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On Diagonalizable Delay-Doppler Channels and Their Diagonalizing Waveforms
Sirui Li, Cheng Du, Yu Zhu
TL;DR
Doubly selective channels couple transmitted symbols, but suitable waveforms can enable one-tap equalization on certain delay-Doppler supports. Under CP-based transmission and orthonormal modulation, this paper characterizes all such supports and derives their corresponding waveforms in closed form.
Problem
The complete family of delay-Doppler supports admitting fixed-waveform diagonalization for low-complexity equalization remains unclear beyond several known examples.
Method
Under CP-based block transmission and orthonormal modulation, the paper explicitly characterizes maximal diagonalizable supports and derives an orthonormal basis for each.
Results
The construction covers rectangular grids, delay-Doppler lines, and sheared grids, with waveforms implementable using parallel FFTs and phase rotations.
Takeaways & Limitations
Zak-OTFS and DAFT-OFDM emerge as special cases of a broader closed-form waveform construction for exact channel diagonalization.
Abstract
from arXiv · showhide
In doubly selective channels, the joint delay and Doppler dispersion generally induces coupling among transmitted symbols, thereby increasing receiver equalization complexity. Nevertheless, by using appropriately designed waveforms, channels with certain delay-Doppler (DD) supports can be diagonalized for one-tap equalization. The whole picture of such DD supports and their corresponding waveforms is still unclear, except for several examples identified in literature. In this paper, under cyclic-prefix (CP)-based block transmission and assuming that the modulation waveforms form an orthonormal basis, we identify all such channel supports by an elementary expression, and derive the corresponding waveforms in closed form.
I. INTRODUCTION · II. SYSTEM MODEL AND DIAGONALIZATION CRITERION · A. Unified Transceiver and DD Channel Model
The paper studies which delay-Doppler supports permit channel diagonalization with fixed orthonormal waveforms under CP-based block transmission. It identifies all maximal supports and derives closed-form waveform bases for one-tap equalization.
- I. INTRODUCTION: Future wireless systems require waveforms robust to delay-Doppler dispersion under high mobility, wide bandwidths, and integrated sensing and communication.Waveform design is presented as fundamental to new wireless systems.
- I. INTRODUCTION: Existing waveforms either exploit channel diversity through DD-domain processing or pursue low-complexity equalization through channel diagonalization.OTFS, ODDM, and Zak-OTFS represent the first class, while DAFT-OFDM represents the second.
- I. INTRODUCTION: The paper adopts the diagonalization-oriented design philosophy because fixed waveforms cannot diagonalize every possible channel across the entire delay-Doppler plane.SVD-based precoding can create parallel subchannels when full channel state information is available, but individual path gains are generally unknown in waveform design.
- II. SYSTEM MODEL AND DIAGONALIZATION CRITERION: The central question is which delay-Doppler supports admit diagonalization and which waveforms correspond to those supports under CP-based transmission with an orthonormal modulation basis.The modulation matrix is square and its columns form an orthonormal basis.
- I. INTRODUCTION: The paper gives an explicit representation of all maximal diagonalizable supports, encompassing rectangular grids, delay-Doppler lines, and sheared grids, while determining their exact number.These supports form a broader family beyond the examples previously identified in the literature.
- I. INTRODUCTION: For every identified support, the paper derives a closed-form orthonormal waveform basis that diagonalizes all channels sharing that support.This construction targets simple channel equalization while preserving the orthonormal-basis assumption.
- A. Unified Transceiver and DD Channel Model: CP-based transmission uses a symbol vector s and a unitary modulation matrix B, with a prefix at least as long as the maximum delay so interblock interference is removed after CP removal.The receiver applies the matched-filter bank B^H, and unitarity keeps the filtered noise white; B^H H B is the effective symbol-domain channel.
- A. Unified Transceiver and DD Channel Model: The sampled doubly selective channel is represented over one N-sample block by a discrete delay-Doppler spreading expansion using cyclic time shifts, Doppler shifts, and Weyl operators.The discrete support is the set of indices u where the spreading coefficient h[u] is nonzero, and additive noise is included in the received block model.
B. Diagonalizable DD Supports
This section characterizes diagonalizable DD supports through unitary modulation bases that diagonalize every channel supported on the set. The characterization reduces to pairwise commutation of the corresponding Weyl operators, with maximal supports identified as Lagrangian submodules.
