Real Discrete Fourier Transform Research Articles

New Algorithm for Real-Valued Fourier Transform

This paper presents a direct algorithm for fast real discrete Fourier transform (RDFT) computing, using the discrete Fourier transform (DFT) conjugate symmetric property to reduce redundancies. In RDFT, all the input and output signals were real, which differed from complex DFT. Therefore, the structure of the proposed algorithm showed only real-data operations. The developed algorithm showed the desired properties, such as in-place computation, regularity, simplicity, and arithmetic operations reduction. The RFFT performance was compared with other related transforms, such as the fast Hartley transform (FHT) for the computation in the radix-2 algorithm. It was found that FHT showed the best performance in terms of arithmetic complexity.

An efficient blind color image watermarking algorithm in spatial domain combining discrete Fourier transform

For achieving the purpose of rapid protection of color image copyright better, a novel color image blind watermarking algorithm is proposed in this paper, which integrates discrete Fourier transform (DFT) in spatial domain. According to the unique feature of the direct current (DC) component of DFT, two procedures of embedding and blind extracting watermark are accomplished in spatial domain utilizing the correlation principle between adjacent pixel blocks and different steps in different channels of color image, without real DFT. The experiment results demonstrate that the presented algorithm has strong robustness and high real time under the premise of better watermark invisibility, and solves the puzzles of slow running-speed and weak robustness of large capacity color image digital watermarking algorithm.

DSP for MATLAB™ and LabVIEW™ III: Digital Filter Design

This book is Volume III of the series DSP for MATLAB™ and LabVIEW™. Volume III covers digital filter design, including the specific topics of FIR design via windowed-ideal-lowpass filter, FIR highpass, bandpass, and bandstop filter design from windowed-ideal lowpass filters, FIR design using the transition-band-optimized Frequency Sampling technique (implemented by Inverse-DFT or Cosine/Sine Summation Formulas), design of equiripple FIRs of all standard types including Hilbert Transformers and Differentiators via the Remez Exchange Algorithm, design of Butterworth, Chebyshev (Types I and II), and Elliptic analog prototype lowpass filters, conversion of analog lowpass prototype filters to highpass, bandpass, and bandstop filters, and conversion of analog filters to digital filters using the Impulse Invariance and Bilinear Transform techniques. Certain filter topologies specific to FIRs are also discussed, as are two simple FIR types, the Comb and Moving Average filters. The entire series consists of four volumes that collectively cover basic digital signal processing in a practical and accessible manner, but which nonetheless include all essential foundation mathematics. As the series title implies, the scripts (of which there are more than 200) described in the text and supplied in code form here will run on both MATLAB™ and LabVIEW™. The text for all volumes contains many examples, and many useful computational scripts, augmented by demonstration scripts and LabVIEW™ Virtual Instruments (VIs) that can be run to illustrate various signal processing concepts graphically on the user's computer screen. Volume I consists of four chapters that collectively set forth a brief overview of the field of digital signal processing, useful signals and concepts (including convolution, recursion, difference equations, LTI systems, etc), conversion from the continuous to discrete domain and back (i.e., analog-to-digital and digital-to-analog conversion), aliasing, the Nyquist rate, normalized frequency, sample rate conversion and Mu-law compression, and signal processing principles including correlation, the correlation sequence, the Real DFT, correlation by convolution, matched filtering, simple FIR filters, and simple IIR filters. Chapter four of Volume I, in particular, provides an intuitive or first principle understanding of how digital filtering and frequency transforms work. Volume II provides detailed coverage of discrete frequency transforms, including a brief overview of common frequency transforms, both discrete and continuous, followed by detailed treatments of the Discrete Time Fourier Transform (DTFT), the z-Transform (including definition and properties, the inverse z-transform, frequency response via z-transform, and alternate filter realization topologies including Direct Form, Direct Form Transposed, Cascade Form, Parallel Form, and Lattice Form), and the Discrete Fourier Transform (DFT) (including Discrete Fourier Series, the DFT-IDFT pair, DFT of common signals, bin width, sampling duration, and sample rate, the FFT, the Goertzel Algorithm, Linear, Periodic, and Circular convolution, DFT Leakage, and computation of the Inverse DFT). Volume IV, the culmination of the series, is an introductory treatment of LMS Adaptive Filtering and applications, and covers cost functions, performance surfaces, coefficient perturbation to estimate the gradient, the LMS algorithm, response of the LMS algorithm to narrow-band signals, and various topologies such as ANC (Active Noise Cancelling) or system modeling, Periodic Signal Removal/Prediction/Adaptive Line Enhancement (ALE), Interference Cancellation, Echo Cancellation (with single- and dual-H topologies), and Inverse Filtering/Deconvolution/Equalization. Table of Contents: Principles of FIR Design / FIR Design Techniques / Classical IIR Design

