Constructing Vector y using Fourier Transform

In summary, to construct the vector y from Fc = y, you need to split the vector c into two components, c(even) and c(odd), and use the formula yj = y(even)j + w^j*y(odd)j and y(j+m) = y(even)j - w^j*y(odd)j, where w is the root to the jth power from the equation w^n = 1.
  • #1
bodensee9
178
0

Homework Statement



Hi, Can someone explain the following?

This is in the book but I wasn't clear. So, I am given the following formulas for constructing the "y" vector from the equation Fc= y, and since I'm doing a fast Fourier transform, I am supposed to construct the "y" from these "half-y's". The formula is

So I am supposed to split the "c" into even and odd components. And then apparently, the jth component of y, yj = y(even)j + w^j*y(odd)j, for j = 0 ... m - 1
and the j+m component of y, y(j+m) = y (even)j - w^jy(odd)j. for j = 0 ... m - 1.
w should be the root to the jth power from the equation w^n = 1.

Can someone explain what that means or how I would go about constructing such a vecto? There are no examples in the book. So, suppose that I rearrange and multiply by a Fourier matrix F and I get half vectors [40000000] (vertical vector), how would I go about finding vector y?

Thanks.
 
Physics news on Phys.org
  • #2
Homework EquationsFc=y, where F is a Fourier matrix and c and y are vectors. The Attempt at a SolutionTo construct the vector y from the equation Fc = y, you need to split the vector c into two components: c(even) and c(odd). You can do this by taking the even-indexed elements of c and putting them into c(even), and the odd-indexed elements of c and putting them into c(odd). Once you have split c into two components, you can use the formula given in your question to construct the vector y. The jth component of y, yj = y(even)j + w^j*y(odd)j, for j = 0 ... m - 1, and the j+m component of y, y(j+m) = y (even)j - w^jy(odd)j. Here, w is the root to the jth power from the equation w^n = 1. For example, suppose that you are given the vector c = [2, 3, 4, 5]. Then, you would split this vector into two components, c(even) = [2, 4] and c(odd) = [3, 5]. Then, you can substitute these two components into the formula for constructing y. For j = 0, the jth component of y, y0 = y(even)0 + w^0*y(odd)0 = 2 + (1)*3 = 5, and the j+m component of y, y(0+2) = y(even)0 - w^0*y(odd)0 = 2 - (1)*3 = -1. Similarly, for j = 1, the jth component of y, y1 = y(even)1 + w^1*y(odd)1 = 4 + (1)*5 = 9, and the j+m component of y, y(1+2) = y(even)1 - w^1*y(odd)1 = 4 - (1)*5 = -1. In this case, the vector y constructed from Fc = y is y = [5, -1, 9, -1].
 

FAQ: Constructing Vector y using Fourier Transform

What is a Fourier Transform?

A Fourier Transform is a mathematical tool used to decompose a function into its individual frequency components. It is commonly used in signal processing and data analysis to understand the underlying patterns and frequencies within a complex waveform.

What is the difference between a Fourier Transform and a Fourier Series?

A Fourier Transform is used to analyze continuous signals or functions, while a Fourier Series is used for analyzing periodic signals. A Fourier Series breaks down a periodic signal into a sum of simple sinusoidal functions, while a Fourier Transform converts a continuous signal into a representation in the frequency domain.

What are the applications of Fourier Transform?

Fourier Transform has a wide range of applications in fields such as engineering, physics, mathematics, and computer science. It is commonly used in signal processing, image processing, quantum mechanics, and data compression.

What is the relationship between Fourier Transform and the Heisenberg Uncertainty Principle?

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know the exact position and momentum of a particle. This principle is related to Fourier Transform, as the transform converts a function in the time domain to its frequency representation, making it impossible to know the exact frequency and time domain of a signal simultaneously.

How is Fourier Transform used in image processing?

Fourier Transform is commonly used in image processing to analyze and manipulate images. It is used to decompose an image into its frequency components, which can then be modified or filtered to enhance certain features in the image. This is often used in applications such as image compression, noise reduction, and image enhancement.

Back
Top