Waveform-Domain Complementary Signal Sets

• WDCSS are collections of sequences whose summed aperiodic autocorrelations vanish at all nonzero lags, ensuring interference-free performance.
• They employ design principles like symmetry, paraunitary filter banks, and algebraic methods to achieve low sidelobe levels, Doppler resilience, and anti-jamming capabilities.
• WDCSS enable practical applications in MIMO radar, secure communications, and low-PMEPR multicarrier modulation while balancing design trade-offs.

Waveform-Domain Complementary Signal Sets (WDCSS) are collections of sequences or waveforms whose summed aperiodic autocorrelation functions result in strict zero sidelobes at all nonzero lags, and which may additionally be engineered to exhibit resilience to Doppler, frequency-selective fading, and adversarial jamming. WDCSS generalize classical Golay complementary pairs to larger families, enable multi-user, multi-antenna, and multi-polarization radar and communication applications, and permit deep algebraic, filter-bank, and waveform-theoretic design methodologies.

1. Mathematical Characterization and Definitions

Let $\{x^{(m)}(n)\}_{m=0}^{M-1}$ be a set of $M$ sequences, each length $L$, where $x^{(m)}(n) \in \mathbb{C}$. The collection forms a WDCSS if the sum of their aperiodic autocorrelations vanishes at all nonzero lags: $$\sum_{m=0}^{M-1} R_{x^{(m)}}(k) = 0 \quad \forall\, k \neq 0$$ where $$R_{x^{(m)}}(k) = \sum_{n=0}^{L-1-k} x^{(m)}(n)\, [x^{(m)}(n+k)]^*$$. The aggregate total energy at zero lag is constant: $\sum_m R_{x^{(m)}}(0) = C$ (Budisin, 2018).

For matrix notation, define $A \in \mathbb{C}^{M \times N}$ with rows $x^{(m)}$, then the column-orthogonality condition $AA^H = M I_N$ ensures waveform-domain complementarity (Su et al., 2024). For continuous waveforms $\{s_i(t)\}$, the analogous property is that the composite autocorrelation sum $$\sum_{i=0}^{M-1} \int s_i(\tau) s_i^*(\tau + t)d\tau$$ is zero for all $t \neq 0$.

WDCSS generalize to complete complementary codes (CCC), where collections of complementary sets are additionally pairwise orthogonal (all summed cross-correlation between sequences in different sets vanish) (Wang et al., 2020).

2. Core Construction Principles

WDCSS construction leverages several distinct principles:

a) Symmetry/Anti-symmetry and Quasi-Orthogonality:

By synthesizing waveform “chips” $f(t)$, $g(t)$ such that $g(t) = y f(-t)$ ($y = \pm 1$), the summed aperiodic autocorrelations of $f$ and $g$ cancel exactly off the mainlobe, provided the pair also achieves quasi-orthogonality in convolution (i.e., small cross-correlation magnitude within the chip duration), yielding a robust zero-sidelobe region (Kadlimatti et al., 2017).

b) Paraunitary Filter Banks and Radix-M Generators:

The paraunitary (PU) filter-bank paradigm extends the Golay-pair generator to $M$-branch constructions using sequences of unitary transforms interleaved with branch-wise delays. The Radix-$M$ generator utilizes cascaded unitary matrices and systematic delays to yield complete WDCSS and efficient matched correlators, generalizing to arbitrary symbol alphabets (polyphase, QAM, etc.) (Budisin, 2018).

c) Prouhet–Thue–Morse and Equal Sums-of-Like-Powers:

WDCSS with built-in Doppler-resilient properties are constructed by mapping codewords via digit-sum index sequences (generalized PTM), ensuring that not only are autocorrelation sidelobes zeroed, but the low-order Taylor coefficients of the ambiguity function are also nulled, achieving range–Doppler sidelobe suppression (Nguyen et al., 2014).

d) Algebraic Trace and Permutation Polynomial Methods:

Using permutation polynomials over finite fields, field-trace functions, and Butson–Hadamard (BH) matrices, one generates large WDCSS and CCCs with provable two-level autocorrelation, tight peak-to-mean envelope power ratio (PMEPR) control, and connections to generalized Reed–Muller codes (Wang et al., 2020).

