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| A '''discrete Hartley transform (DHT)''' is a [[List of Fourier-related transforms|Fourier-related transform]] of discrete, periodic data similar to the [[discrete Fourier transform]] (DFT), with analogous applications in signal processing and related fields. Its main distinction from the DFT is that it transforms real inputs to real outputs, with no intrinsic involvement of [[complex number]]s. 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 [[Ralph Hartley|R. V. L. Hartley]] in 1942.
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| Because there are fast algorithms for the DHT analogous to the [[fast Fourier transform]] (FFT), the DHT was originally proposed by [[Ronald N. Bracewell|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 or outputs can ordinarily be found with slightly fewer operations than any corresponding algorithm for the DHT (see below).
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| == Definition ==
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| Formally, the discrete Hartley transform is a linear, invertible [[function (mathematics)|function]] ''H'' : '''R'''<sup>''n''</sup> <tt>-></tt> '''R'''<sup>''n''</sup> (where '''R''' denotes the set of [[real number]]s). The ''N'' real numbers ''x''<sub>0</sub>, ...., ''x''<sub>''N''-1</sub> are transformed into the ''N'' real numbers ''H''<sub>0</sub>, ..., ''H''<sub>''N''-1</sub> according to the formula
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| :<math>H_k = \sum_{n=0}^{N-1} x_n \left[ \cos \left( \frac{2 \pi}{N} n k \right) + \sin \left( \frac{2 \pi}{N} n k \right) \right] | |
| \quad \quad
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| k = 0, \dots, N-1 </math>.
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| The combination <math>\cos(z) + \sin(z) \!</math> <math>= \sqrt{2} \cos(z-\frac{\pi}{4})</math> is sometimes denoted <math>\mathrm{cas}(z) \!</math>, and should be contrasted with the <math>e^{-iz} = \cos(z) - i \sin(z) \!</math> that appears in the DFT definition (where ''i'' is the [[imaginary number|imaginary unit]]).
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| As with the DFT, the overall scale factor in front of the transform and the sign of the sine term are a matter of convention. Although these conventions occasionally vary between authors, they do not affect the essential properties of the transform.
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| == Properties ==
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| The transform can be interpreted as the multiplication of the vector (''x''<sub>0</sub>, ...., ''x''<sub>''N''-1</sub>) by an ''N''-by-''N'' [[matrix (math)|matrix]]; therefore, the discrete Hartley transform is a [[linear operator]]. The matrix is invertible; the inverse transformation, which allows one to recover the ''x''<sub>''n''</sub> from the ''H''<sub>''k''</sub>, is simply the DHT of ''H''<sub>''k''</sub> multiplied by 1/''N''. That is, the DHT is its own inverse ([[Involution (mathematics)|involutary]]), up to an overall scale factor.
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| The DHT can be used to compute the DFT, and vice versa. For real inputs ''x''<sub>''n''</sub>, the DFT output ''X''<sub>''k''</sub> has a real part (''H''<sub>''k''</sub> + ''H''<sub>''N-k''</sub>)/2 and an imaginary part (''H''<sub>''N-k''</sub> - ''H''<sub>''k''</sub>)/2. Conversely, the DHT is equivalent to computing the DFT of ''x''<sub>''n''</sub> multiplied by 1+''i'', then taking the real part of the result.
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| As with the DFT, a cyclic [[convolution]] '''z''' = '''x'''*'''y''' of two vectors '''x''' = (''x''<sub>''n''</sub>) and '''y''' = (''y''<sub>''n''</sub>) to produce a vector '''z''' = (''z''<sub>''n''</sub>), all of length ''N'', becomes a simple operation after the DHT. In particular, suppose that the vectors '''X''', '''Y''', and '''Z''' denote the DHT of '''x''', '''y''', and '''z''' respectively. Then the elements of '''Z''' are given by:
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| :<math> \begin{matrix}
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| Z_k & = & \left[ X_k \left( Y_k + Y_{N-k} \right)
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| + X_{N-k} \left( Y_k - Y_{N-k} \right) \right] / 2
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| \\
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| Z_{N-k} & = & \left[ X_{N-k} \left( Y_k + Y_{N-k} \right)
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| - X_k \left( Y_k - Y_{N-k} \right) \right] / 2
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| \end{matrix}
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| </math>
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| where we take all of the vectors to be periodic in ''N'' (''X''<sub>''N''</sub> = ''X''<sub>0</sub>, etcetera). Thus, just as the DFT transforms a convolution into a pointwise multiplication of complex numbers (''pairs'' of real and imaginary parts), the DHT transforms a convolution into a simple combination of ''pairs'' of real frequency components. The inverse DHT then yields the desired vector '''z'''. In this way, a fast algorithm for the DHT (see below) yields a fast algorithm for convolution. (Note that this is slightly more expensive than the corresponding procedure for the DFT, not including the costs of the transforms below, because the pairwise operation above requires 8 real-arithmetic operations compared to the 6 of a complex multiplication. This count doesn't include the division by 2, which can be absorbed e.g. into the 1/''N'' normalization of the inverse DHT.)
