Fast Hartley transform pruning
TL;DR: It is shown that for many applications, such as interpolation and convolution of signals, a significant number of zeros are padded to the nonzero valued samples before the transform is computed, and significant savings can be obtained by pruning the FHT algorithm.
Abstract: The discrete Hartley transform (DHT) is discussed as a tool for the processing of real signals. Fast Hartley transform (FHT) algorithms which compute the DHT in a time proportional to N log/sub 2/ N exist. In many applications, such as interpolation and convolution of signals, a significant number of zeros are padded to the nonzero valued samples before the transform is computed. It is shown that for such situations, significant savings in the number of additions and multiplications can be obtained by pruning the FHT algorithm. The modifications in the FHT algorithm as a result of pruning are developed and implemented in an FHT subroutine. The amount of savings in the operation is determined. >
Citations
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TL;DR: It is shown that the DCT, DST, DHT and their inverse transforms share an almost identical lattice structure so that they are very suitable for VLSI implementation for high-speed applications.
Abstract: Unified efficient computations of the discrete cosine transform (DCT), discrete sine transform (DST), discrete Hartley transform (DHT), and their inverse transforms employing the time-recursive approach are considered. Unified parallel lattice structures that can dually generate the DCT and DST simultaneously as well the DHT are developed. These structures can obtain the transformed data for sequential input time recursively with a throughput rate of one per clock cycle. The total number of multipliers required is a linear function of the transform size N, with no constraint on N. The resulting architectures are regular, modular, and without global communication so that they are very suitable for VLSI implementation for high-speed applications. It is shown that the DCT, DST, DHT and their inverse transforms share an almost identical lattice structure. The tradeoff between time and area for the block data processing is considered. >
97 citations
TL;DR: The authors provide a theoretical justification by showing that any discrete transform whose basis functions satisfy the fundamental recurrence formula has a second-order autoregressive structure in its filter realization and extend these time-recursive concepts to multi-dimensional transforms.
Abstract: An optimal unified architecture that can efficiently compute the discrete cosine, sine, Hartley, Fourier, lapped orthogonal, and complex lapped transforms for a continuous stream of input data that arise in signal/image communications is proposed. This structure uses only half as many multipliers as the previous best known scheme (Liu and Chiu, 1993). The proposed architecture is regular, modular, and has only local interconnections in both data and control paths. There is no limitation on the transform size N and only 2N-2 multipliers are needed for the DCT. The throughput of this scheme is one input sample per clock cycle. The authors provide a theoretical justification by showing that any discrete transform whose basis functions satisfy the fundamental recurrence formula has a second-order autoregressive structure in its filter realization. They also demonstrate that dual generation transform pairs share the same autoregressive structure. They extend these time-recursive concepts to multi-dimensional transforms. The resulting d-dimensional structures are fully-pipelined and consist of only d 1D transform arrays and shift registers. >
75 citations
Cites background from "Fast Hartley transform pruning"
...…discrete sinusoidal transforms, the discrete cosine transform (DCT) [6]-[9], the discrete sine transform (DST) [9], [lo], the discrete Hartley transform (DHT) [27], [28], [26], [6], and the discrete Fourier transform (DFT) [.3] are widely used because of their efficient performance [SI, [20]-[24]....
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TL;DR: This paper is based on the cosine and sine symmetric implementation of the discrete Hartley transform (DHT), which is the key in reducing the computational complexity of the FBNLMS by 33% asymptotically (with respect to multiplications).
