Fourier Transform Homework: Solving P(t) with E(t_1) & E(t_2)

In summary, the conversation revolved around the process of Fourier transforming an expression and how to compare the terms on the left and right-hand sides of the equation. The individual suggests rewriting the expressions until they can be compared, but is unsure of what to do with the integrals over t1 and t2.
  • #1
Niles
1,866
0

Homework Statement


Hi

I wish to Fourier transform the following expression

[tex]
P(t) = \int\limits_{ - \infty }^\infty {dt_1 dt_2 \chi (t - t_1 ,t - t_2 )E(t_1 )E(t_2 )}
[/tex]

What I do is the following

[tex]
\int\limits_{ - \infty }^\infty {P(\omega )e^{ - i\omega t} } = \int\limits_{ - \infty }^\infty {dt_1 dt_2 d\omega _1 d\omega _2 \,\chi (\omega _1 ,\omega _2 )E(\omega _1 )E(\omega _2 )e^{ - i\omega _1 (t - t_1 )} e^{ - i\omega _2 (t - t_2 )} e^{ - i\omega _1 t_1 } e^{ - i\omega _2 t_2 } }
[/tex]

I'm pretty sure I need to keep rewriting the expressions on the LHS and RHS until I reach a point, where I can compare the terms to each other. But do you have a hint for what I need to do from here?

Cheers,
Niles.
 
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  • #2
Ok, so what we have is

[tex]
\int\limits_{ - \infty }^\infty {d\omega P(\omega )e^{ - i\omega t} } = \int\limits_{ - \infty }^\infty {dt_1 dt_2 d\omega _1 d\omega _2 \,\chi (\omega _1 ,\omega _2 )E(\omega _1 )E(\omega _2 )e^{ - i\omega _1 (t - t_1 )} e^{ - i\omega _2 (t - t_2 )} e^{ - i\omega _1 t_1 } e^{ - i\omega _2 t_2 } }
[/tex]
[tex]
\int\limits_{ - \infty }^\infty {d\omega P(\omega )e^{ - i\omega t} } = \int\limits_{ - \infty }^\infty {dt_1 dt_2 d\omega _1 d\omega _2 \,\chi (\omega _1 ,\omega _2 )E(\omega _1 )E(\omega _2 )e^{ - i\omega _1 t} e^{ - i\omega _2 t}}
[/tex]

But this seems a little odd, because what am I supposed to do about the integral over t1 and t2?
 

1. What is Fourier Transform?

Fourier Transform is a mathematical concept that allows us to analyze a signal in terms of its frequency components. It decomposes a function of time into its constituent frequencies, which can be represented as a sum of sine and cosine waves.

2. How is Fourier Transform useful in solving P(t) with E(t1) and E(t2)?

Fourier Transform is useful in solving P(t) with E(t1) and E(t2) because it allows us to convert the signals from the time domain to the frequency domain. This makes it easier to analyze and manipulate the signals using mathematical operations, such as multiplication and convolution, which are essential in solving the given problem.

3. What are the steps involved in solving P(t) with E(t1) and E(t2) using Fourier Transform?

The steps involved in solving P(t) with E(t1) and E(t2) using Fourier Transform are as follows:

  • Take the Fourier Transform of E(t1) and E(t2) to obtain their corresponding frequency representations.
  • Multiply the two frequency representations to get the frequency representation of P(t).
  • Take the inverse Fourier Transform of the resulting frequency representation to obtain P(t).

4. What is the role of the Fourier Transform in signal processing?

The Fourier Transform is a fundamental tool in signal processing as it allows us to analyze signals in terms of their frequency components. This is useful in various applications, such as filtering, compression, and noise reduction, where manipulating the frequency components of a signal can improve its quality or extract useful information.

5. Are there any limitations of Fourier Transform?

Yes, there are limitations to Fourier Transform. One of the main limitations is that it assumes that the signal is periodic and infinite. This may not be the case in real-world signals, which can be non-periodic and have finite durations. Additionally, the Fourier Transform does not consider the time information of the signal, and thus, may not be suitable for time-sensitive applications.

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