Fourier Transform of Rect fn

In summary, the conversation discusses the application of Fourier transform to non-periodic functions, specifically the Rect function. The difference between Fourier series and Fourier transforms is clarified, and an example is given to show how the transform of {1} is a delta function. The conversation ends with a suggestion to use A as a large positive real number in the computation.
  • #1
mr_whisk
7
0
Hi all,

How is the Fourier transform applied to non-periodic functions, such as the Rect function?

Any help would be greatly appreciated,

Cheers,

Jamie :)
 
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  • #2
I think you are confusing Fourier series, which apply to periodic functions, and Fourier transforms, which apply to integrable (in some sense) functions.
 
  • #3
OK, i can see why my post appeared to sound like that, but I know what the differnce is.

What i mean is, say, how would you show that the FT transform of {1} is a delta function??

Cheers
 
  • #4
Let A be a ( large ) positive real number

The transform you specified will lead you to integrate exp( - i * w * t ) with t ranging from -A to A, and letting A approaching +∞. When A approaches +∞, this will give you a function similar to one of these :

http://en.wikipedia.org/wiki/Impulse_function#Representations_of_the_delta_function

I let you try the computation and identify which one corresponds to the Fourier transform of 1.
 

What is the Fourier Transform of a Rectangular Function?

The Fourier Transform of a Rectangular Function is a mathematical operation that transforms a rectangular function from its time domain to its frequency domain. It decomposes the rectangular function into a sum of sine and cosine waves with different frequencies and amplitudes.

Why is the Fourier Transform of a Rectangular Function important?

The Fourier Transform of a Rectangular Function is important because it allows us to analyze signals and systems in the frequency domain. This can help us understand the different frequency components present in a signal and how they contribute to its overall behavior.

What are the properties of the Fourier Transform of a Rectangular Function?

The Fourier Transform of a Rectangular Function has several important properties, including linearity, time shifting, frequency shifting, and convolution. These properties allow us to manipulate signals and systems in the frequency domain and make predictions about their behavior in the time domain.

How is the Fourier Transform of a Rectangular Function calculated?

The Fourier Transform of a Rectangular Function is calculated using a mathematical formula that involves integrating the rectangular function multiplied by a complex exponential function. This integral can be solved analytically or numerically using various techniques such as Fourier series or the Fast Fourier Transform algorithm.

What are some real-world applications of the Fourier Transform of a Rectangular Function?

The Fourier Transform of a Rectangular Function has many practical applications, including signal processing, image processing, and data compression. It is also used in fields such as audio and video engineering, telecommunications, and medical imaging to analyze and manipulate signals and images in the frequency domain.

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