DSP for MATLAB™ and LabVIEW™ II: Discrete Frequency Transforms

This book is Volume II of the series DSP for MATLAB™ and LabVIEW™. This volume provides detailed coverage of discrete frequency transforms, including a brief overview of common frequency transforms, both discrete and continuous, followed by detailed treatments of the Discrete Time Fourier Transform (DTFT), the z -Transform (including definition and properties, the inverse z -transform, frequency response via z-transform, and alternate filter realization topologies (including Direct Form, Direct Form Transposed, Cascade Form, Parallel Form, and Lattice Form), and the Discrete Fourier Transform (DFT) (including Discrete Fourier Series, the DFT-IDFT pair, DFT of common signals, bin width, sampling duration and sample rate, the FFT, the Goertzel Algorithm, Linear, Periodic, and Circular convolution, DFT Leakage, and computation of the Inverse DFT). The entire series consists of four volumes that collectively cover basic digital signal processing in a practical and accessible manner, but which nonetheless include all essential foundation mathematics. As the series title implies, the scripts (of which there are more than 200) described in the text and supplied in code form here will run on both MATLAB™ and LabVIEW™. The text for all volumes contains many examples, and many useful computational scripts, augmented by demonstration scripts and LabVIEW™ Virtual Instruments (VIs) that can be run to illustrate various signal processing concepts graphically on the user's computer. Volume I consists of four chapters that collectively set forth a brief overview of the field of digital signal processing, useful signals and concepts (including convolution, recursion, difference equations, LTI systems, etc), conversion from the continuous to discrete domain and back (i.e., analog-to-digital and digital-to-analog conversion), aliasing, the Nyquist rate, normalized frequency, sample rate conversion and Mu-law compression, and signal processing principles including correlation, the correlation sequence, the Real DFT, correlation by convolution, matched filtering, simple FIR filters, and simple IIR filters. Chapter 4 of Volume I, in particular, provides an intuitive or "first principle" understanding of how digital filtering and frequency transforms work, preparing the reader for the present volume (Volume II). Volume III of the series covers digital filter design (FIR design using Windowing, Frequency Sampling, and Optimum Equiripple techniques, and Classical IIR design) and Volume IV, the culmination of the series, is an introductory treatment of LMS Adaptive Filtering and applications.

The DFT: An Owner's Manual for the Discrete Fourier Transform

Preface 1. Introduction. A Bit of History An Application Problems 2. The Discrete Fourier Transform (DFT). Introduction DFT Approximation to the Fourier Transform The DFT-IDFT pair DFT Approximations to Fourier Series Coefficients The DFT from Trigonometric Approximation Transforming a Spike Train Limiting Forms of the DFT-IDFT Pair Problems 3. Properties of the DFT. Alternate Forms for the DFT Basic Properties of the DFT Other Properties of the DFT A Few Practical Considerations Analytical DFTs Problems 4. Symmetric DFTs. Introduction Real sequences and the Real DFT (RDFT) Even Sequences and the Discrete Cosine Transform (DST) Odd Sequences and the Discrete Sine Transform (DST) Computing Symmetric DFTs Notes Problems 5. Multi-dimensional DFTs. Introduction Two-dimensional DFTs Geometry of Two-Dimensional Modes Computing Multi-Dimensional DFTs Symmetric DFTs in Two Dimensions Problems 6. Errors in the DFT. Introduction Periodic, Band-limited Input Periodic, Non-band-limited Input Replication and the Poisson Summation Formula Input with Compact Support General Band-Limited Functions General Input Errors in the Inverse DFT DFT Interpolation - Mean Square Error Notes and References Problems 7. A Few Applications of the DFT. Difference Equations - Boundary Value Problems Digital Filtering of Signals FK Migration of Seismic Data Image Reconstruction from Projections Problems 8. Related Transforms. Introduction The Laplace Transform The z- Transform The Chebyshev Transform Orthogonal Polynomial Transforms The Discrete Hartley Transform (DHT) Problems 9. Quadrature and the DFT. Introduction The DFT and the Trapezoid Rule Higher Order Quadrature Rules Problems 10. The Fast Fourier Transform (FFT). Introduction Splitting Methods Index Expansions (One ---> Multi-dimensional) Matrix Factorizations Prime Factor and Convolution Methods FFT Performance Notes Problems Glossary of (Frequently and Consistently Used) Notations References.

