Rectangular Fourier Transform and its Properties

In summary, a Rectangular Fourier transform is a mathematical tool used to represent a function as a sum of sine and cosine waves. It is calculated by taking the integral of the function multiplied by a complex exponential function and has many applications in fields such as signal processing and image analysis. Its main limitation is that it assumes the function being transformed is periodic and the calculation can become computationally intensive for large data sets.
  • #1
0xDEADBEEF
816
1
Is there a name for a transformation using the orthonormal base

[tex]s_k(x)=\lceil \sin kx \rceil,\: c_k(x) = \lceil \cos kx \rceil \quad ?[/tex]

So basically a Fourier transform or Fourier series using periodic rectangles. What are the properties? Is there some kind of convolution theorem?
 
Mathematics news on Phys.org
  • #2
I found some answers. The Walsh-transform looks very similar. I noticed that the functions are not orthogonal so sign(sin(kx)) and sign(cos(kx)) is probably a better choice.
 

What is a Rectangular Fourier transform?

A Rectangular Fourier transform is a mathematical tool used to represent a function as a sum of sine and cosine waves. It is often used in signal processing and image analysis to analyze and manipulate signals in the frequency domain.

How is a Rectangular Fourier transform calculated?

A Rectangular Fourier transform is calculated by taking the integral of the function multiplied by a complex exponential function. This integral is evaluated over a specific range of values, typically from negative infinity to infinity.

What is the difference between a Rectangular Fourier transform and a Fourier series?

A Rectangular Fourier transform is used to represent a continuous function, while a Fourier series is used to represent a periodic function. Additionally, a Fourier series uses a discrete set of frequencies, while a Rectangular Fourier transform uses a continuous range of frequencies.

What are the applications of a Rectangular Fourier transform?

A Rectangular Fourier transform has many applications in various fields, including signal processing, image analysis, and data compression. It is also used in the design and analysis of filters, as well as in solving differential equations and boundary value problems.

Are there any limitations to using a Rectangular Fourier transform?

One limitation of using a Rectangular Fourier transform is that it assumes the function being transformed is periodic, which may not always be the case in real-world applications. Additionally, the calculation of the transform can become computationally intensive for large data sets.

Similar threads

I Fourier transform of rectangular pulses
    • Calculus
Replies
5
Views
2K
Fourier transform radial component of magnetic field
    • Advanced Physics Homework Help
Replies
5
Views
1K
A few questions to make sure I understand Fourier series
    • Calculus and Beyond Homework Help
Replies
3
Views
284
Solving modified heat equation
    • Calculus and Beyond Homework Help
Replies
5
Views
267
Fourier series and the shifting property of Fourier transform
    • Calculus and Beyond Homework Help
Replies
6
Views
914
MHB Fourier Transform of Periodic Functions
    • General Math
Replies
5
Views
3K
What is the Fourier transform of cos(2πf0x - π/4)?
    • General Math
Replies
2
Views
1K
Fourier transform: duality property and convolution
    • Calculus and Beyond Homework Help
Replies
1
Views
786
I On deriving the (inverse) Fourier transform from Fourier series
    • Topology and Analysis
Replies
4
Views
287
Fourier sine and cosine transforms of Heaviside function
    • Calculus and Beyond Homework Help
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top