Fourier Transform of cosine and rect

Fourier transform and provides the equation they used, which includes a substitution with Euler's formula and the use of the delta function. They clarify that the equation is not for convolutions, but for modulation in imaging studies. They also mention that 'rect' stands for the rectangle function. In summary, Vbump is seeking confirmation on their correct use of Euler's formula in their Fourier transform equation for a function used in imaging studies. They clarify that the equation is not for convolutions and mention the use of the rectangle function.
  • #1
vkl
2
0

Homework Statement


Just wanted to check if I did the Fourier transform of a somewhat long function correctly


Homework Equations



f(x) = (1+cos([tex]\frac{2pix}{w}[/tex]))rect2([tex]\frac{x}{w}[/tex])
they're not convolutions, just a modulation equation used in imaging studies
'rect' is rectangle function

The Attempt at a Solution


Euler's formula used to substitute in for cos(ax) with ((e^(iax) + e^(-iax)))/2
i=imaginary
the resultant Fourier Transform:

[tex]\hat{F}[/tex](k) = ([tex]\delta[/tex](x)+([tex]\delta[/tex](k-[tex]\frac{1}{W}[/tex])+[tex]\delta[/tex](k+[tex]\frac{1}{W}[/tex])))(w)sinc([tex]\pi[/tex]kw)

where '[tex]\delta[/tex]' stands for the delta function

Thanks in advance for the help

V
 
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  • #2
bump
any takers?

V
 

What is the Fourier Transform of cosine?

The Fourier Transform of cosine is a mathematical tool used to decompose a function into its constituent frequencies. It represents the function in terms of sinusoidal waves of different amplitudes and frequencies, providing a more detailed understanding of the function's frequency components.

What is the Fourier Transform of a rect function?

The Fourier Transform of a rect function, also known as a rectangular function, is a function that has a constant value of 1 within a certain interval and 0 elsewhere. Its Fourier Transform is a sinc function, which is a sinusoidal wave with infinite oscillations that decay towards 0 as the frequency increases.

What is the relationship between the Fourier Transform of cosine and rect?

The Fourier Transform of a cosine function and a rect function are closely related. In fact, the Fourier Transform of a rect function can be derived from the Fourier Transform of a cosine function by taking its convolution with a sinc function.

What are the applications of Fourier Transform of cosine and rect?

The Fourier Transform of cosine and rect functions has various applications in signal processing, image processing, and data analysis. It is used to filter out specific frequencies from a signal, to compress images, and to analyze the frequency components of a dataset, among other things.

What are the limitations of using Fourier Transform of cosine and rect?

The Fourier Transform of cosine and rect functions assumes that the signal or function is periodic, which may not always be the case in real-world applications. It also requires the function to be integrable, which can be a limitation for some functions. Additionally, it does not work well with functions that have discontinuities or sharp edges.

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