Fast algorithms for the discrete Fourier preprocessing transforms

It is pointed out that the two-stage Fourier algorithms are useful in a variety of applications in signal/image processing and recognition. The first stage, known as the DFPT (discrete Fourier preprocessing transform), has the potential of very fast and low-cost implementation, say, in a VLSI chip, as well as high-quality performance in applications without the second stage. The DFT (discrete Fourier transform), the RDFT (real DFT), or any other DFPT can be obtained if the second stage is completed. Since the second stage consists of independent blocks of circular correlations, they can be computed in fast sequential or parallel architectures. There are two groups of fast algorithms for the computation of DFPTs. The first group involves the representation of a DFPT in terms of skew-circular correlations (SCCs), which are then computed by fast SCC algorithms or parallel architectures such as semisystolic arrays. The second group involves the fast computation of a DFPT, including the DFT and the RDFT, by the implementationally simplest DFPT, such as class 2, case V DFPT. >

Fast computation of real discrete Fourier transform for any number of data points

In many applications, it is desirable to have a fast algorithm (RFFT) for the computation of the real discrete Fourier transform (RDFT) for any number of data points N. To achieve this, the two-factor Cooley-Tukey decimation-in-time and decimation-in-frequency RFFT algorithms are first developed and expressed in terms of matrix factorization using Kronecker products. This is generalized to any number of factors with arbitrary radices. Each factor M involves the computation of the size-M RDFT, which is carried out by the best size-M RFFT algorithm available. The RFFT algorithm for the case where M is a prime number is also developed. The RFFT algorithms are more efficient in the number of operations when the factors are arranged in a certain order, unlike the Cooley-Tukey complex FFT algorithms. which have the same number of operations for any order of the factors. >

Nonlinear matched filtering

A new class of nonlinear matched filters is discussed. These filters involve the transformation of the signal spectrum and the filter transfer function through a nonlinearity before they are multiplied in the transform domain. The resulting filter structures can be considered to be analogous to three-layer neural nets. They have better performance in terms of signal discrimination and lack of false correlation signals and artifacts than previously known filters. The matched filters are further subdivided into two major classes according to whether the filtering is based on a discrete Fourier transform (DFT) or a real discrete Fourier transform (RDFT). The DFT and the RDFT are approximations to the complex and real Fourier transforms, respectively. The RDFT-based filtering gives better performance in terms of signal discrimination and lack of false correlation signals and artifacts than the DFT-based filtering.

Representation of real discrete Fourier transform in terms of a new set of functions based upon Möbius inversion

The trigonometric functions sin(2πn/N) and cos(2πn/N) are transformed into a new set of basis functions using Mobius inversion of certain types of series. The new basis functions are number theoretic series. They are used to represent the real discrete Fourier transform (RDFT) in terms of 2 matrices of factorization. The first matrix, with elements 1, -1 and 0 is obtained by replacing cos(2πk/N) and sin(2πk/N) by μ(k/N + 1/4) and μ(k/N), where μ(x) is the bipolar rectangular wave function. The second matrix is block-diagonal where each block is a circular correlation and consists of the new basis functions. Some applications of the new representation are discussed.

Real discrete Fourier transform

The real discrete Fourier transform (RDFT) corresponds to the Fourier series for sampled periodic signals with sampled periodic frequency responses just as discrete Fourier transform (DFT) corresponds to the complex Fourier series for the same type of signals. RDFT has better performance than DFT in data compression and filtering for all signals in the sense that Pearl's measure for RDFT is less than Pearl's measure for DFT by an amount ΔW. RDFT also has better performance than DFT in the computation of real convolution because of the reduced number of operations, and the fact that forward and inverse transforms can be implemented with the same signal flowgraph, thereby facilitating hardware and software design.