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Journal ArticleDOI

16-Point Reversible Integer Discrete Fourier Transform With 12 Control Bits

01 Feb 2010-IEEE Transactions on Signal Processing (IEEE)-Vol. 58, Iss: 2, pp 912-916
TL;DR: This correspondence discusses the reversible integer 16-point discrete Fourier transform (RiDFT) which uses integer operations with control bits and the integer approximation of the transform with eight control bits with additional two lifting schemes.
Abstract: This correspondence discusses the reversible integer 16-point discrete Fourier transform (RiDFT) which uses integer operations with control bits The decomposition of the RiDFT is based on the paired representation, when the Fourier transform is split recursively into a set of short transforms of orders 8, 4, 2, and 1 Control bits allow for inverting the integer approximations of multiplications by twiddle factors The proposed 16-point RiDFT uses 16 operations of real multiplication and 62 additions The integer approximation of the transform with eight control bits with additional two lifting schemes, which requires two more multiplications, is also considered
Citations
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Proceedings ArticleDOI
23 Aug 2010
TL;DR: The 2-D reversible integer discrete Fourier transform (RiDFT), which is based on the concept of the paired representation of the 2- D image, which is referred to as the unique2-D frequency and 1-D time representation, is described.
Abstract: This paper describes the 2-D reversible integer discrete Fourier transform (RiDFT), which is based on the concept of the paired representation of the 2-D image, which is referred to as the unique 2-D frequency and 1-D time representation. The 2-D DFT of the image is split into a minimum set of short transforms, and the image is represented as a set of 1-D signals. The paired 2-DDFT involves a few operations of multiplication that can be approximated by integer transforms, such as one-point transforms with one control bit. 24 control bits are required to perform the 8×8-point RiDFT, and 264 control bits for the 16×16-point 2-D RiDFT of real inputs. The fast paired method of calculating the 1-D DFT is used. The computational complexity of the proposed 2-D RiDFTs is comparative with the complexity of the fast 2-D DFT.

2 citations


Cites methods from "16-Point Reversible Integer Discret..."

  • ...With four additional control bits, the inverse integer 16-point DFT can be performed (see [12] for detail)....

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References
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Journal ArticleDOI
TL;DR: Two approaches to build integer to integer wavelet transforms are presented and the precoder of Laroiaet al., used in information transmission, is adapted and combined with expansion factors for the high and low pass band in subband filtering.

1,269 citations


"16-Point Reversible Integer Discret..." refers methods in this paper

  • ...We now consider for comparison the application of the three-step lifting schemes [5], [6] for integer approximation of two rotations which represent multiplications by factors and , instead of the multiplications and with control bits....

    [...]

Journal ArticleDOI
TL;DR: Numerical simulations suggest that the exact SR-LS and SRD-LS estimates outperform existing approximations of the SR- LS and SRd-LS solutions as well as approximated solutions which are based on a semidefinite relaxation.
Abstract: We consider least squares (LS) approaches for locating a radiating source from range measurements (which we call R-LS) or from range-difference measurements (RD-LS) collected using an array of passive sensors. We also consider LS approaches based on squared range observations (SR-LS) and based on squared range-difference measurements (SRD-LS). Despite the fact that the resulting optimization problems are nonconvex, we provide exact solution procedures for efficiently computing the SR-LS and SRD-LS estimates. Numerical simulations suggest that the exact SR-LS and SRD-LS estimates outperform existing approximations of the SR-LS and SRD-LS solutions as well as approximations of the R-LS and RD-LS solutions which are based on a semidefinite relaxation.

538 citations

Journal ArticleDOI
TL;DR: In this article, a concept of integer fast Fourier transform (IntFFT) for approximating the discrete Fourier Transform (DFT) is introduced, where the lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures.
Abstract: A concept of integer fast Fourier transform (IntFFT) for approximating the discrete Fourier transform is introduced. Unlike the fixed-point fast Fourier transform (FxpFFT), the new transform has the properties that it is an integer-to-integer mapping, is power adaptable and is reversible. The lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations. Split-radix FFT is used to illustrate the approach for the case of 2/sup N/-point FFT, in which case, an upper bound of the minimal dynamic range of the internal nodes, which is required by the reversibility of the transform, is presented and confirmed by a simulation. The transform can be implemented by using only bit shifts and additions but no multiplication. A method for minimizing the number of additions required is presented. While preserving the reversibility, the IntFFT is shown experimentally to yield the same accuracy as the FxpFFT when their coefficients are quantized to a certain number of bits. Complexity of the IntFFT is shown to be much lower than that of the FxpFFT in terms of the numbers of additions and shifts. Finally, they are applied to noise reduction applications, where the IntFFT provides significantly improvement over the FxpFFT at low power and maintains similar results at high power.

165 citations

Book
31 Jul 2003
TL;DR: This reference presents a more efficient, flexible, and manageable approach to unitary transform calculation and examines novel concepts in the design, classification, and management of fast algorithms for different transforms in one-, two-, and multidimensional cases.
Abstract: This reference presents a more efficient, flexible, and manageable approach to unitary transform calculation and examines novel concepts in the design, classification, and management of fast algorithms for different transforms in one-, two-, and multidimensional cases. Illustrating methods to construct new unitary transforms for best algorithm selection and development in real-world applications, the book contains a wide range of examples to compare the efficacy of different algorithms in a variety of one-, two-, and three-dimensional cases. Multidimensional Discrete Unitary Transforms builds progressively from simple representative cases to higher levels of generalization.

97 citations

Journal ArticleDOI
TL;DR: A two-dimensional (2-D) integer discrete cosine transform is proposed, which needs only integer operations and shifts and is nonseparable and requires a far fewer number of operations than that used by the corresponding row-column 2-D integer discrete Cosine transform.
Abstract: A method is proposed to factor the type-II discrete cosine transform (DCT-II) into lifting steps and additions. After approximating the lifting matrices, we get a new type-II integer discrete cosine transform (IntDCT-II) that is float-point multiplication free. Based on the relationships among the various types of DCTs, we can generally factor any DCTs into lifting steps and additions and then get four types of integer DCTs, which need no float-point multiplications. By combining the polynomial transform and the one-dimensional (1-D) integer cosine transform, a two-dimensional (2-D) integer discrete cosine transform is proposed. The proposed transform needs only integer operations and shifts. Furthermore, it is nonseparable and requires a far fewer number of operations than that used by the corresponding row-column 2-D integer discrete cosine transform.

83 citations