Discrete Affine Fourier Transform

• DAFT is a unitary, parameterized generalization of the DFT that employs chirp multiplications to modulate signal phases for optimal delay-Doppler tiling.
• It underpins advanced multicarrier modulation schemes such as AFDM, OCDM, and DAFT-spread AFDMA, enhancing robustness in high-mobility channels.
• The transform supports fast O(N log N) algorithms and efficient channel equalization by uniquely mapping delay-Doppler shifts to nonoverlapping indices.

The discrete affine Fourier transform (DAFT) is a unitary, parameterized generalization of the discrete Fourier transform (DFT) characterized by two real chirp parameters. The DAFT is constructed as a sequence of “chirp” quadratic phase multiplications and DFT operations, and plays a central role in modern waveform design for high-mobility multicarrier wireless communications. By tuning its parameters, DAFT enables optimal tiling of the delay-Doppler domain, providing a versatile foundation for advanced modulation schemes such as Affine Frequency Division Multiplexing (AFDM), Orthogonal Chirp Division Multiplexing (OCDM), and variants of filter bank modulations. DAFT’s algebraic structure further supports new signal processing tools, including PAPR reduction, convolution theorems, and efficient channel equalization under doubly-dispersive (delay- and Doppler-spread) channels.

1. Formal Definition and Mathematical Structure

Let $N$ denote the transform length, $\mathbf{s} = [s_0, ..., s_{N-1}]^T$ the input vector, and $\mathbf{S} = [S_0, ..., S_{N-1}]^T$ the output. The DAFT introduces two real parameters, $c_1$ (input-side chirp rate) and $c_2$ (output-side chirp rate). The forward DAFT is given by

$$S_m = \sum_{n=0}^{N-1} \frac{1}{\sqrt{N}} e^{-j2\pi c_1 n^2} e^{-j\frac{2\pi}{N} n m} e^{-j2\pi c_2 m^2} s_n, \quad m=0,1,\ldots,N-1$$

or, equivalently, in matrix notation:

$$\mathbf{S} = \mathbf{A} \mathbf{s}, \quad \mathbf{A} = \boldsymbol{\Lambda}_{c_2} \, \mathbf{F} \, \boldsymbol{\Lambda}_{c_1}$$

where $\mathbf{F}$ is the unitary DFT matrix, $$[\mathbf{F}]_{m,n} = e^{-j\frac{2\pi}{N} m n} / \sqrt{N}$$, and $$\boldsymbol{\Lambda}_c = \mathrm{diag}(e^{-j2\pi c n^2})_{n=0}^{N-1}$$.

The inverse DAFT, exploiting the unitarity of $\mathbf{A}$, is:

$$\mathbf{s} = \mathbf{A}^H \mathbf{S} = \boldsymbol{\Lambda}_{c_1}^H \, \mathbf{F}^H \, \boldsymbol{\Lambda}_{c_2}^H \mathbf{S}$$

This quadratic-phase ("chirp") structure generalizes the DFT ($c_1 = c_2 = 0$), and for $c_1 = c_2 = 1/(2N)$ recovers the discrete Fresnel transform integral to OCDM (Bemani et al., 2021, Bemani et al., 2022, Zhang et al., 2024).

2. Parameterization and Relation to Time–Frequency Geometry

The parameters $(c_1, c_2)$ impart quadratic phase rotations in the time and frequency (i.e., DFT) domains, effectively "tilting" the time–frequency grid. $c_1$ modulates the input with a time-quadratic (chirp) phase pre-DFT; $c_2$ imposes a quadratic phase post-DFT. The resulting DAFT basis comprises orthogonal chirps with linear instantaneous frequency sweeps, in contrast to the DFT’s constant-frequency sinusoidal basis.

A key unifying concept is the “chirp slope” parameter

$k = 2N\,c_1,$

which determines the tilt/slope of each subcarrier in the time–frequency plane:

• $k = 0$: Standard OFDM (DFT)
• $k = 1$: OCDM (discrete Fresnel transform)
• $k \approx 2\alpha_{\max} + 1$: AFDM (optimal for delay-Doppler separation in high-mobility channels)

Special attention must be given to choosing $c_1$, as overlapping delay-Doppler path indices in the DAFT domain are avoided by enforcing

$c_1 = \frac{2\alpha_{\max} + 1}{2N}$

and further constraints to prevent modulo-$N$ index collisions under "underspread" channel conditions ($l_{\max},\,\alpha_{\max} \ll N$) (Bemani et al., 2021, Bemani et al., 2022, Zhang et al., 2024).

