of creating a mathematical abstraction of a PA's nonlinear dynamics, for which the Zadoff-Chu sequence serves as an ideal excitation signal. Its flat frequency response and wide bandwidth stimulate the amplifier uniformly across the spectrum, revealing frequency-dependent memory effects. The captured input-output data pairs are used to train Volterra series or neural network models that form the inverse predistorter function.