The Chirp Z-Transform¶
Let \(X \in \mathbb{C}^{N}\).
The length-\(M\) Chirp Z-Transform \(\text{CZT}_{N}^{M}\{X\} \in \mathbb{C}^{M}\) of parameters \(A, W \in \mathbb{C}^{*}\) is defined as:
\(\text{CZT}_{N}^{M}\) can be efficiently computed using \(\text{(i)DFT}\) in \(\mathcal{O}(L \log L)\) operations, where \(L \ge N + M - 1\) can be arbitrarily chosen.
For a \(D\)-dimensional Chirp Z-Transform, namely a \((M_1, M_2, \ldots, M_D)\) length transform of \(X \in \mathbb{C}^{N_1 \times N_2 \times \cdots \times N_D}\), the above operation can be performed one dimension at a time.
Implementation Notes¶
czt() can be used to compute \(\text{CZT}_{N}^{M}\) as
defined above, with \(L \ge N + M - 1\) optimally chosen.
cztn() can be used to compute \(\text{CZT}_{N_1, N_2,
\ldots, N_D}^{M_1, M_2, \ldots, M_D}\), with \(L_d \ge N_d + M_d - 1\) optimally chosen for
\(d = 1, 2, \ldots, D\). In our implementation, we opt for a more efficient approach than
applying the 1D \(\text{CZT}\) along each dimension.