Simple inverse fourier transforms

[mathrocks]

Can someone please tell my what the inverse Fourier transform of t*u(t) is??
I've been looking at tables but there isn't anything for just t...

 

[LeBrad]

Do you know the property relating the signal in time to a signal differentiated in the frequency domain?

 

[piercas]

The inverse Fourier transform of t*u(t) is a challenging question, as it depends on the definition of the Fourier transform being used. In general, the inverse Fourier transform of t*u(t) would be a complex-valued function, as t*u(t) is a complex-valued function in the frequency domain.

One approach to finding the inverse Fourier transform would be to use the definition of the Fourier transform, which states that the inverse Fourier transform of a function F(x) is given by the integral of F(x) multiplied by the complex exponential e^(2πixf) over all frequencies f. In this case, we would need to calculate the inverse Fourier transform of t*u(t) by integrating t*u(t) multiplied by e^(2πixf) over all frequencies f.

Another approach would be to use the properties of the Fourier transform, such as linearity and time-shifting, to break down the function into simpler components that have known inverse Fourier transforms. For example, we could rewrite t*u(t) as t*(1*u(t)), where 1 is the unit step function. Then, we can use the time-shifting property to shift the unit step function by t units, resulting in the inverse Fourier transform of t*u(t) being a combination of the inverse Fourier transforms of t and u(t).

In summary, the inverse Fourier transform of t*u(t) is a complex-valued function that can be calculated using the definition of the Fourier transform or by breaking it down into simpler components using the properties of the Fourier transform. It is important to carefully consider the definition and properties being used in order to accurately determine the inverse Fourier transform.