3. Canonical Waveform Families and Engineering Approaches

WDCSS implementation admits diverse waveform bases:

• Discrete Frequency-Coding LFM/PLFM Chips:

Discretized linear or piecewise-linear frequency-modulated (LFM/PLFM) chips (with parameters such as slope/chirp rate) are mapped into complementary sets by modulating Golay sequences; symmetry and quasi-orthogonality yield adjacent cross-correlation sidelobes -30 dB or below, and concatenation enables utilization over a shared frequency band (Kadlimatti et al., 2017).

• Polyphase/Constellation Sets via PU/DFT Matrices:

PU-generated WDCSS support polyphase, QAM, or Eisenstein (hexagonal) alphabets by appropriate unitary matrix selection. These code families are systematically engineered for arbitrary $M$ and length $L$ using digit permutations to influence spectral properties (e.g., PAPR) (Budisin, 2018).

• Phase-Code Sets for Anti-Jamming:

Walsh–Hadamard-derived $\pm 1$ phase-coded WDCSS are deployed for interrupted-sampling repeater jamming (ISRJ) rejection, ensuring that matched filtering produces single-block nonzero response precisely at true target returns; in this regime, at least $D \ge N$ orthogonal waveforms are necessary for length-$N$ codes (Su et al., 2024).

• Golay-Derived Doppler-Resilient Sets:

Null-space-based weighting of transmission and reception for pulse trains of Golay codes yields WDCSS with range-sidelobe free ambiguity over prescribed Doppler intervals—solved as basis selection or coordinate-descent in the null space of Doppler-sampled constraints (Wang et al., 2021).

4. Applications: MIMO, Radar, Antijamming, and Communications

WDCSS find application predominantly in high-integrity active sensing and communication environments:

• MIMO Radar:

In multi-input, multi-output radar, each array element transmits a distinct WDCSS waveform. Mutual zero-sidelobe regions suppress range-sidelobe interference both in the spatial domain and under frequency-selective fading. Quasi-orthogonal code-sets constructed from Golay pairs and their mates further double the effective code pool without escalating cross-correlation (Kadlimatti et al., 2017).

• Antijamming and Electronic Countermeasures:

Phase-coded WDCSS in conjunction with waveform-domain adaptive matched filtering (WD-AMF) enable suppression of ISRJ and related coherent threats. When D≥N, side-lobe levels at jammer-induced “false peaks” are strictly bounded by the jammer duty cycle, and mainlobe SNR is preserved even under high jamming-to-signal ratios (Su et al., 2024).

• Range-Doppler Sidelobe Suppression:

PU and PTM-based designs, as well as null-space constraint methods, permit the construction of codes with range-sidelobe immunity across specified Doppler intervals. These methods extend naturally to fully polarimetric architectures: joint design of transmit and receive weighting achieves mutual orthogonality over time, frequency, and polarization domains (Wang et al., 2021, Nguyen et al., 2014).

• Low-PMEPR Multicarrier Modulation:

Algebraically structured WDCSS produced by BH matrices, permutation polynomials, and trace mappings find application in OFDM frameworks, reducing PMEPR and facilitating multiuser/multichannel multiplexing with natural error-correcting capability (Wang et al., 2020).

5. Key Performance Metrics and Design Trade-offs

WDCSS code sets are evaluated by several rigorous metrics:

• Peak Sidelobe Level (PSL):

$$|\mathrm{max}_{\tau\neq 0}\sum_m R_{x^{(m)}}(\tau)|$$, guided by symmetry/anti-symmetry and quasi-orthogonality bounds. Values of $\lesssim -30$ dB are achievable with DFCW chips (Kadlimatti et al., 2017).

