# Fast Type-II Hartley Transform Algorithms for Short-Length Input Sequences
## 1 Introduction  
Among other orthogonal transforms, the discrete Hartley transform (DHT) is a well-known and well-tested tool in digital signal and image processing [1, 2, 3]. This transform is used in smart grid applications [4], signal processing [5], digital image watermarking [6, 7, 8], audio watermarking [9], image compression and encryption [10, 11], image enhancement [12], OFDM communication systems [13, 14, 15], face recognition [16], audio classification [17, 18], etc. Its advantage is primarily that it can successfully supplement the discrete Fourier transform in many applications [1, 2, 3]. For example, in audio classification applications, features extracted from the representation of signals in the frequency domain are typically focused on the magnitude spectral content, while the phase spectral content is ignored [19]. The conventional Fourier phase spectrum is a highly discontinuous function. To extract the features in classification applications, function continuity is required. Thus, the Hartley phase spectrum was introduced as an alternative to the Fourier phase spectrum. The former encapsulates the phase content of the signal more efficiently if the sources of phase discontinuities are detected and overcome [20]. The latter is because the Hartley phase spectrum does not suffer from the phase \"wrapping ambiguities\" introduced due to the inverse tangent function employed in the Fourier phase spectrum computation [19].  
Compared to wavelet analysis, the Hartley transform provides only global frequency information, although the ability to analyze signals in terms of both time and frequency may be obtained with windowing applied. As a result of windowing, a short DHT is obtained [21]. The DHT allows for efficient signal denoising, being a fixed global transform. Moreover, the traditional DHT can be adapted to detect fine signal details by using windows of different sizes [8; 22]. Although DHT [10; 11] is exploited in image compression infrequently, it is widely applied in audio and image watermarking [6; 7; 8; 22] to enhance robustness against different noise attacks. Also, a significant increase in computing efficiency by saving arithmetic operations and memory storage is achieved [22].  
To date, four types of discrete Hartley transforms have been formulated [23]. However, the most popular and most developed transform is still DHT-I [1; 10; 13; 14; 15]. The other transforms from this family have been studied much less. One important, but not the main, advantage of discrete orthogonal transforms is the huge number of so-called fast algorithms that have been designed for them, allowing for the acceleration of their implementation [23]. The direct computation of the DHT-II of an \(N\)-point sequence requires \(N^{2}\) arithmetic operations. Therefore, fast DHT-II algorithms have been introduced to reduce the computational complexity and implementation cost [24; 25; 26; 27; 28; 29].  
It should be noted that the papers dealing with the efficient implementation of the DHT-II transform considered the case of large lengths of input data sequences [24; 25; 26; 27; 28; 29]. However, short-length DHT-II algorithms are of particular interest, since these transforms are considered typical modules in the synthesis of more complex algorithms [23]. Once constructed, short-length DHT-II algorithms can be successfully applied in various projects to unify the development process of the final product. There are many papers devoted to the efficient implementation of traditional short-length trigonometric transforms, but they mainly concern cosine or sine transforms [30; 31; 32; 33]. Then, the development of fast DHT-II algorithms for short-length input sequences becomes relevant. Below, the current state of the problem is analyzed to select an approach to the development of such fast DHT-II algorithms.