Abstract: The least mean squared (LMS) algorithm and its variants have been the most often used algorithms in adaptive signal processing. However the LMS algorithm suffers from a high computational complexity, especially with large filter lengths. The Fourier transform-based block normalized LMS (FBNLMS) reduces the computation count by using the discrete Fourier transform (DFT) and exploiting the fast algorithms for implementing the DFT. Even though the savings achieved with the FBNLMS over the direct-LMS implementation are significant, the computational requirements of FBNLMS are still very high, rendering many real-time applications, like audio and video estimation, infeasible. The Hartley transform-based BNLMS (HBNLMS) is found to have a computational complexity much less than, and a memory requirement almost of the same order as, that of the FBNLMS. This paper is based on the cosine and sine symmetric implementation of the discrete Hartley transform (DHT), which is the key in reducing the computational complexity of the FBNLMS by 33% asymptotically (with respect to multiplications). The parallel implementation of the discrete cosine transform (DCT) in turn can lead to more efficient implementations of the HBNLMS.
11 citations
Cites background from "Fast Hartley transform pruning"
...Further manipulation is made easy if we notice that is a submatrix of , where ....
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19 Dec 2005
TL;DR: This paper proposes an alternate instance of padding zeros to the data sequence that results in computational cost reduction to O(pNlog2 N) and can be used to achieve non-uniform upsampling that would zoom-in or zoom-out a particular frequency band.
Abstract: The classical Cooley-Tukey fast Fourier transform (FFT) algorithm has the computational cost of O(Nlog2N) where N is the length of the discrete signal. Spectrum resolution is improved through padding zeros at the tail of the discrete signal, if (p -1)N zeros are padded (where p is an integer) at the tail of the data sequence, the computational cost through FFT becomes O(pNlog2pN). This paper proposes an alternate instance of padding zeros to the data sequence that results in computational cost reduction to O(pNlog2 N). It has been noted that this modification can be used to achieve non-uniform upsampling that would zoom-in or zoom-out a particular frequency band, in addition, it may be used for pruning the spectrum, which would reduce resolution of an unimportant frequency band
11 citations
Journal Article•
TL;DR: Simulation results for the application of a bandwidth efficient algorithm (mapping algorithm) to an image transmission system that considers three different real valued transforms to generate energy compact coefficients show that the system performs its best when discrete cosine transform is used.
Abstract: In this paper we present simulation results for the application of a bandwidth efficient algorithm (mapping algorithm) to an image transmission system. This system considers three different real valued transforms to generate energy compact coefficients. First results are presented for gray scale and color image transmission in the absence of noise. It is seen that the system performs its best when discrete cosine transform is used. Also the performance of the system is dominated more by the size of the transform block rather than the number of coefficients transmitted or the number of bits used to represent each coefficient. Similar results are obtained in the presence of additive white Gaussian noise. The varying values of the bit error rate have very little or no impact on the performance of the algorithm. Optimum results are obtained for the system considering 8x8 transform block and by transmitting 15 coefficients from each block using 8 bits. Keywords—Additive white Gaussian noise channel, mapping algorithm, peak signal to noise ratio, transform encoding.
5 citations
Cites methods from "Fast Hartley transform pruning"
...7 PSNR vs number of coefficients for mapping technique for DCT, DST and DHT in the presence of AWGN....
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...6 PSNR vs number of bits for mapping algorithm for DCT, DST and DHT in the presence of AWGN....
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...The image undergoes a real valued image transformation such as DCT, DST and DHT....
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...2 PSNR vs number of bits for mapping algorithm for DCT, DST and DHT under noiseless conditions....
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...The performance of this algorithm is observed for different real valued transformations such as discrete cosine transform (DCT)[3-5], discrete sine transform (DST)[5-7] and discrete Hartley transform (DHT)[6,8]....
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References
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TL;DR: The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable and promises to speed up Fourier-transform calculations.
