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papernuke1
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Is there a time domain function whose Fourier transform is the Dirac delta with no harmonics? I.e. a single frequency impulse
johnqwertyful said:Couldn't you just take the inverse Fourier transform?
Shyan said:That will be the chicken-egg problem!
papernuke1 said:Oh! I did that and it's it's a constant function, thanks
The Dirac delta function, also known as the unit impulse function, is a mathematical function that is defined as zero everywhere except at the origin, where it is infinite. It is a distribution that is commonly used to model point-like sources or idealized point particles in physics and engineering.
A function whose Fourier transform is Dirac delta represents a perfect point source in the frequency domain. This means that the function has infinite amplitude at a single frequency and zero amplitude at all other frequencies. In other words, the function contains all possible frequencies with equal importance, which can be useful in signal processing and filtering applications.
The Dirac delta function is closely related to the impulse response of a system. In fact, the impulse response of a system can be obtained by taking the inverse Fourier transform of the system's transfer function, which is the Fourier transform of the Dirac delta function. This means that the impulse response of a system describes how the system responds to an impulse input, making it a useful tool in system analysis and design.
No, a function can only have a Dirac delta Fourier transform at a single point. This is because the Dirac delta function is defined as infinite at a single point and zero everywhere else. If a function has a Fourier transform that is Dirac delta at multiple points, it would essentially be infinite at multiple frequencies, which is not mathematically possible.
The Dirac delta function has many applications in physics, engineering, and signal processing. Some examples include modeling point sources in electromagnetic fields, analyzing the behavior of linear systems, and representing signals with sudden changes or impulses. It also plays a crucial role in the theory of distributions, which is important in many areas of mathematics and physics.