Chirp-Zak Transform in Delay-Doppler Applications

• CZT is a discrete transform that integrates quadratic-phase chirp modulation with Zak folding to localize signal energy in the delay-Doppler domain.
• It enables efficient multiplexing, robust channel estimation, and low-complexity implementation through sparse kernel structures and FFT-based computations.
• CZT underpins the Orthogonal Chirp Delay-Doppler Division Multiplexing framework, delivering enhanced bit error rate performance and spectral efficiency in challenging wireless channels.

The Chirp–Zak Transform (CZT) is a discrete linear transformation that synergistically combines quadratic-phase ("chirp") modulation and the discrete Zak transform to achieve localized and orthogonal signal spread in the delay–Doppler (DD) domain. Developed in the context of multi-carrier modulation for high-mobility wireless communications, CZT underpins the Orthogonal Chirp Delay–Doppler Division Multiplexing (CDDM) framework. Its defining feature is the ability to map data symbols onto maximally localized and mutually orthogonal DD atoms, facilitating efficient multiplexing, robust channel estimation, and low-complexity implementation, notably in scenarios with severe multipath and Doppler effects (Bai et al., 22 Nov 2025).

1. Formal Definition and Core Transform

For a length-$N$ discrete sequence $x[n]$, with $N = M_D N_D$ ($M_D$ delay bins, $N_D$ Doppler bins), the CZT comprises two sequential operations:

1. Chirp Modulation: Each sample is modulated by a discrete chirp waveform (discrete fractional Fourier transform kernel):

$$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$

The chirped sequence is

$$s_c[q] = \sum_{w=0}^{N-1} x[w] \, \varphi_w(q), \quad q = 0, \ldots, N-1.$$

2. Zak Folding: The discrete Zak transform is then applied:

$$\mathcal{CZ}\{x\}[m, n] = \frac{1}{\sqrt{N_D}} \sum_{k=0}^{N_D-1} s_c(m + k M_D) \exp\left\{ -j 2\pi \frac{n k}{N_D} \right\},$$

with $m = 0, \ldots, M_D-1$ and $n = 0, \ldots, N_D-1$. The result is an $x[n]$0 DD domain array.

The per-symbol CZT kernel is defined by

$x[n]$1

The complete DD representation is then

$x[n]$2

With proper choice of $x[n]$3 and chirp phases, the transform is unitary.

2. Orthogonality and Matrix Structure

CZT forms an orthonormal basis over the DD array:

$x[n]$4

Consequently, the CZT can be represented as a tall unitary matrix $x[n]$5, with

$x[n]$6

Each CZT kernel is extremely sparse, having exactly one nonzero per delay row, with the nonzero's Doppler index determined by

$x[n]$7

This structure allows the CZT to be factorized explicitly as

$x[n]$8

where $x[n]$9 is the $N = M_D N_D$0 DFT matrix, and $N = M_D N_D$1 is an $N = M_D N_D$2 sparse selection matrix with entries $N = M_D N_D$3.

3. Comparison to the Classical Zak Transform

Both CZT and the discrete Zak transform (DZT) map a length-$N = M_D N_D$4 sequence to an $N = M_D N_D$5 array:

• The DZT "folds" $N = M_D N_D$6 into the DD domain,

$N = M_D N_D$7

• The CzT preconditions the sequence by chirp modulation, then applies the Zak folding.

The quadratic-phase spreading in CZT ensures each input symbol is distributed over the entire time axis yet localized in both delay and Doppler by the chirp autocorrelation. DD atoms created via CZT exhibit ideal localization properties, a key advantage over standard DZT for joint delay–Doppler processing.

4. Computational Aspects and FFT-Based Algorithms

An efficient implementation relies on two properties: sparsity of the kernels and fast Fourier transforms.

• Forward CZT: For each $N = M_D N_D$8, only $N = M_D N_D$9 out of $M_D$0 elements are nonzero per kernel; these are determined and precomputed. The transformation proceeds by summing $M_D$1 input symbols' contributions at their assigned DD bins (cost $M_D$2) and then performing an $M_D$3-point DFT per delay row ($M_D$4), for total

$M_D$5

per frame.

