Timeline for What does the exponential term in the Fourier transform mean?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 31, 2020 at 11:18 | comment | added | Bob K | @Olli - Thanks for the editorial help with my deltas. I thought it didn't look quite right, but I didn't realize why. | |
| Aug 31, 2020 at 4:48 | comment | added | Olli Niemitalo | I don't think it makes sense to argue how many frequencies a general signal contains, without agreeing about what "reasonable" decomposition into periodic functions is meant. A frequency is then just a shorthand expression for a periodic component of a frequency. A reasonable decomposition will not, for example, include components that completely cancel one another, or components that are identical. | |
| Aug 31, 2020 at 4:12 | history | edited | Olli Niemitalo | CC BY-SA 4.0 |
Change δ(ω0) to δ(ω−ω0)
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| Aug 26, 2020 at 1:14 | comment | added | Cedron Dawg | You seemed to have missed the point of my roots example. A pure real tone can be said to have a frequency, singular. This does not mean it is not composed of two complex pure tones, each having its own frequency. When the pure real tone is modulated by a pure complex tone, the fact that there are actually two frequencies within the cosine becomes readily apparent. The other point, you are not "adding" or "subtracting" information in your examples. There is no change in the number of parameters describing the system. (We have reached the chat prompt which means I am done commenting.) | |
| Aug 26, 2020 at 0:43 | comment | added | Bob K | Frequency is not defined by Fourier mathematics any more than air pressure is defined by a barometer. And my casual use of Dirac delta is intentional, so as not to lose the wheat in the chaff. | |
| Aug 26, 2020 at 0:42 | comment | added | Bob K | $e^{ix} = \cos(x) + i\sin(x)\ $ is the addition of information (to $\cos(x)$). $e^{ix} + e^{-ix} = 2\cos(x) + i\sin(x) - i\sin(x) = 2\cos(x)\ $ is subtraction/loss of information. | |
| Aug 26, 2020 at 0:32 | comment | added | Cedron Dawg | Sort of similar. Q: How many roots does $x^2$ have? A: Two, they are both zero. | |
| Aug 26, 2020 at 0:10 | comment | added | Cedron Dawg | This isn't about function decomposition. You could have just as readily said $f(x) = x^2 = x^{3/2} x^{1/2}$ for just as specious of an argument. The phrase "contains both frequencies" is in context of the FT (continuous in this case). If $cos$ only had one frequency there would only be one non-zero value in the spectrum. | |
| Aug 26, 2020 at 0:00 | comment | added | Bob K | @Cedron - Consider a function $f(x) = x^2 +ix$. $\ $And $\ \therefore\ f(-x) = x^2 -ix$ $\ x^2 = \tfrac{1}{2}(f(x) + f(-x))\ $ Should we conclude that $x^2$ is something more than just a function on the real number line? It is secretly made of two complex functions? If so, which two?... because I could just as easily have defined $f(x)$ as $x^2 +ix^3$. | |
| Aug 25, 2020 at 22:31 | comment | added | Cedron Dawg | "That does not mean $cos(\omega_0 t)$ contains both frequencies, because $\omega_0$ can have only one value." No. The cosine is the sum of two complex pure tones of opposite frequencies (two distinct values). What you can't tell is the sign of $\omega_0$. Either is a valid interpretation, similar to picking a square root. So by convention, frequencies for real valued pure tones are considered positive. | |
| Aug 25, 2020 at 22:25 | history | edited | lennon310 | CC BY-SA 4.0 |
added 4 characters in body
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| Aug 25, 2020 at 22:07 | review | Late answers | |||
| Aug 25, 2020 at 22:24 | |||||
| Aug 25, 2020 at 21:48 | history | answered | Bob K | CC BY-SA 4.0 |