Mathematica 2d Fourier Transform
Francisca Noggles
You can add noconds=False to the transform routines to find under which conditions the transformation integral converges. As asmeurer says, we literally compute the defining integral, so you will never see delta functions coming up. In the case of the fourier transform of cos, the conditions are a convoluted way of saying "never", which unfortunately sympy does not recognise. (I.e. the algorithm says something like "the integral is zero if blah", and blah never holds.)
For the fourier transform of the step function, the conditions seem to be saying that this works if z has negative argument (angle), not too big. Note that this is indeed when the transform integral converges (b/c you need to pick up a falling exponential term over the positive reals). I don't have time to think about whether the computation is correct in this case.
I am practicing computing Fourier transform integrals, and am wondering if Wolfram Alpha or Mathematica has any ability to show the steps it takes to get to a solution. Once I struggle enough, I'd just like to see how Mathematica computed it -- to learn and perhaps even see a better way.
I believe that Mathematica concentrates on being able to find integrals of, sometimes very difficult, problems and the methods and algorithms it uses to do this do not lend themselves to showing the steps that you might take to do this by hand. Thus I don't think you will get the steps that you are looking for with either WolframAlpha or Mathematica.
There is one possibility that you might explore. There is a free package at that you might be able to download and use within your Mathematica session. That has an option to show the steps that it used to perform the integration. I don't know if the results will be what you are looking for, but you could try it and see. One limitation is that I believe it only does indefinite integration and you are probably looking at definite integrals for your fourier transform. But you might still be able to get some insight into the process and be able to apply that.
Thanks Bill -- I will see if I can get Rubi working for Mathematica. Sometimes I get stuck, and perhaps I can use that package to help me see the next step -- that is what I want to use something like this for. Eventually, I hope to ditch it all together once I have more practice.
The Fourier transform is without any doubt an essential tool in applications such as signal and image processing.It is an integral transform that can be discretized in an algorithm called the Discrete Fourier Transform (DFT).Basically, that means replacing the integral by a numerical quadrature formula.A smart implementation turns this into a Fast Fourier Transform (FFT) procedure that has become awidely used numerical computational techniquethat is stable (because it is an orthogonal transform) and fast (the typical order is N log N for signals of size N).So it may be surprising that after decades of intensive use and analysis, something new is added.
The one-dimensional Fourier transform can be considered as a rotation over ninety degrees in an orthogonaltime-frequency representation of the data. The original signal given as a function of time on a time axisis transformed into its spectral contentsthat is as a function of frequency and the frequency axis is orthogonal to the time axis.In optical systems, certain lens combinations can transform (rotate) the data over any angle,which is called a fractional Fourier transform (FrFT).The classical Fourier transforms is a special case of the FrFT, which in turn is a special caseof an even more general linear fractional transforms (LFT).The latter kind of transforms has applications in quantum theory.The FrFT is like computing a fractional power of the ordinary Fourier transform.The extended Fourier transform (XFT) discussed in this book emerged by defining a discrete version ofthe Fourier transform in a slightly different way than what is done in the classical DFT.In the XFT, it is implemented as a special case of a discrete version of the FrFT.The result is a procedure that is as fast as the FFT, but slightly more accurate.
In the first introductory chapter, the ordinary DFT is recalled and it is illustrated that for non-periodic functionsand when the N is not very large, some errors will occur as a consequence of the discretization.The DFT occurs as a unitary matrix that multiplies the data vector of the sampled signal to generate the frequency vector representation of the same signal.In this introduction also the two-dimensional transform is considered.It is illustrated that the effect of a translation in the transform, is a cyclic shift of the image, but it also has some other side effects.
The next two chapters give a survey of many applications where the XFT can be used, mainly (partial)differential equations, and that includes usual derivatives as well as fractional derivatives (and integrals),and other fractional transforms (Laplace, Hilbert, derivative,...) and the generalization to fractional Fourier transformsand linear canonical transforms.The two appendices describe the mathematica and the matlab codes for the implementation of the XFT.
The book gives a concise survey of problems with classical DFT and introduces the XFT as an alternative.The performance is clearly illustrated with many applications and above all, the code to compute theXFT (and related algorithms) are provided both in mathematica and matlab, so that it is possible to immediatelystart experimenting with the methods.Recommended for everyone who uses Fourier transforms in a computational context and wants to learnabout its extended XFT alternative and the theory behind it.
The wavelet transform is an useful mathematical tool. It is a mapping of a time signal to the time-scale joint representation. The wavelet transform is generated from a wavelet function by dilation and translation. This wavelet function satisfies an admissible condition so that the original signal can be reconstructed by the inverse wavelet transform. In this study, we firstly give some basic properties of the weighted variable exponent amalgam spaces. Then we investigate the convergence of the θ-means of f in these spaces under some conditions. Finally, using these results the convergence of the inverse continuous wavelet transform is considered in these spaces.
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