Timeline for Predicting the peak "amplitude" of a damped sine wave in the frequency spectrum with FFT
Current License: CC BY-SA 4.0
24 events
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| Mar 16 at 2:24 | comment | added | Piyush Grover | You are looking for Laplace transform of $Sin(\omega t)$ at $s= \alpha+i\omega$. Simply consult the Laplace transform table | |
| Mar 15 at 17:34 | history | edited | ACR | CC BY-SA 4.0 |
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| Mar 15 at 17:17 | review | Close votes | |||
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| Mar 15 at 16:00 | comment | added | ACR | Downvoting without stating any legitimate reason serves no purpose at all nor affects anything. | |
| Mar 14 at 23:22 | vote | accept | ACR | ||
| Mar 14 at 23:22 | history | edited | ACR | CC BY-SA 4.0 |
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| Mar 14 at 20:52 | history | edited | ACR | CC BY-SA 4.0 |
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| Mar 14 at 20:06 | history | edited | ACR | CC BY-SA 4.0 |
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| Mar 14 at 19:14 | comment | converted from answer | Troy Handlovic |
I simulated in MATLAB using the built-in fft command for the Fourier transform where the exponentially dampened signal is defined as A*exp(-alpha* t) .*sin(2 * pi * frequency * t). Regardless of the size of the alpha value, there is no shift in the frequency domain at all.
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| Mar 14 at 15:37 | vote | accept | ACR | ||
| Mar 14 at 16:50 | |||||
| Mar 14 at 12:17 | comment | added | ACR | Yes, but the figures are made from FFT for sampled data and finite duration, this is why I removed the $X(\omega)$ | |
| Mar 14 at 9:28 | answer | added | Carlo Beenakker | timeline score: 4 | |
| Mar 14 at 6:14 | comment | added | Michael Engelhardt | Ok, now in early versions of your post you had $X(\omega)$. So, isn't the maximal value of the magnitude of $X(\omega)$ exactly what you're looking for? | |
| Mar 14 at 5:15 | comment | added | ACR | Let us look at the figures, D (raw signal) and E (two sided frequency spectrum obtained via FFT of the raw signal). The y axis of E is the magnitude of the FFT. Looking at E, we have two peaks. I want to "predict" their maximum y-value, provided I know all the values in $x(t)$, i.e., I know $A$, and $\alpha$ and $\omega_0$. In spectroscopic terms, the maximum y-value would be termed as peak height. | |
| Mar 14 at 4:44 | comment | added | Michael Engelhardt | We seem to be going in circles here. What is a "peak height of the $x(t)$ after FFT"? Do you want $A$ given $X(\omega )$? Or do you want the maxima of $X(\omega )$ given $x(t)$? Can you please specify what you want in terms of objects in your formulae/figures instead of wording which I, frankly, find unintelligible? | |
| Mar 14 at 3:56 | comment | added | ACR | The FFT spectrum is all correct. There is no issue in calculations there. My interest is in knowing how can I predict the peak heights of the $x(t)$ after FFT. I cannot find a relation that can allow us to predict the peak height in the frequency spectrum. In this simple sine case it was $A/2$. In the exponential case $\alpha$ is involved. | |
| Mar 14 at 2:24 | comment | added | Michael Engelhardt | Oh, so you're simply interested in calculating the Fourier transform of your $x(t)$? Well - you had an expression for $X(\omega )$ in earlier versions of your question ... but that is not what you want? Are you saying that your graphs E) or F) don't agree with that $X(\omega )$? Might that be because your FFT literally calculates the Fourier transform of your graph D) over the shown finite interval of $5s$, pretending that that graph periodically repeats outside that interval? That would of course be different from the Fourier transform over the whole real (half-)axis ... | |
| Mar 13 at 20:11 | comment | added | ACR | @MichaelEngelhardt, I have now added the figure and hopefully the query is clear. You can see that we have peaks now in the frequency spectrum and they have a corresponding height. In the simple sine case, their height was A/2 but how can we predict the exponentially decaying sine wave case. | |
| Mar 13 at 20:09 | history | edited | ACR | CC BY-SA 4.0 |
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| Mar 13 at 18:44 | comment | added | Michael Engelhardt | I still don't understand what you're trying to find. What is a "peak height" or a "magnitude frequency spectrum"? Can you point out which property of your original expression $x(t)$ you're interested in? And again, what are the given data that you want to derive said property from? | |
| Mar 13 at 17:28 | history | edited | ACR | CC BY-SA 4.0 |
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| Mar 13 at 17:25 | comment | added | ACR | I think the key question is that if we have an exponentially decaying sine wave with amplitude A, how can I predict the peak heights (=magnitude frequency spectrum) in the FFT. For a non-decaying sine wave with amplitude A, in the frequency spectrum, the peak height is also A/2. | |
| Mar 13 at 16:48 | comment | added | Michael Engelhardt | I find it hard to make out what exactly you're asking here. What is given, what are you trying to determine? Are you saying that you have $X(\omega )$ at a discrete set of points (from a FFT), and you're trying to determine $A$? | |
| Mar 13 at 16:31 | history | asked | ACR | CC BY-SA 4.0 |