Integer-Order Discrete Hankel Transform (DHT) and Fast Hankel Transform (FHT)

In signal processing, discrete Hankel forward and inverse transforms are fundamental operations commonly used for converting signals between spatial/radial domains and frequency domains. The integer-order discrete Hankel transform serves as one of the most essential mathematical tools in this category.

The integer-order discrete Hankel forward transform converts a given function f(r) into a new function g(k), where k represents the frequency variable. This transformation enables better understanding of frequency-domain characteristics and properties of functions. In computational implementations, this typically involves Bessel function evaluations and discrete summation algorithms that handle radial symmetry in data.

The fast Hankel transform (FHT) and its inverse represent optimized computational implementations of the discrete Hankel transforms. These algorithms employ techniques similar to FFT-based approaches, utilizing recurrence relations and orthogonal properties of Bessel functions to achieve O(N log N) complexity. They are particularly valuable for processing large datasets where they maintain computational accuracy while significantly reducing processing time compared to direct summation methods.

Therefore, for applications requiring frequent signal processing operations, mastering both the integer-order discrete Hankel transforms and their fast implementations becomes crucial for efficient and accurate computational workflows.