Why do Fourier transforms use complex numbers?
Jan 10, 2013 · In this domain, the signal has two "properties" - magnitude and phase. If we want to get only the signal's "power" in a specific frequency bin, we indeed only need to take the absolute value of the Fourier transform, which is real. But, the Fourier transform gives the phase of each frequency as well.
harmonic analysis - Fourier transform of a complex exponential ...
Jun 6, 2017 · Fourier transform of a complex exponential. Ask Question Asked 7 years, 6 months ago. Modified 5 years, 8 ...
Inverse Fourier Transform gives a complex function when it …
Jul 27, 2015 · For this reason I think (contrary to what the title suggests) that the inverse Fourier transform should yield a complex function, it just so happens that this result can sometimes lie in the subset of real functions. This point has been made clear in the comments.
integration - Complex Frequency Shifting in Fourier Transform ...
Is there a more general relation where $\omega_0 \in \mathbb{C}$? I know that, physically speaking, having a complex frequency shift (or complex time shift) does not have a meaning, but mathematically it could transform a sinusoidal function, for example, into a hyperbolic one. In short, is there a quick mathematical trick to doing a complex shift?
Complex Fourier Transform - Mathematics Stack Exchange
The discrete Fourier transform in its original formulation is for complex inputs and outputs. And since all Fourier transforms are complex linear, you can combine the DFT of the real part and the DFT of the imaginary part to obtain the DFT of the non-separated complex input vector.
What is the spectrum of the Fourier transform operator?
I am asked to find the spectrum of the Fourier transform operator $\mathcal F:L^2(\mathbb R)\to L^2(\mathbb R)$, where $\mathcal Fu(\omega)=\int_\mathbb R \frac 1{\sqrt{2\pi}}u(x)e^{-ix\omega} dx$. I know that $\mathcal F^4=\mathrm{Id}$, and I read this thread already Two questions in spectral theory: the spectrum of the Fourier transform and ...
integration - Fourier transform of a real function is real ...
The definition of Fourier transform is that famous formula and will not necessarily produce real coefficients for a real function. But we should know that if the Fourier transform exists for a real function, we can also write that function with only real trigonometric functions.
Fourier Transform: why time reversal in freq domain is the same as ...
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complex analysis - Why does Fourier Transform use a negative …
$\begingroup$ Using a Fourier transform with a positive exponent would simply give you a reflection of the current theory. The main advantage of having a minus in the current definition is that it becomes compatible with the standard version of Bochner's theorem for example.
complex analysis - Fourier transform on $1/(x^2+a^2)
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