Constructing u(t * omega) from U(t, omega)

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In summary, the conversation is about converting a 2D function to a 1D function through integration over a constant variable. The function in question is the discrete Gabor transform of a sampled signal, stored in a 2D matrix. The question is how to efficiently perform numerical integration over a constant variable using the 2D matrix.
  • #1
nkinar
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Hello---

I am reading a paper on numerical methods which requires a 2D function to be converted to a 1D function. Let U(t, omega) be the discrete Gabor transform of a sampled signal, where t is time (seconds) and omega is the angular frequency. U(t, omega) is stored in a 2D m-by-n matrix.

Now U(t, omega) must be converted to u(chi) = u(t * omega), where chi = (t * omega), by integration over constant chi.

How do I efficiently perform numerical integration over constant chi, given the 2D matrix U(t, omega)?
 
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  • #2
Do you mean you want to perform

[tex] \int_a^b U(t, \chi / t) \, dt [/tex]

numerically?
 
  • #3
Hello CompuChip--

Thank you very much for your response! Yes, I think that I would like to numerically perform the integration that you describe using U(t, omega) as a 2D m-by-n matrix. How would I proceed?

Why do you write (chi/t) as an argument to U(t, omega)?
 

What is the purpose of constructing u(t * omega) from U(t, omega)?

The purpose of constructing u(t * omega) from U(t, omega) is to model a signal in the time domain using its frequency domain representation. This allows for analysis and manipulation of signals using mathematical operations in the time domain.

What is the mathematical process involved in constructing u(t * omega) from U(t, omega)?

The mathematical process involves taking the inverse Fourier transform of U(t, omega) to obtain the signal u(t) in the time domain. Then, the signal is multiplied by the complex exponential e^(j * omega * t) to obtain u(t * omega).

How is the frequency domain representation U(t, omega) obtained?

The frequency domain representation U(t, omega) is obtained by taking the Fourier transform of the signal u(t) in the time domain. This involves decomposing the signal into its individual frequency components.

What information can be obtained from the frequency domain representation U(t, omega)?

The frequency domain representation U(t, omega) provides information about the frequency components present in the signal. This includes the amplitude, phase, and frequency of each component.

What are the advantages of using the frequency domain representation U(t, omega)?

The frequency domain representation U(t, omega) allows for easier analysis and manipulation of signals using mathematical operations such as convolution and filtering. It also provides a more intuitive understanding of the frequency components present in a signal.

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