I need to transform a function from Fourier domain to Laplace domain. This function's equivalent in time domain is too much complicated and MATLAB can only estimate it with considerable errors. On the other hand, this function includes some derivatives in time domain that should be applied to an unknown signal. MATLAB cannot distinguish derivative operators in the inverse Fourier transform, so gives me a function which is only dependent to time and the dependence to the derivative of the unknown input signal cannot be considered in inverse Fourier transform. In other words, "d/dt"operator with no function to be differentiated and similar higher order derivative operators could not be distinguished by MATLAB in this way. I know that the dynamic behavior of the system is applied with convolution integral in time domain but I cannot include convolution in my simulated system. As a result, I decided to find this function's equivalent in Laplace domain, then find the Taylor series expantion for this function in Laplace domain, and then replace S^1, S^2, S^3, and etc with d/dt, d^2/dt^2, d^3/dt^3, and etc. So I need to transform this function from Fourier domain to Laplace directly as the first step while I could not find any way except transforming it from Fourier to time domain and then transforming it from time to Laplace.
I would appreciate it if you could also suggest any other solutions for solving this problem.
Dear Dr. Harald Kirchsteiger
Thank you for your answer. It's a continues time function. You are right. If s is purely imaginary, the Laplace domain function can be found by replacing each j \omega with s. Indeed I didn't ask my question correctly. Let's tell more details about my problem.
The mentioned frequency domain function describes the impedance of a nonlinear electric element. I wanted to assume this function as a transfer function between voltage and current and simulate this element's behavior in the time domain in a nonlinear power-electronic circuit which results in applying arbitrary voltage waveform to the element which is not single-frequency and sinusoidal. At first, I found the Taylor expansion of this function in the Fourier domain. The coefficients of the Taylor series were complex. When I transformed this function to the time domain by assuming each "j \omega" as "d/dt", Some extra "j"s remained in the time domain function that I couldn't deal with them. If the voltage was single-frequency or at least contained finite number of frequencies, It would be possible to find the real current by finding the phase of each sinusoidal waveform of current based on imaginary to real part ratio of each sinusoidal response of the system. But the voltage isn't in this form.
By replacing each j \omega with s, and then finding the Taylor series and transforming it from Laplace to time, the same function that I had previously found in time domain would appear and the imaginary part of time-domain function would not be eliminated. I was hopeful that by using Laplace transform I won't face a complex relation in time domain. Indeed, my main problem is not being able to find the real waveform of the current in the time domain while the mathematical tools of MATLAB that I know, only give me complex responses that could not be dealt with due to not being the system single-frequency.
Thank you for your time.
Fourier and Laplace domains are similar. Fourier analysis is a particular case of Laplace. In Laplace domain the frequency variable is the complex valus sigma + j * omega = sigma + j * (2*pi*f), while in Fourier domain the frequency variable is simply f. So, In Fourier analysis your system's response is along the 'frequency' axis, f (or omega, if you prefer), while in Laplace analysis your system's response is ove the complex plane with sigma being in one direction and omega perpendicular to that. Laplace response is the surface of a 3D plot where the 'floor is the plane sigma + j * omega (think of a cake or a circus and your response is the canvas). Fourier response is a 'cut' of that cake.
Dear Dr. Fernando Soares Schlindwein
Thank you for your comprehensive and intuitive explanation about the relation between Laplace and Fourier Analysis. According to your descriptions, not only it is not possible to transfer a function from Fourier domain to Laplace in MATLAB, but also it is mathematically impossible unless applying some special restricting conditions to the system. Indeed, the system response in the Laplace domain needs some extra information that could not be extracted from Fourier domain transfer function. Approving Dr. Kirchsteiger's words about being it impossible to generalize from Fourier transform to Laplace transform.
So I need to find another solution for solving this problem!
Thank's again for the answer that you provided for me.
The way to say 'thank you' in RG is to click 'recommend'. I appreciated you writing back.
Best of luck!
Fernando
Of course Dr. Schlindwein.
I had forgotten to do it. Thank you for reminding me.
Best regards.
I have not still understood why, but the problem was solved by calculating the Taylor series of the Fourier-domain function at zero point. I had calculated the Taylor series of the Fourier-domain function at a nonzero point and this resulted in appearing complex coefficients for all of the nomials of the Taylor expansion both in Fourier and time domains. When I find the Taylor series at zero point, amazingly, the coefficients of the nomials with even powers of \omega become real and the coefficients of the nomials with odd powers become imaginary. Therefore, each of the terms can be written as a real coefficient of (j \omega)^n. When such a system in Fourier domain is transformed to time domain, a pure real relation including the derivatives of the input signal is achieved in time domain.
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