why in fourier transform we have to multiple with e^{-j\omegat}

[Osawa_Odessa]

Hello everybody,
I would like to know why in fourier transform we have to multiple with \[e^{-j\omega\ t}\]?:smile:

 

[jeeudr]

The short answer would be: that is the definition of the Fourier transform.

The longer answer is below:
You can think of the Fourier transform as a sort of conversion between two bases (from time to frequency). Imagine that you are resolving a vector (in 2D space) into its X and Y components, you would be doing a dot product operation with the x and y unit vectors and obtain the coefficient values. The Fourier transformation can be thought to be something similar. Think of your transform operation as converting from a time basis to frequency basis. You are resolving your function into components expressed in the basis of frequency vectors.

 

[Osawa_Odessa]

You can think of the Fourier transform as a sort of conversion between two bases (from time to frequency). Imagine that you are resolving a vector (in 2D space) into its X and Y components, you would be doing a dot product operation with the x and y unit vectors and obtain the coefficient values. The Fourier transformation can be thought to be something similar. Think of your transform operation as converting from a time basis to frequency basis. You are resolving your function into components expressed in the basis of frequency vectors.

Hello jeeudr,
I like the opinion but it still seem not clear to me. Could you explain in more detail or reommend me some sites or material that I should read?

 

[ashiir]

Re: why in fourier transform we have to multiple with e^{-j\omega t}

From my understanding (that means it may be wrong).
The Fourier transform is a transformation method which objectively transform a function in normal basis into a polar basis.
This transformation resulting in representation of the signal from time domain to be frequency domain function.
The exponential term exp(j*w*t) represents the polar mapping of the function in the transformation process.

 

[abdulwahababid]

Re: why in fourier transform we have to multiple with e^{-j\omega t}

The sinusoidal time function that a phasor represents is obtained by multiplying the phasor by e^ jwt

 

[jeeudr]

I suggest that you take a look at this: https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/

It is not exactly the same as what I said, but nevertheless, it is a good way to look at things.