Discrete Fourier Transform question

[kakolukia786]

Hi, I am learning Fourier transformation by my own. I am reading a book "Fourier Transformation" by R. Bracewell. In chapter 11, in examples of discrete Fourier transforms, it gives for N =2, {1 0} transforms to 1/2{1 1}. I can do this in MATLAB but I can't figure it out how to do it by hand. Searching over the internet, I came across some material but it did not help. Can someone explain me how to get those transforms. Thanks

 

[DrClaude]

Have you looked at http://en.wikipedia.org/wiki/Discrete_Fourier_transform#Definition?

 

[marcusl]

Bracewell's is an excellent book. The result you quote should be obvious except, perhaps, for the normalization in front which would usually be used for the inverse DFT (the forward DFT would have the factor 1). I don't have this book here but look at his definition of the DFT and check the normalization.

 

[ProfMimi]

Computing a discrete Fourier transform is basically the same as computing the coefficients of a Fourier series (except for the normalization factor). If you are confused by this simple question, then reviewing how to compute Fourier series might help.

 

[alphy]

Hello,

The discrete Fourier transform is a mathematical tool used to analyze signals and data in the frequency domain. It is commonly used in engineering, physics, and other scientific fields.

To understand how to calculate the discrete Fourier transform by hand, it is important to first understand the concept of complex numbers and their representation in the complex plane. The discrete Fourier transform involves complex numbers and their manipulation, so a basic understanding of complex numbers is necessary.

To calculate the discrete Fourier transform of a signal, you will need to follow a specific formula, which is essentially a summation of the signal multiplied by a complex exponential function. This formula can be found in your textbook or online.

To calculate the transform for N=2, you will need to use the formula for N=2, which is:

X(k) = (1/N) ∑x(n)e^(-j2πnk/N)

Where X(k) is the discrete Fourier transform of the signal x(n) and n ranges from 0 to N-1.

In the example given in your book, N=2 and the signal is {1,0}, so the formula becomes:

X(k) = (1/2) ∑x(n)e^(-j2πnk/2)

Substituting in the values for n=0 and n=1, we get:

X(0) = (1/2)(1*e^(-j2π*0*0/2) + 0*e^(-j2π*0*1/2))

X(1) = (1/2)(1*e^(-j2π*1*0/2) + 0*e^(-j2π*1*1/2))

Simplifying, we get:

X(0) = (1/2)(1 + 0) = 1/2

X(1) = (1/2)(1 + 0) = 1/2

Therefore, the discrete Fourier transform for N=2 and the signal {1,0} is {1/2, 1/2}.

I hope this explanation helps you understand how to calculate the discrete Fourier transform by hand. It can be a bit complicated at first, but with practice and a solid understanding of complex numbers, you will be able to perform these calculations easily. Good luck with your studies!