Representations of the Fourier's integral

In summary, there are three representations for Fourier series and at least two Fourier integral representations. There is also a representation for Fourier integral in amplitude/phase notation, and the coefficients A(ω) and B(ω) are not the sine and cosine transforms. Finally, you can relate F(ω) with A(ω) and B(ω) using the complex Fourier transform.
  • #1
Jhenrique
685
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If exist 3 representations for Fourier series (sine/cosine, exponential and amplite/phase) and at least two Fourier integral that I know
[tex]f(t)=\int_{0}^{\infty }A(\omega)cos(\omega t) + B(\omega)sin(\omega t)d\omega[/tex]
[tex]f(t)=\int_{-\infty }^{+\infty }\frac{e^{+i\omega t}}{\sqrt{2\pi}} F(\omega)d\omega[/tex]
So, exist too a representation for Fourier integral in amplitude/phase notation? How is it?
Other question: ##A(\omega)## and ##B(\omega)## are the sine and cosine transforms?
And another ask: I can relates F(ω) with A(ω) and B(ω), like we do in series Fourier ##\left (|c_{n}|=\frac{1}{2}\sqrt{a_{n}^{2} + b_{n}^{2}} \right )##?
 
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  • #2


Hello, thank you for your questions. Yes, there is also a representation for Fourier integral in amplitude/phase notation. It is given by:

f(t)=\int_{-\infty }^{+\infty }A(\omega)\cos(\omega t + \phi(\omega))d\omega

where A(ω) is the amplitude and φ(ω) is the phase shift of the signal at frequency ω.

To answer your second question, A(ω) and B(ω) are not necessarily the sine and cosine transforms. They are the coefficients of the Fourier series expansion in the sine/cosine representation. In the exponential representation, they are the real and imaginary parts of the complex Fourier coefficients.

Finally, to relate F(ω) with A(ω) and B(ω), you can use the following relationship:

F(\omega) = \frac{1}{\sqrt{2\pi}}\left(A(\omega) - iB(\omega)\right)

This is known as the complex Fourier transform. So, you can find the amplitude and phase information from the complex Fourier transform. I hope this helps clarify things for you. Let me know if you have any further questions.
 

FAQ: Representations of the Fourier's integral

What is the Fourier's integral?

The Fourier's integral is a mathematical tool used to represent a function as a combination of sine and cosine waves with different frequencies, amplitudes, and phases.

What are the applications of the Fourier's integral?

The Fourier's integral has a wide range of applications in various fields such as signal processing, image processing, data compression, and solving differential equations.

What is the difference between the Fourier's integral and Fourier series?

The Fourier's integral is used to represent a continuous function whereas the Fourier series is used to represent a periodic function. The Fourier series also uses a discrete set of frequencies while the Fourier's integral uses a continuous range of frequencies.

How is the Fourier's integral calculated?

The Fourier's integral involves converting a function from the time or spatial domain to the frequency domain using the Fourier transform. The inverse Fourier transform is then used to convert the frequency domain representation back to the time or spatial domain.

What is the significance of the Fourier's integral in science and engineering?

The Fourier's integral is an essential tool in science and engineering as it allows us to analyze and understand complex phenomena by breaking them down into simpler components. It also helps in solving various mathematical problems and has numerous practical applications.

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