# Fixed-point error analysis of fast Hartley transform

TL;DR: In this paper, a fixed-point error analysis has been carried out for the fast Hartley transform (FHT) and the results are compared with the FFT error-analysis results.

About: This article is published in Signal Processing.The article was published on 1990-03-01. It has received 11 citations till now. The article focuses on the topics: Hartley transform & Discrete Hartley transform.

##### Citations

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TL;DR: It is found that the fixed-point-error characteristics of the row-column approach for 2D DCTs is very similar to that of their 1D counterparts, which shows one of the algorithms is better than others in terms of average SNR performance.

Abstract: A fixed-point-error analysis for several 1D fast DCT algorithms is presented. For comparison, a direct-form approach is also included in the investigation. A statistical model is used as the basis for predicting the fixed-point error in implementing the algorithms, and a suitable scaling scheme is selected to avoid overflow. Closed-form expressions for both the mean and variance for fixed-point error are derived and compared with experimental results. Simulation results show close agreement between theory and experiment, validating the analysis. The results show that one of the algorithms is better than others in terms of average SNR performance. Based on the 1D analysis, attempts are made to investigate the fixed-point-error analysis of the two-column approach for 2D DCT. It is found that the fixed-point-error characteristics of the row-column approach for 2D DCTs is very similar to that of their 1D counterparts. >

46 citations

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TL;DR: Suitable scaling schemes are chosen for the Lee's and the Hou's fast DCT algorithms, and the relative fixed-point roundoff error analyses are carried out, and it is shown that in DCT and for N>16 stage-by-stage scaling of Hou's algorithm has the best performance, whereas in inverse DCT, the global scaling of either algorithms has thebest performance.

Abstract: Suitable scaling schemes are chosen for the Lee's and the Hou's (1984) fast DCT algorithms, and the relative fixed-point roundoff error analyses are carried out, respectively. The average output signal-to-noise ratio are then calculated, and it is shown that in DCT and for N>16 stage-by-stage scaling of Hou's algorithm has the best performance, whereas in inverse DCT, the global scaling of either algorithms has the best performance. >

16 citations

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TL;DR: In this article, a fixed-point error analysis for well-known fast l-D IDCT algorithms, such as Lee, Hou, and Vetterli, is presented.

Abstract: In this paper, a fixed-point error analysis for well-known fast l-D IDCT algorithms, such as Lee, Hou, and Vetterli, are presented. For a comparison purpose, a direct-form method is also included in our investigation. Based on the l-D analysis, the fixed-point error analysis of the row-column method and the Cho-Lee algorithm are also investigated for 2-D IDCT. Closed-form expressions for the rounding error variances are derived and compared with the experimental results. There is a close agreement between the theory and experiment, demonstrating that the analysis presented in this paper is valid. In addition, we also discuss the minimum word length to satisfy requirements for the implementation of 8/spl times/8 IDCT. >

15 citations

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TL;DR: A relationship between the range of the twiddle factor and the dimension of the discrete cosine transform is first derived, whence a suitable scaling model is chosen for the DCT algorithm and the average output signal-to-noise ratio is calculated.

Abstract: In this paper, a fixed-point round-off error analysis of the discrete cosine transform (DCT) has been carried out. A relationship between the range of the twiddle factor and the dimension of the DCT is first derived, whence a suitable scaling model is chosen for the DCT algorithm and the average output signal-to-noise ratio is calculated.

9 citations

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TL;DR: Efficient implementation of the FHT on different DSP processors is considered, instead of counting the required arithmetic operations, the necessary number of instruction cycles for an implementation of FHT is used as a measure.

3 citations

##### 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

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01 Aug 1984TL;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

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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

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Rice University

^{1}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

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IBM

^{1}TL;DR: In this article, an analysis of the fixed-point accuracy of the power of two, fast Fourier transform algorithm is presented, which leads to approximate upper and lower bounds on the root-mean-square error.

Abstract: This paper contains an analysis of the fixed-point accuracy of the power of two, fast Fourier transform algorithm. This analysis leads to approximate upper and lower bounds on the root-mean-square error. Also included are the results of some accuracy experiments on a simulated fixed-point machine and their comparison with the error upper bound.

164 citations