• Inverse CZT: Recovery exploits the sparsity pattern: each $M_D$6 is reconstructed from $M_D$7 nonzero $M_D$8 pairs using the known chirp phases, with cost $M_D$9.
• Summary Table: CZT Implementation Workflow

Step	Operation	Complexity
Forward CZT	Sparse sum + $N_D$0-DFTs/row	$N_D$1
Inverse CZT	Sparse correlation per symbol	$N_D$2

The transform avoids explicit storage of large dense matrices; all operations are based on sparse data and fast DFTs.

5. Application in Orthogonal Chirp Delay–Doppler Division Multiplexing (CDDM)

Within CDDM, the CZT is used to spread $N_D$3 data symbols into a DD-domain data matrix $N_D$4, providing each symbol with full DD diversity. Subsequent synthesis of the time-domain multicarrier waveform follows:

$N_D$5

where $N_D$6 is a pulse-shaping filter, e.g., square-root raised-cosine (SRRC), energy normalized.

Thanks to chirp spreading, symbols maintain orthogonality and maximal DD localization under the DD channel. Pulse-shaping further reduces out-of-band emissions. The overall system achieves bit error rate (BER) and spectral efficiency improvements versus conventional schemes under both perfect and imperfect channel state information (Bai et al., 22 Nov 2025).

6. Superimposed Pilot Scheme for Joint Channel and DD Estimation

CZT enables a superimposed sparse pilot structure for simultaneous estimation of channel coefficients and DD shifts in sparse $N_D$7-path channels. The procedure involves:

• Embedding: A single pilot index $N_D$8 is reserved with known $N_D$9; its CZT kernel is replaced with a "reduced" version (λ nonzeros), typically chosen diagonally in the DD grid.
• Detection: Post demodulation, the received DD grid $$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$0 is thresholded to locate pilot impulses, directly yielding path delay and Doppler indices $$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$1.
• Gain Estimation: Path gains $$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$2 are recovered by chirp-correlating the corresponding DD bins, exploiting orthogonality:

$$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$3

where $$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$4 encodes the known phase rotation due to the path.

This strategy achieves normalized mean square error (NMSE) and estimation accuracy equivalent to orthogonal delay-Doppler division multiplexing (ODDM), but with reduced computational complexity (Bai et al., 22 Nov 2025).

7. Practical Guidelines for Parameter Selection

• Delay and Doppler Grid Resolution: Set $$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$5 and $$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$6 for proper coverage.
• Total Data Symbols: $$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$7, matching the number of symbols per frame.
• Chirp Rate: For $$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$8, the chirp coefficient is always $$\varphi_i(w) = \exp\left\{ j \frac{\pi}{4} - j \frac{\pi}{N} (w - i)^2 \right\}, \quad w, i = 0, \ldots, N-1.$$9. For AFDM-based chirps, tuning $$s_c[q] = \sum_{w=0}^{N-1} x[w] \, \varphi_w(q), \quad q = 0, \ldots, N-1.$$0 and $$s_c[q] = \sum_{w=0}^{N-1} x[w] \, \varphi_w(q), \quad q = 0, \ldots, N-1.$$1 ensures DD guard intervals.
• Pilot Length: Increasing λ improves channel estimation but increases pilot interference. A typical setting is $$s_c[q] = \sum_{w=0}^{N-1} x[w] \, \varphi_w(q), \quad q = 0, \ldots, N-1.$$2 with ≈30% of pilot power.
• Computational Complexity: Total per-frame complexity is $$s_c[q] = \sum_{w=0}^{N-1} x[w] \, \varphi_w(q), \quad q = 0, \ldots, N-1.$$3; for fixed $$s_c[q] = \sum_{w=0}^{N-1} x[w] \, \varphi_w(q), \quad q = 0, \ldots, N-1.$$4, $$s_c[q] = \sum_{w=0}^{N-1} x[w] \, \varphi_w(q), \quad q = 0, \ldots, N-1.$$5 and $$s_c[q] = \sum_{w=0}^{N-1} x[w] \, \varphi_w(q), \quad q = 0, \ldots, N-1.$$6 are balanced based on the application's delay and Doppler spread.

A plausible implication is that proper parameter balancing directly affects performance and computational cost in CDDM deployments.

For further details, including explicit proofs, extended algorithms, and numerical experiments comparing BER, out-of-band emissions (OOBE), and NMSE with OFDM, OTFS, AFDM, and ODDM, see (Bai et al., 22 Nov 2025).