• Integrated Sidelobe Level (ISL):

$\sum_{\tau\neq 0}|\sum_m R_{x^{(m)}}(\tau)|^2$, typically $-38$ dB or lower for optimized constructions (Kadlimatti et al., 2017).

• PMEPR:

Upper bounded by $p^k$ when WDCSS reside in generalized Reed–Muller cosets; crucial for multicarrier transmissions (Wang et al., 2020).

• Doppler Null Order:

PTM construction yields Doppler resilience to order $D$, but at the cost of exponentially lengthened codes: $L = M^{D+1}$ (Nguyen et al., 2014).

• Jamming Suppression Ratio:

Bounded by $-10\log_{10}\epsilon$ where $\epsilon$ is the jammer duty cycle; WDCSS+WD-AMF demonstrates range peak sidelobe levels exceeding $50$ dB even under severe ISRJ (Su et al., 2024).

Trade-offs arise between code length, Doppler tolerance, set size, modulation alphabet, PMEPR, and decoding complexity; permutation selection, unitary matrix choice, and code selection from mates/bases directly influence realized performance.

6. Algorithmic and Implementation Aspects

WDCSS construction and utilization involve both algebraic and algorithmic elements:

• Explicit Algorithmic Steps (example: DFCW-based WDCSS) (Kadlimatti et al., 2017):
  1. Choose base Golay pair and discrete frequency chip parameters.
  2. Generate DFCW chips for desired set size.
  3. Form coded waveforms via chip-convolution and temporal concatenation.
  4. Receiver: two-branch matched filtering, delay, and summation.

• PU-Filter and RM-G Construction (Budisin, 2018):

  • Cascaded unitary and delay blocks generate large, efficiently correlatable sets; matched filtering reduces from $O(L)$ per sample to $O(KM^2)$ operations.
• Null-Space Methods for Doppler Resilience (Wang et al., 2021):
  • Formulate Doppler-Vandermonde constraints, extract null space, optimize SNR over basis, and separately decode sign/weight patterns.
• Phase-Code WDCSS Design via Hadamard Matrices (Su et al., 2024):
  • Construct D≥N orthogonal sequences via partial column selection; yield straightforward hardware implementation due to binary nature.
• Algebraic Construction via Permutation Polynomials (Wang et al., 2020):
  • Leverage trace and PP parametrizations; systematic generation of $p$-ary sets linked with Reed–Muller codes; analytical assessment of Hamming distance and PMEPR.

7. Limitations, Open Problems, and Future Directions

Current WDCSS approaches face several intrinsic and practical limitations:

• Doppler Sensitivity:

Pure phase-coded WDCSS show limited inherent Doppler tolerance; dedicated waveform engineering (e.g., frequency stepping or Z-complementary design) is needed for high-speed/maneuvering scenarios (Su et al., 2024).

• Complexity of Optimal Filter/Code Selection:

Null-space SNR maximization is a nonconvex problem; practical implementations rely on coordinate descent, basis selection, or repeated randomized heuristics (Wang et al., 2021).

• Code Length vs. Resilience Trade-off:

Construction methods such as PTM or algebraic BH-matrix approaches incur exponential growth in set length for increasing null order or set size (Nguyen et al., 2014).

• Implementation Overhead:

For high-Dimensional applications (e.g., $D = 256$), real-time execution of adaptive matched filters may challenge available signal processing hardware (Su et al., 2024).

• Open Research Questions:
  • Analytical selection of “optimal” offsets in the algebraic constructions to further minimize PMEPR or maximize error correction.
  • Extension of WDCSS to zero-correlation-zone (ZCZ) sets and higher-order correlation criteria (Wang et al., 2020).
  • Hardware-efficient, adaptive realization of WD-AMF filtering for large set sizes (Su et al., 2024).
  • Direct experimental validation on wideband or highly dynamic radars under real-world electronic attack scenarios.

Ongoing developments target the synthesis of WDCSS with simultaneous optimality in correlation, PMEPR, Doppler/localization robustness, and implementation efficiency, extending their utility across radar, communication, and secure synchronization domains.