Question on Definition of Fourier Transform

In summary, the discussion centers around the definition of the Fourier Transform in a physics class and the professor's unconventional approach of switching the negative signs in the forward and reverse kernels. While this may not make a theoretical difference, it causes practical issues with software functions and tables of transforms. There is no apparent reason for this deviation from the generally accepted formulas, and it is advised to bring up the potential difficulties with the professor and software programs.
  • #1
nickmai123
78
0
I have a question, specifically to physics people, on their definition of the Fourier Transform (and its inverse by proxy). I'm an EE and math person, so I've done a lot of analysis of (real/complex) and work with (signal processing) the transform.

In a physics class I'm taking, the professor defined the transform with the negative signs in the forward and reverse kernels flipped; this is against any and all known conventions that I'm aware of. Theoretically, it doesn't make a difference since its a simple substitution, but in practice, it causes really screws things up like using an FFT function built into MATLAB and Mathematica. Also, the tables of transforms and properties are all flipped around.

That is, in the physics class, the forward (and reverse) transforms are as follows:

[tex]\mathcal{F}\left\{x\left(t\right)\right\}(f) = X\left(f\right) = \int_{-\infty}^{\infty}x\left(t\right)e^{j2\pi f t}dt[/tex]

[tex]\mathcal{F}^{-1}\left\{X\left(f\right)\right\}(t) = x\left(t\right) = \int_{-\infty}^{\infty}X\left(f\right)e^{-j2\pi f t}df[/tex]

Is there a reason for this? I can't think of a physical situation where defining the transforms as such is more advantageous than the generally accepted formulas.
 
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  • #2
There is no reason. Which has + and which has - in the exponent is purely a matter of convention. You might ask your professor why he switched - it may have been an oversight. You should bring up the possible problems with MATLAB and Mathematica.
 

FAQ: Question on Definition of Fourier Transform

1. What is the Fourier Transform?

The Fourier Transform is a mathematical operation that decomposes a signal or function into its constituent frequencies. It is often used in signal processing and engineering to analyze and manipulate signals.

2. How is the Fourier Transform defined?

The Fourier Transform is defined as the mathematical relationship between a function in the time domain and its representation in the frequency domain. It is typically denoted by the symbol F and is expressed as F(f(t)) = F(ω) = ∫f(t)e^(-iωt)dt, where f(t) is the function in the time domain, ω is the frequency, and e^(-iωt) is the complex exponential function.

3. What is the difference between the Fourier Transform and the Fourier Series?

The Fourier Transform is a continuous representation of a signal or function in the frequency domain, whereas the Fourier Series is a discrete representation. The Fourier Series is used for periodic signals, while the Fourier Transform can be applied to both periodic and non-periodic signals.

4. What are some applications of the Fourier Transform?

The Fourier Transform has a wide range of applications in fields such as signal processing, engineering, physics, and mathematics. It is used for image and audio compression, filtering, spectral analysis, and solving differential equations, among others.

5. Are there any limitations or drawbacks to using the Fourier Transform?

One limitation of the Fourier Transform is that it assumes a signal or function is infinite in duration, which may not always be the case in real-world applications. Additionally, the Fourier Transform may not accurately represent signals with abrupt changes or discontinuities. Other limitations include the complexity of calculations and the need for an understanding of advanced mathematical concepts.

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