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What is the concept of negative frequency?
Negative frequency is an idea associated with complex exponentials. A single sine wave can be broken down into two complex exponentials (‘spinning numbers’), one with a positive exponent and one with a negative exponent. That one with the negative exponent is where you get the concept of a negative frequency.
Why is the FFT symmetric about the middle?
The reason for the mirroring is because I use an FFT on real numbers (real FFT). The normal FFT as everyone knows works on complex numbers. Hence the imaginary part is “set” to 0 in the real FFT, resulting in a mirroring around the middle (or technically speaking the mirroring is around 0 and N/2).
Is the Fourier transform symmetric?
When we take the the Fourier Transform of a real function, for example a one-dimensional sound signal or a two-dimensional image we obtain a complex Fourier Transform. This Fourier Transform has special symmetry properties that are essential when calculating and/or manip- ulating Fourier Transforms.
Why is the FFT mirrored in the negative half?
Real signals are “mirrored” in the real and negative halves of the Fourier transform because of the nature of the Fourier transform. The Fourier transform is defined as the following- H (f) = ∫ h (t) e − j 2 π f t d t Basically it correlates the signal with a bunch of complex sinusoids, each with its own frequency.
What’s the difference between positive and negative frequency?
Comparing the negative frequency version with the positive frequency version shows that the cosine is the same while the sine is inverted. They are still 90 degrees out of phase with each other, though, allowing them to catch any signal component at that (negative) frequency.
Why is the FFT ” mirrored ” in signal processing stack?
EDIT: Specifically, the negative frequency correlation is the conjugate of the positive frequency correlation (due to the inverted imaginary sine component) for real signals. In mathematical terms, this is, as Dilip pointed out, the following-
Why is the FFT ” mirrored ” in the Fourier transform?
Real signals are “mirrored” in the real and negative halves of the Fourier transform because of the nature of the Fourier transform. The Fourier transform is defined as the following-