scholarly journals Chirp Signal Transform and Its Properties

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mio Horai ◽  
Hideo Kobayashi ◽  
Takashi G. Nitta
Keyword(s):  
Fourier Transform ◽  
Differential Equations ◽  
Discrete Fourier Transform ◽  
Chirp Signal ◽  
Constant Amplitude ◽  
Continuous Version ◽  
Fundamental Properties ◽  
Chirp Signals ◽  
Zero Autocorrelation

The chirp signalexp(iπ(x-y)2)is a typical example of CAZAC (constant amplitude zero autocorrelation) sequence. Using the chirp signals, the chirp z transform and the chirp-Fourier transform were defined in order to calculate the discrete Fourier transform. We define a transform directly from the chirp signals for an even or odd numberNand the continuous version. We study the fundamental properties of the transform and how it can be applied to recursion problems and differential equations. Furthermore, whenNis not prime and  N=ML, we define a transform skippedLand develop the theory for it.

Continuous quaternion fourier and wavelet transforms

2014 ◽  
Vol 12 (04) ◽  
pp. 1460003 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino ◽  
Rémi Vaillancourt
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Equation ◽  
Wavelet Transforms ◽  
Quaternion Algebra ◽  
Two Dimensional ◽  
Quaternion Fourier Transform ◽  
Fundamental Properties

A two-dimensional (2D) quaternion Fourier transform (QFT) defined with the kernel [Formula: see text] is proposed. Some fundamental properties, such as convolution, Plancherel and vector differential theorems, are established. The heat equation in quaternion algebra is presented as an example of the application of the QFT to partial differential equations. The wavelet transform is extended to quaternion algebra using the kernel of the QFT.

Numerical Solution of Nonlinear Kaup-Kupershmit Equation, KdV-KdV and Hirota-Satsuma Systems

2012 ◽  
Vol 13 (7-8) ◽  
Author(s):  
Akbar Mohebbi
Keyword(s):  
Fourier Transform ◽  
Linear Operator ◽  
Differential Equations ◽  
Numerical Solution ◽  
Discrete Fourier Transform ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Time Stepping

AbstractIn this work we investigate the numerical solution of Kaup-Kupershmit (KK) equation, KdV-KdV and generalized Hirota-Satsuma (HS) systems. The proposed numerical schemes in this paper are based on fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original partial differential equations (PDEs) with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which will be solved with fourth order time-stepping methods. After transforming the equations to a system of ODEs, the linear operator in KK and HS equation is diagonal but in KDV-KDV equation is not diagonal. However for KDV-KDV system which is the focus of this paper, we show that the exponential of linear operator and related inverse matrix have definite structure which enable us to implement the methods such as diagonal case. Comparing numerical solutions with exact traveling wave solutions demonstrates that those methods are accurate and readily implemented.

scholarly journals The Formalization of Discrete Fourier Transform in HOL

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Zhiping Shi ◽  
Yupeng Zhang ◽  
Yong Guan ◽  
Liming Li ◽  
Jie Zhang
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Order Logic ◽  
Theorem Prover ◽  
Formal Definition ◽  
Accurate Analysis ◽  
Higher Order Logic ◽  
Safety Critical ◽  
Fundamental Properties ◽  
Definition Of

Traditionally, Discrete Fourier Transform (DFT) is performed with numerical or symbolic computation, which cannot guarantee 100% accurate analysis which may be necessary for safety-critical applications. Machine theorem proving is one of the formal methods that perform accurate analysis with completeness to some extent. This paper proposes the formalization of DFT in a higher-order logic theorem prover named HOL. We propose the formal definition of DFT and verify the fundamental properties of DFT. Two case studies are presented to illustrate usefulness and correctness of the formalized DFT, including formal verifications of Fast Fourier Transform (FFT) and cosine frequency shift.

APPLICATION OF UNCONVENTIONAL METHODS FOR FREQUENCY ANALYSIS IN ACOUSTICS

Akustika ◽  
2020 ◽  
Vol 36 (36) ◽  
pp. 25-32
Author(s):  
Jaroslav Smutný ◽  
Dušan Janoštík ◽  
Viktor Nohál
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Frequency Analysis ◽  
Mathematical Analysis ◽  
Measured Data ◽  
Application Area ◽  
Acoustic Response ◽  
Recommendations For Practice ◽  
Unconventional Methods ◽  
Frequency Area

The goal of this study is to familiarize a wider professional public with not fully known procedures suitable for processing measured data in the frequency area. Described is the use of the so-called Multi-taper method to analyze the acoustic response. This transformation belongs to a group of nonparametric methods outgoing from discrete Fourier transform, and this study includes its mathematical analysis and description. In addition, the use of respective method in a specific application area and recommendations for practice are described.

Sign in / Sign up

Export Citation Format

Share Document