# Integral computation over Fourier/Convolution

• divB
In summary, the conversation discusses the approach to solving a given integral using Fourier transforms and Parseval's theorem. However, there is a misunderstanding in applying Parseval's theorem, which leads to different solutions between the conversation participants and Wolfram Alpha. The correct solution can be obtained by evaluating the Fourier transform at f = 0.
divB
Hi,

I just tried to solve the following integral:

$$\int_{-\infty}^{\infty} \frac{2}{1+(2\pi t)^2} \mathrm{sinc}(2t) dt$$

My approach is: Convert both to Fourier domain and the multiplication becomes a convolution. Because of Parsevals theorem, I can either integrate in time or frequency domain. Because of linearity, I can then put the outer integral inside:

$$\int_{f=-\infty}^{\infty} \int_{\nu=-\infty}^{\infty} e^{-|\nu|} \frac{1}{2} \Pi\left(\frac{f-\nu}{2}\right) d\nu df \\ = \int_{\nu=-\infty}^{\infty} e^{-|\nu|} \frac{1}{2} \underbrace{\int_{f=-\infty}^{\infty}\Pi\left(\frac{f-\nu}{2}\right) df}_{2} d\nu$$

It can be clearly seen, that only the double-sided exponential stays whose integral is 2.
So my total solution is 2.

However, Wolfram alpha gives me (e-1)/e:

http://www.wolframalpha.com/input/?i=integral+2%2F%281%2B%282*pi*t%29^2%29+*+sin%28pi*2*t%29%2F%28pi*2*t%29%2Ct%2C-infinity%2Cinfinity

Can anyone explain where I did wrong?

Thanks,
div

PS: The sinc is defined as normalized, i.e. $\mathrm{sinc}(t)=\sin(\pi t)/\pi t$ ...

divB said:
Because of Parsevals theorem, I can either integrate in time or frequency domain.
This is your fundamental problem. Parseval's theorem would apply if you were integrating the magnitude squared of your function, but you are not doing that.

In general, if your original function is $x(t)$, then
$$\int_{-\infty}^{\infty} x(t) dt = \left. \int_{-\infty}^{\infty} x(t) e^{-ift} dt \right|_{f = 0} = \hat{x}(0)$$
where I have assumed the following definition for the Fourier transform:
$$\hat{x}(f) = \int_{-\infty}^{\infty} x(t) e^{-ift} dt$$
Thus, integration in the time domain does not correspond to integration in the frequency domain, but rather to evaluation of the Fourier transform at $f = 0$.

Oh, of course, thanks!

Do you have a pointer on how to calculate this integral with Fourier transform theorems?

Thanks!
divB

As jbunniii implied, you want to evaluate
$$F\left[\frac{2}{1+(2\pi t)^2} \text{sinc }2t\right]_{f = 0}$$ where F[ ] denotes the Fourier transform. What's the Fourier transform of that product?

Oh, right, didn't realize that!
Thanks a lot, got it!

## 1. What is integral computation over Fourier/Convolution?

Integral computation over Fourier/Convolution is a mathematical process that involves evaluating the integral (area under the curve) of a function that has been transformed using Fourier or convolution operations. This allows for the analysis and manipulation of signals and systems in the frequency domain.

## 2. How is integral computation over Fourier/Convolution used in science?

Integral computation over Fourier/Convolution is used in a variety of scientific fields, including signal processing, image processing, and physics. It allows for the analysis and manipulation of complex signals and systems, making it a valuable tool in understanding and modeling real-world phenomena.

## 3. What are some applications of integral computation over Fourier/Convolution?

The applications of integral computation over Fourier/Convolution are vast and diverse. Some common examples include noise reduction in audio signals, image enhancement in photography, and frequency analysis in physics experiments. It is also used in data compression, radar and sonar systems, and medical imaging.

## 4. How does integral computation over Fourier/Convolution differ from traditional integration?

Integral computation over Fourier/Convolution differs from traditional integration in that it operates on functions that have been transformed using Fourier or convolution operations. This allows for the analysis and manipulation of signals and systems in the frequency domain, which can often provide insights and solutions that traditional integration cannot.

## 5. What are the advantages of using integral computation over Fourier/Convolution?

Some advantages of using integral computation over Fourier/Convolution include the ability to analyze and manipulate complex signals and systems, the potential for more efficient and accurate solutions compared to traditional integration, and the wide range of applications in various scientific fields. It also allows for a deeper understanding of the frequency characteristics of a signal or system, which can be useful in problem-solving and experimentation.

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