Transform decomposition method of pruning the FHT algorithms
Introduction
The concept of pruning the fast algorithm was first introduced by Markel [1], and applied to the input of radix-2 decimation-in-frequency (DIF) FFT flow graph. Later, Skinner [2] showed that the application of Markel's pruning algorithm to the input of radix-2 decimation-in-time (DIT) FFT flow graph is more efficient. Subsequently, Sreenivas and Rao [3] combined Skinner's and Markel's pruning strategies to prune both the input and the output of the FFT flow graph. In 1993, Sorensen and Burrus [4] proposed a new method, called the transform decomposition (TD) method, for pruning the input as well as output of the FFT flow graph, which was proved to be much more efficient than the existing methods of pruning.
Narayanan and Prabhu [5] explored the possibility of pruning the input of the radix-2 DIF FHT flow graph, similar to the method followed by Markel. We shall call the algorithm they have proposed as NAP algorithm. Although the NAP algorithm is computationally efficient, we find that its programming complexity is quite high, since the structure of the pruned FHT flow graph is very different and much more difficult to handle than its FFT counterpart.
In this paper, we apply the TD method to prune the output of the FHT flow graph and calculate the reduction in the number of computations. The paper is organized as follows: Section 2 gives a brief review of the existing method of pruning the FHT flow graph. Section 3 introduces the TD method of pruning the FHT flow graph. Then, its computational complexity is analyzed and compared with the existing method. Section 4 gives the implementation details and Section 5 highlights the conclusions drawn.
Section snippets
Narayanan and Prabhu (NAP) pruning algorithm
For clarity, we briefly review the NAP method [5]. The pruned flow graph, unlike its FFT counterpart, does not repeat itself in the subsequent stages [1], [2], [3]. In the case of FFT, a block pruned at any stage gives rise to two similar blocks in the next stage. The new blocks thus formed are half the size of the previous blocks. However, such a compact structure does not exist in the case of FHT. This makes its analysis difficult [5].
In the NAP pruning algorithm, pruning the first stage
Transform decomposition
The idea of transform decomposition (TD) method was first introduced by Sorensen and Burrus [4] for pruning the FFT flow graphs. In this section, we describe the method of pruning the FHT flow graph (using the TD method), when the number of output points is limited. A 2D mapping on n (0≤n≤N−1) is defined as in [6]:where N=N1N2. This mapping is used to convert a one-dimensional sequence into a two-dimensional sequence. The discrete Hartley transform (DHT) of an N
Implementation
The implementation of SRFHT algorithm and the TD method of pruning the FHT algorithm have been coded in C language. The basic idea behind writing the SRFHT code has been taken from [8], [9]. The SRFHT, unlike any other FHT algorithm (e.g. radix-2 or radix-4), does not proceed in stages [7]. This is because one N-point block gives rise to one N/2-point and two N/4-point blocks. Hence, we need to keep track of the blocks to be processed at each stage. An efficient method for computing the Split
Conclusions
The existing method of pruning the FHT algorithm has been briefly reviewed in the paper. The transform decomposition (TD) method has been used to prune the output of the FHT algorithm, thus achieving tremendous reduction in the number of computations. Implementation of the SRFHT algorithm has been done using C programming and this has been utilised in the TD method to prune the FHT algorithm.
Acknowledgements
The authors wish to thank the two anonymous reviewers whose comments and criticisms have greatly enhanced the presentation of this paper.
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