Multidimensional vector radix FHT algorithms

In this paper, efficient multidimensional (M-D) vector radix (VR) decimation-in-frequency and decimation-in-time fast Hartley transform (FHT) algorithms are derived for computing the discrete Hartley transform (DHT) of any dimension using an appropriate index mapping and the Kronecker product. The proposed algorithms are more effective and highly suitable for hardware and software implementations compared to all existing M-D FHT algorithms that are derived for the computation of the DHT of any dimension. The butterflies of the proposed algorithms are based on simple closed-form expressions that allow easy implementations of these algorithms for any dimension. In addition, the proposed algorithms possess properties such as high regularity, simplicity and in-place computation that are highly desirable for software and hardware implementations, especially for the M-D applications. A close relationship between the M-D VR complex-valued fast Fourier transform algorithms and the proposed M-D VR FHT algorithms is established. This type of relationship is of great significance for software and hardware implementations of the algorithms, since it is shown that because of this relationship and the fact that the DHT is an alternative to the discrete Fourier transform (DFT) for real data, a single module with a little or no modification can be used to carry out the forward and inverse M-D DFTs for real- or complex-valued data and M-D DHTs. Thus, the same module (with a little or no modification) can be used to cover all domains of applications that involve the DFTs or DHTs.     

New radix-(2/spl times/2/spl times/2)/(4/spl times/4/spl times/4) and radix-(2/spl times/2/spl times/2)/(8/spl times/8/spl times/8) DIF FFT algorithms for 3-D DFT

In this paper, new three-dimensional (3-D) radix-(2/spl times/2/spl times/2)/(4/spl times/4/spl times/4) and radix-(2/spl times/2/spl times/2)/(8/spl times/8/spl times/8) decimation-in-frequency (DIF) fast Fourier transform (FFT) algorithms are developed and their implementation schemes discussed. The algorithms are developed by introducing the radix-2/4 and radix-2/8 approaches in the computation of the 3-D DFT using the Kronecker product and appropriate index mappings. The butterflies of the proposed algorithms are characterized by simple closed-form expressions facilitating easy software or hardware implementations of the algorithms. Comparisons between the proposed algorithms and the existing 3-D radix-(2/spl times/2/spl times/2) FFT algorithm are carried out showing that significant savings in terms of the number of arithmetic operations, data transfers, and twiddle factor evaluations or accesses to the lookup table can be achieved using the radix-(2/spl times/2/spl times/2)/(4/spl times/4/spl times/4) DIF FFT algorithm over the radix-(2/spl times/2/spl times/2) FFT algorithm. It is also established that further savings can be achieved by using the radix-(2/spl times/2/spl times/2)/(8/spl times/8/spl times/8) DIF FFT algorithm.     

A new radix-2/8 FFT algorithm for length-q/spl times/2/sup m/ DFTs

In this paper, a new radix-2/8 fast Fourier transform (FFT) algorithm is proposed for computing the discrete Fourier transform of an arbitrary length N=q/spl times/2/sup m/, where q is an odd integer. It reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FFT algorithms. It is shown that the arithmetic complexity (multiplications+additions) of the proposed algorithm is, in most cases, the same as that of the existing split-radix FFT algorithm. The basic idea behind the proposed algorithm is the use of a mixture of radix-2 and radix-8 index maps. The algorithm is expressed in a simple matrix form, thereby facilitating an easy implementation of the algorithm, and allowing for an extension to the multidimensional case. For the structural complexity, the important properties of the Cooley-Tukey approach such as the use of the butterfly scheme and in-place computation are preserved by the proposed algorithm.     

A blind audio watermarking technique based on a parametric quantization index modulation

In this paper, we propose an efficient transform-based blind audio watermarking technique by introducing a parametric quantization index modulation (QIM). Theoretical expressions for the signal to watermark ratio and probability of error are derived and then used in an optimization technique based on the Lagrange multipliers method to find the optimal values for the parameters of the parametric QIM that ensure the imperceptibility while maximizing the robustness under an additive white Gaussian noise (AWGN) attack. Moreover, a fast scheme for the implementation of the proposed watermarking technique is developed and an efficient procedure is suggested to find the interval for the best selection of the watermark embedding positions that provide a good trade-off between the effects of high and low pass filtering attacks. The parameters of the resulting optimal parametric QIM coupled with the embedding positions constitute a highly robust secret key for the proposed watermarking technique. We also carry out several experiments to show the usefulness of the theoretical analysis presented in the paper and compare the proposed technique with other existing QIM-based watermarking techniques by considering known attacks such as AWGN, re-quantization, resampling, low/high pass filtering, amplitude scaling and common lossy compressions.

     

A new split-radix FHT algorithm for length-q*2/sup m/DHTs

In this paper, a new split-radix fast Hartley transform (FHT) algorithm is proposed for computing the discrete Hartley transform (DHT) of an arbitrary length N=q*2/sup m/, where q is an odd integer. The basic idea behind the proposed FHT algorithm is that a mixture of radix-2 and radix-8 index maps is used in the decomposition of the DHT. This idea and the use of an efficient indexing process lead to a new decomposition different from that of the existing split-radix FHT algorithms, since the existing ones are all based on the use of a mixture of radix-2 and radix-4 index maps. The proposed algorithm reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FHT algorithms. It is shown that the arithmetic complexity (multiplications+additions) of the proposed algorithm is, in almost all cases, the same as that of the existing split-radix FHT algorithm for length- q*2/sup m/ DHTs. Since the proposed algorithm is expressed in a simple matrix form, it facilitates an easy implementation of the algorithm, and allows for an extension to the multidimensional case.     

A new proposed algorithm of arbitrary radix for the computation of the 2D DFT

In this paper, we propose a new approach for computing 2D FFT's that are suitable for implementation on a systolic array architecture. Our algorithm is derived in this paper from a Cooley decimation‐in‐time algorithm by using an appropriate indexing process. It is proved that the number of multiplications necessary to compute our proposed algorithm is significantly reduced while the number of additions remains almost identical to that of conventional 2D FFT's. Comparison results show the good performance of the proposed 2D FFT algorithm against the row‐column FFT transform. Copyright © 1999 John Wiley & Sons, Ltd.