3. Core Algebraic Properties

The DAFT is a unitary transformation ($\mathbf{A}^H \mathbf{A} = \mathbf{I}$), ensuring invertibility and energy preservation ($\|\mathbf{s}\|_2 = \|\mathbf{S}\|_2$). Columns of $\mathbf{A}$ form an orthonormal set in $\mathbb{C}^N$. These properties guarantee that signal structure and diversity are preserved under transmission, a critical aspect for communications over time-varying multipath channels. The DAFT also admits a generalized convolution theorem: circular convolution in the input domain corresponds to pointwise multiplication in the DAFT (chirped) domain up to a phase, paralleling the classical DFT result but crucially accommodating the chirp structure (Nafchi et al., 2020).

4. Input–Output Relations and Channel Diagonalization

When used as a front end for transmission over a doubly-dispersive channel, the DAFT admits efficient input–output analysis. For a discrete-time model,

$r_n = \sum_{l=0}^{L-1} s_{n-l} g_n(l) + w_n$

with $g_n(l)$ modeling multipath and Doppler shifts, the DAFT domain input–output relationship becomes

$$\mathbf{y} = \mathbf{A H A}^H \mathbf{x} + \widetilde{\mathbf{w}}$$

Under proper parameter selection ($c_1,\,c_2$), each path maps to a unique index (“tap”) in the DAFT domain: $\mathrm{loc}_i = \alpha_i + 2N c_1 l_i$ for path $i$. Full delay-Doppler separation is achievable, resulting in a DAFT domain effective channel with nonoverlapping single-tap support per path, facilitating low-complexity equalization and full diversity (Bemani et al., 2021, Bemani et al., 2022, Zhang et al., 2024).

5. Applications in Multicarrier Communications

DAFT serves as the foundation for several recently-developed waveforms:

Waveform	DAFT Parameterization	Distinguishing Characteristic
OFDM	$c_1 = 0, c_2 = 0$	Sinusoidal carriers, best for static/slow channels
OCDM	$c_1 = c_2 = 1/(2N)$	Chirped carriers, delay diversity
AFDM	$c_1 = (2\alpha_{\max} + 1)/(2N), c_2 \ll 1$ or irrational	Full delay-Doppler diversity for high-mobility

DAFT is instrumental in AFDM, where it delivers full diversity under time/frequency selectivity by enabling delay and Doppler shifts to map orthogonally in the transform domain. DAFT is also employed in DAFT-spread AFDMA to reduce PAPR—an effect achieved by matching DAFT parameters according to subcarrier allocation and ensuring block repetition of symbols in time (Tao et al., 2024). With proper parameter choices, DAFT-based modulation features lower PAPR than classic O-AFDMA, facilitating efficient power amplifier design (Senger et al., 26 Nov 2025).

6. Implementation Complexity and Fast Algorithms

DAFT can be implemented as “chirp–multiply → FFT → chirp–multiply”, maintaining $O(N \log N)$ computational complexity. Each transform involves two diagonal matrix multiplications (per-sample chirp phase multiplications) and one FFT/IFFT. This allows for direct integration in modern digital baseband architectures at OFDM-like cost (Zhang et al., 2024, Nafchi et al., 2020).

7. Extensions, Generalizations, and Algebraic Aspects

DAFT generalizes to more abstract settings, such as transforms associated with the (double) affine Hecke algebra (DAHA). In these contexts, the DAFT kernel arises as "spherical functions" or deformations of discrete cosine transforms with explicit orthogonality and Plancherel formulas, providing spectral decompositions for algebraically motivated difference operators (Diejen et al., 2012). This algebraic viewpoint connects the DAFT to a broader family of harmonic analysis tools, including discrete analogues of the canonical and fractional Fourier transforms.

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DAFT establishes a framework for flexible, robust, and efficient signal representations in environments characterized by strong time-frequency dispersion, unifying previously disparate multicarrier and chirp-based schemes under a single, parameterized, and algebraically tractable transformation (Bemani et al., 2021, Bemani et al., 2022, Tao et al., 2024, Zhang et al., 2024, Nafchi et al., 2020, Senger et al., 26 Nov 2025, Diejen et al., 2012).