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| == Fast algorithms ==
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| Just as for the DFT, evaluating the DHT definition directly would require O(''N''<sup>2</sup>) 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 FFT algorithm|Cooley-Tukey]] to [[Prime-factor FFT algorithm|Prime-Factor]] to Winograd (Sorensen et al., 1985) to [[Bruun's FFT algorithm|Bruun's]] (Bini & Bozzo, 1993), has a direct analogue for the discrete Hartley transform. (However, a few of the more exotic FFT algorithms, such as the QFT, have not yet been investigated in the context of the DHT.)
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| 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|power-of-two]] sizes ''N'', is the subject of the United States [[software patent|patent]] number 4,646,256, issued in 1987 to [[Stanford University]]. Stanford placed this patent in the public domain in 1994 (Bracewell, 1995).
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| As mentioned above, DHT algorithms are typically slightly less efficient (in terms of the number of [[floating-point]] operations) than the corresponding DFT algorithm (FFT) specialized for real inputs (or outputs). This was first argued by Sorensen et al. (1987) and Duhamel & Vetterli (1987). The latter authors obtained what appears to be the lowest published operation count for the DHT of power-of-two sizes, employing a split-radix algorithm (similar to the [[split-radix FFT algorithm|split-radix FFT]]) that breaks a DHT of length ''N'' into a DHT of length ''N''/2 and two real-input DFTs (''not'' DHTs) of length ''N''/4. In this way, they argued that a DHT of power-of-two length can be computed with, at best, 2 more additions than the corresponding number of arithmetic operations for the real-input DFT.
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| On present-day computers, performance is determined more by [[CPU cache|cache]] and [[CPU pipeline]] considerations than by strict operation counts, and a slight difference in arithmetic cost is unlikely to be significant. Since FHT and real-input FFT algorithms have similar computational structures, neither appears to have a substantial ''a priori'' speed advantage (Popovic and Sevic, 1994). As a practical matter, highly optimized real-input FFT libraries are available from many sources (e.g. from CPU vendors such as [[Intel]]), whereas highly optimized DHT libraries are less common.
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| On the other hand, the redundant computations in FFTs due to real inputs are more difficult to eliminate for large [[prime number|prime]] ''N'', despite the existence of O(''N'' log ''N'') complex-data algorithms for such cases, because the redundancies are hidden behind intricate permutations and/or phase rotations in those algorithms. In contrast, a standard prime-size FFT algorithm, [[Rader's FFT algorithm|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 (Frigo and Johnson, 2005). On the other hand, a non-DHT-based adaptation of Rader's algorithm for real-input DFTs is also possible (Chu & [[C. Sidney Burrus|Burrus]], 1982).
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| ==References==
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| * R. N. Bracewell, "Discrete Hartley transform," ''J. Opt. Soc. Am.'' '''73''' (12), 1832–1835 (1983).
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| * R. N. Bracewell, "The fast Hartley transform," ''[[Proc. IEEE]]'' '''72''' (8), 1010–1018 (1984).
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| * R. N. Bracewell, ''The Hartley Transform'' (Oxford Univ. Press, New York, 1986).
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| * R. N. Bracewell, "Computing with the Hartley Transform," ''Computers in Physics'' '''9''' (4), 373–379 (1995).
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| * R. V. L. Hartley, "A more symmetrical Fourier analysis applied to transmission problems," ''[[Proc. IRE]]'' '''30''', 144–150 (1942).
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| * H. V. Sorensen, D. L. Jones, C. S. Burrus, and M. T. Heideman, "On computing the discrete Hartley transform," ''IEEE Trans. Acoust. Speech Sig. Processing'' '''ASSP-33''' (4), 1231–1238 (1985).
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| * H. V. Sorensen, D. L. Jones, M. T. Heideman, and C. S. Burrus, "Real-valued fast Fourier transform algorithms," ''IEEE Trans. Acoust. Speech Sig. Processing'' '''ASSP-35''' (6), 849–863 (1987).
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| * Pierre Duhamel and Martin Vetterli, "Improved Fourier and Hartley transform algorithms: application to cyclic convolution of real data," ''IEEE Trans. Acoust. Speech Sig. Processing'' '''ASSP-35''', 818–824 (1987).
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| * Mark A. O'Neill, "Faster than Fast Fourier", '''Byte''' '''13'''(4):293-300, (1988).
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| * J. Hong and M. Vetterli and P. Duhamel, "Basefield transforms with the convolution property," ''Proc. IEEE'' '''82''' (3), 400-412 (1994).
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| * D. A. Bini and E. Bozzo, "Fast discrete transform by means of eigenpolynomials," ''Computers & Mathematics (with Applications)'' '''26''' (9), 35–52 (1993).
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| * Miodrag Popović and Dragutin Šević, "A new look at the comparison of the fast Hartley and Fourier transforms," ''IEEE Trans. Signal Processing'' '''42''' (8), 2178-2182 (1994).
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| * Matteo Frigo and Steven G. Johnson, "[http://fftw.org/fftw-paper-ieee.pdf The Design and Implementation of FFTW3]," ''Proc. IEEE'' '''93''' (2), 216–231 (2005).
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| * S. Chu and C. Burrus, "A prime factor FTT <nowiki>[</nowiki>''sic''<nowiki>]</nowiki> algorithm using distributed arithmetic," '' IEEE Transactions on Acoustics, Speech, and Signal Processing'' '''30''' (2), 217–227 (1982).
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| [[Category:Discrete transforms]]
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| [[Category:Fourier analysis]]
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