- Definition and criterion: A DD support S is diagonalizable when a unitary matrix B depending only on S makes B^H H B diagonal for every channel supported within S.Diagonal effective channel matrices decouple modulation symbols and enable independent one-tap equalization.
- Definition and criterion: Because channel coefficients are arbitrary, diagonalizability is equivalent to B^H W_N(u) B being diagonal for every u ∈ S.The columns of B therefore form a common orthonormal eigenbasis of the indexed Weyl operators.
- Definition and criterion: A nonempty DD support admits such a basis if and only if its corresponding Weyl operators commute pairwise.The Weyl-operator commutativity condition can be expressed directly using the associated delay-Doppler indices.
- Maximal supports: Any DD set satisfying pairwise commutation contains at most N elements.This cardinality bound motivates the definition of supports attaining the maximum.
- Maximal supports: Lagrangian submodules are exactly the maximal diagonalizable DD supports.The section introduces an elementary representation of these submodules without invoking the Chinese remainder theorem used in prior work.
III. IDENTIFICATION OF MAXIMAL DIAGONALIZABLE DD SUPPORTS · A. Elementary Representation
Theorem 1 identifies maximal diagonalizable delay-Doppler supports through Lagrangian submodules, stating that the associated family satisfies L_N = E_N.
- A. Elementary Representation: Theorem 1 presents the maximal diagonalizable DD supports.The theorem is stated in the elementary representation subsection.
- A. Elementary Representation: The theorem introduces N : L as a Lagrangian submodule.This notation begins the theorem’s characterization.
- A. Elementary Representation: The elementary representation therefore combines Lagrangian-submodule notation with the equality L_N = E_N.The statement links the introduced family to the theorem’s characterization.
- A. Elementary Representation: The theorem defines the relevant family using Lagrangian submodules of Z2.The family is described as consisting of all such submodules.
- A. Elementary Representation: The identified family satisfies the elementary equality L_N = E_N.This equality is the theorem’s stated result.
- A. Elementary Representation: The theorem’s proof is deferred to Appendix A.No proof details are provided in this passage.
B. Parameter Interpretation and Special Cases
The parameters β, d, and M determine the shear, number and spacing of delay slices, and Doppler spacing of the diagonalizable DD grid. The framework includes Zak-OTFS, DAFT-OFDM, and a numerical sheared-support example as special cases.
- Parameter roles: β controls grid shear, while d sets the number and spacing of parallel DD lines and M = N/d sets Doppler spacing within each slice.These parameters jointly determine the grid geometry.
- Special cases: Zak-OTFS diagonalizes the aligned rectangular support with d = Nτ, M = Nν, and β = 0.The support geometry is illustrated in Fig. 1(a).
- Special cases: For integer α satisfying gcd(α, N) = 1, CP-compatible DAFT-OFDM diagonalizes the cyclic DD line with d = 1, M = N, and β = [2α]N.The cyclic DD-line geometry is illustrated in Fig. 1(c).
- Special cases: With N = 24 and (d, M, β) = (4, 6, 1), the construction produces the sheared grid shown in Fig. 1(b).DAFT-OFDM generally uses a chirp-periodic prefix, which becomes an ordinary CP for the integer chirp parameter considered.
C. Counting Lagrangian Submodules … B. Matrix Implementation and Special Cases
The paper counts all Lagrangian submodules and then derives closed-form orthonormal waveforms that diagonalize the corresponding delay-Doppler supports. Their matrix form enables an implementation with parallel FFTs and includes Zak-OTFS and DAFT-OFDM as special cases.
- C. Counting Lagrangian Submodules: Theorem 1’s representation determines the exact number of Lagrangian submodules in Z2.
- C. Counting Lagrangian Submodules: Corollary 1 expresses the count using the unique prime factorization of every positive integer N > 1.
- IV. DESIGN OF DIAGONALIZING WAVEFORMS: The paper identifies all maximal diagonalizable delay-Doppler supports before designing their corresponding transmit waveforms.
- A. Closed-Form Waveforms: For each ρ ∈ Zd and m ∈ ZM, unit-modulus coefficients γρ,m define the closed-form waveform basis ψρ,m.
- A. Closed-Form Waveforms: Bd,β is unitary and diagonalizes every channel supported on Ld,β, with uniqueness up to column permutation.
- B. Matrix Implementation and Special Cases: The matrix implementation applies symbol phase rotations, row-wise M-point DFTs, and quadratic-phase rotations, requiring d parallel M-point FFTs.