Abstract: The discrete Hartley transform (DHT) resembles the discrete Fourier transform (DFT) but is free from two characteristics of the DFT that are sometimes computationally undesirable. The inverse DHT is identical with the direct transform, and so it is not necessary to keep track of the +i and −i versions as with the DFT. Also, the DHT has real rather than complex values and thus does not require provision for complex arithmetic or separately managed storage for real and imaginary parts. Nevertheless, the DFT is directly obtainable from the DHT by a simple additive operation. In most image-processing applications the convolution of two data sequences f1 and f2 is given by DHT of [(DHT of f1) × (DHT of f2)], which is a rather simpler algorithm than the DFT permits, especially if images are. to be manipulated in two dimensions. It permits faster computing. Since the speed of the fast Fourier transform depends on the number of multiplications, and since one complex multiplication equals four real multiplications, a fast Hartley transform also promises to speed up Fourier-transform calculations. The name discrete Hartley transform is proposed because the DHT bears the same relation to an integral transform described by Hartley [ HartleyR. V. L., Proc. IRE30, 144 ( 1942)] as the DFT bears to the Fourier transform.
465 citations
01 Aug 1984
TL;DR: The Fast Hartley Transform (FHT) is as fast as or faster than the Fast Fourier Transform (FFT) and serves for all the uses such as spectral analysis, digital processing, and convolution to which the FFT is at present applied.
Abstract: A fast algorithm has been worked out for performing the Discrete Hartley Transform (DHT) of a data sequence of N elements in a time proportional to Nlog 2 N. The Fast Hartley Transform (FHT) is as fast as or faster than the Fast Fourier Transform (FFT) and serves for all the uses such as spectral analysis, digital processing, and convolution to which the FFT is at present applied. A new timing diagram (stripe diagram) is presented to illustrate the overall dependence of running time on the subroutines composing one implementation; this mode of presentation supplements the simple counting of multiplies and adds. One may view the Fast Hartley procedure as a sequence of matrix operations on the data and thus as constituting a new factorization of the DFT matrix operator; this factorization is presented. The FHT computes convolutions and power spectra distinctly faster than the FFT.
455 citations
Book•
01 Jan 1986TL;DR: The author describes the fast algorithm he discovered for spectral analysis and indeed any purpose to which Fourier Transforms and the Fast Fourier Transform are normally applied.
Abstract: The author describes the fast algorithm he discovered for spectral analysis and indeed any purpose to which Fourier Transforms and the Fast Fourier Transform are normally applied.
437 citations
TL;DR: A complete set of fast algorithms for computing the discrete Hartley transform is developed, including decimation-in-frequency, radix-4, split radix, prime factor, and Winograd transform algorithms.
Abstract: The discrete Hartley transform (DHT) is a real-valued transform closely related to the DFT of a real-valued sequence. Bracewell has recently demonstrated a radix-2 decimation-in-time fast Hartley transform (FHT) algorithm. In this paper a complete set of fast algorithms for computing the DHT is developed, including decimation-in-frequency, radix-4, split radix, prime factor, and Winograd transform algorithms. The philosophies of all common FFT algorithms are shown to be equally applicable to the computation of the DHT, and the FHT algorithms closely resemble their FFT counterparts. The operation counts for the FHT algorithms are determined and compared to the counts for corresponding real-valued FFT algorithms. The FHT algorithms are shown to always require the same number of multiplications, the same storage, and a few more additions than the real-valued FFT algorithms. Even though computation of the FHT takes more operations, in some situations the inherently real-valued nature of the discrete Hartley transform may justify this extra cost.
275 citations
Journal Article•
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TL;DR: It is shown that for situations in which the relative number of zero-valued samples is quite large, significant time-saving can be obtained by pruning the FFT algorithm.
Abstract: There are basically four modifications of the N=2Mpoint FFT algorithm developed by Cooley and Tukey which give improved computational efficiency. One of these, FFT pruning, is quite useful for applications such as interpolation (in both the time and frequency domain), and least-squares approximation with trignometric polynomials. It is shown that for situations in which the relative number of zero-valued samples is quite large, significant time-saving can be obtained by pruning the FFT algorithm. The programming modifications are developed and shown to be nearly trivial. Several applications of the method for speech analysis are presented along with Fortran programs of the basic and pruned FFT algorithm. The technique described can also be applied effectively for evaluating a narrow region of the frequency domain by pruning a decimation-in-time algorithm.
270 citations