discrete Hartley transform
(DHT) is a Fourier-related transform of discrete, periodic data similar to the discrete_Fourier_transform (DFT), with analogous applications in signal processing etcetera. Its main distinction from the DFT is that it transforms real inputs to real outputs, with no intrinsic involvement of complex_numbers. Just as the DFT is the discrete analogue of the continuous Fourier_transform, the DHT is the discrete analogue of the continuous Hartley transform, introduced by R. V. L. Hartley in 1942. Because there are fast algorithms for the DHT analogous to the fast_Fourier_transform (FFT), the DHT was originally proposed by R. N. Bracewell in 1983 as a more efficient computational tool in the common case where the data are purely real. It was subsequently argued, however, that specialized FFT algorithms for real inputs/outputs can ordinarily be found with fewer operations than any corresponding algorithm for the DHT.Definition
Formally, the discrete Hartley transform is a linear, invertible ). The ''n'' real numbers ''x''0, ...., ''x'n''-1 are transformed into the ''n'' real numbers ''h''0, ..., ''h'n''-1 according to the formula :h_j = \sum_{k=0}^{n-1} x_k \left[ \cos \left( \frac{2 \pi}{n} j k \right) + \sin \left( \frac{2 \pi}{n} j k \right) \right] \quad \quad j = 0, \dots, n-1 where π is imaginary_unit). Like for the DFT, the overall scale factor in front of the transform and the sign of the sine term are a matter of convention, and differ in some treatments, but do not affect the essential properties.Properties
The transform can be interpreted as the multiplication of the vector (''x''0, ...., ''x'n''-1) by an ''n''-by-''n'' matrix; therefore, the discrete Hartley transform is a linear_operator. The matrix is invertible; the inverse transformation, which allows one to recover the ''x'k'' from the ''h'j'', is simply the DHT of ''h'j'' multiplied by 1/''n''. That is, the DHT is its own inverse, up to an overall scale factor. The DHT can be used to compute the DFT, and vice versa. For real inputs ''x'k'', the DFT output ''f'j'' has a real part (''h'j'' + ''h'n-j'')/2 and an imaginary part (''h'n-j'' - ''h'j'')/2. Conversely, the DHT is equivalent to computing the DFT of ''x''''k'' multiplied by 1+''i'', then taking the real part of the result. As with the DFT, a cyclic convolution z = xFast Algorithms
Just as for the DFT, evaluating the DHT definition directly would require O(''n''2) arithmetical operations (see Big_O_notation). There are fast algorithms similar to the FFT, however, that compute the same result in only O(''n'' log ''n'') operations. Nearly every FFT algorithm, from Cooley-Tukey to Prime-Factor to Winograd, has a direct analogue for the discrete Hartley transform. (However, a few of the more exotic FFT algorithms, such as Bruun's and the QFT, have not yet been investigated in the context of the DHT.) In particular, the DHT analogue of the Cooley-Tukey algorithm is commonly known as the fast Hartley transform (FHT) algorithm, and was first described by Bracewell in 1984. This FHT algorithm, at least when applied to power-of-two sizes ''n'', is the subject of the United States patent number 4,646,256, issued in 1987 to Stanford_University. As mentioned above, DHT algorithms are typically less efficient (in terms of the number of floating-point operations) than the corresponding DFT algorithm specialized for real inputs (or outputs), as demonstrated by Sorensen et al. in 1987. To illustrate this, the following table lists the lowest known operation counts (real multiplications + additions) for the DHT and the DFT, for power-of-two sizes, as achieved by the split-radix Cooley-Tukey FHT/FFT algorithm in both cases: size n DHT (split-radix FHT) DFT (split-radix FFT) for real inputs 40+8=80+6=6 82+22=242+20=22 1612+64=7610+60=70 3242+166=20834+164=198 64124+416=54098+420=518 128330+998=1328258+1028=1286 256828+2336=3164642+2436=3078 5121994+5350=73441538+5636=7174 10244668+12064=167323586+12804=16390 Lowest known operation counts (real mults + adds) for power-of-two DHT and DFT algorithms. (Note that, depending on implementation details, some of the multiplications can be traded for additions or vice versa.) On the other hand, the redundant computations in FFTs due to real inputs are much more difficult to eliminate for large Rader's_algorithm can be directly applied to the DHT of real data, for roughly a factor of two less computation than that of the equivalent complex FFT. Thus, the DHT may provide the most efficient practical pathway to computing real-data DFTs of large prime sizes (Frigo and Johnson, 2003). ----References
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