- B. Matrix Implementation and Special Cases: The resulting complexity is Cd,β,Γ = O(N log M + NI{β̸ = 0} + NI{Γd,M̸ = IN}), determined by FFT length M and nontrivial diagonal phases.
- B. Matrix Implementation and Special Cases: With d = Nτ, M = Nν, β = 0, the construction reduces to the Zak-OTFS waveform; with d = 1 and β = [2α]N, it becomes DAFT-OFDM.
V. CONCLUSION
Under CP-based block transmission with orthonormal modulation waveforms, the paper identifies all maximal DD supports enabling path-gain-independent exact channel diagonalization and derives their corresponding waveforms.
- V. CONCLUSION: The paper identifies all maximal diagonalizable DD supports for doubly selective channels without dependence on path gains.The analysis assumes CP-based block transmission and an orthonormal modulation-waveform basis.
- V. CONCLUSION: The corresponding diagonalizing waveforms are derived in closed form and can be implemented using parallel FFTs and phase rotations.
- V. CONCLUSION: Zak-OTFS and DAFT-OFDM emerge as special cases of the resulting waveform construction.
APPENDIX A PROOF OF THEOREM 1
The proof establishes equality between the two families of submodules by proving both set inclusions. It first verifies that each L_d,β is Lagrangian, then represents every Lagrangian submodule as some L_d,β.
- E_N ⊆ L_N: For L_d,β ∈ E_N, the generators commute pairwise and L_d,β contains exactly N elements, so it is Lagrangian and belongs to L_N.With M = N/d, the delay and Doppler coordinates yield M and d distinct values, respectively, giving |L_d,β| = Md = N.
- L_N ⊆ E_N: For L ∈ L_N, its delay projection is dZ_N for a unique divisor d | N, and its zero-delay subgroup has the structure needed to recover M = N/d.The first isomorphism theorem gives L/K ∼= dZ_N, while |L| = N implies |dZ_N| = M.
- L_N ⊆ E_N: Since L_d,β has N elements and contains L, both sets are equal, proving L_N ⊆ E_N and completing the theorem.The first inclusion already established |L_d,β| = N.
APPENDIX B PROOF OF COROLLARY 1
The proof establishes that Theorem 1’s parametrization is one-to-one and counts its admissible parameters using the standard divisor-sum formula.
- Injectivity: Equal delay projections force d = d′, proving equality of the delay parameters.The argument uses dZ_N = d′Z_N to conclude d = d′.
- Injectivity: For fixed d, the difference β′ − β lies in the zero-delay subgroup, so β′ = β in Z_M.With M = N/d, the difference is a multiple of M.
- Counting: Each divisor d | N contributes N/d choices of β, yielding the standard divisor-sum count.The passage states this count immediately after establishing one-to-one parametrization.
APPENDIX C PROOF OF THEOREM 2
The proof constructs a common eigenbasis for W_N(0, M) and W_N(d, β), using vectors supported on residue classes and satisfying the second eigenvector condition. It then normalizes these vectors and establishes dM = N orthonormal waveforms forming B_d,β.
- Common eigenbasis: A common eigenbasis of W_N(0, M) and W_N(d, β) suffices because L_d,β is generated by these two operators.This reduces diagonalization to constructing simultaneous eigenvectors.
- Common eigenbasis: Each eigenvector of W_N(0, M) is nonzero only at indices n = dj + ρ, with eigenvalue µ_ρ = e2πiρ/d.The construction fixes ρ ∈ Z_d and writes c[j] ≜ x[dj + ρ].
- Waveform construction: Imposing W_N(d, β)x = λx yields a recurrence across the M residue-class samples, whose repeated application determines the waveform coefficients.The recurrence is evaluated at n = dj + ρ, with j − 1 interpreted modulo M.
- Waveform construction: Choosing c[0] = M −1/2γ_ρ,m produces the closed-form waveforms in (20).The normalization choice fixes the waveform family indexed by ρ and m.
- Orthonormality: The resulting waveforms are orthogonal across distinct ρ because their supports are disjoint, and orthogonality for fixed ρ follows from their inner products.The construction yields d choices of ρ and M choices of m.
- Orthonormality: dM = N orthonormal waveforms are obtained, forming the desired waveform B_d,β of C^N.The count establishes a complete orthonormal waveform basis.