# fourier transform

## 1. in words

Given a function \(f(t)\) over time, the fourier transform gives us a function \(\hat{f}(\xi)\), where \(\xi\) is a frequency. \(\hat{f}(\xi)\) is a complex number where:

- the magnitude is the power of \(\xi\) in \(f\)
- \(\theta\) is the phase of \(\xi\) in \(f\)

The two representations contain equivalent information

## 2. compared to fourier series

- A fourier series represents a periodic function as an infinite sum of sines and cosines
- What if the function being represented is not periodic, but it is only non-zero over a finite interval? Then, we can consider the extension of the function: just repeat the function over and over again and find the fourier series of that
- All that to say, if you want the fourier series representation of something, it needs to be periodic, or you will have to consider an extension of the function which is periodic
- Even if the function is periodic, there are some conditions to be met: see math overflow answer
- In contrast, the fourier transform needs (?) a non-periodic function (see article 1)
- but this math overflow answer mentions a way to deal for the transform to deal with periodic functions

- wikipedia mentions a relationship between the fourier series of \(f\) and the fourier transform of \(f\), where \(f\) is zero except for at an interval \(I\). Consider the fourier
*series*taking some interval \(T\) the contains \(I\) and making an extension. As you push \(T\) to infinity, you will reach the fourier transform

## 3. discrete fourier transform vs discrete time fourier transform

- discrete time fourier transform is a continuous function of frequency
- When starting, I was always confused what the meaning of \(k\) was. In the discrete fourier transform (DFT), \(X_k\) is the convolution of the function with \(f_{k}(n) = \exp(\frac{2\pi i n k}{N})\). The frequency associated with \(X_k\) is \(\frac{k}{N}\). \(k\) cycles are completed in the \(N\) samples. What is the "true" frequency? That is, once I've finished my DFT, I have the amplitudes for a bunch of \(k\) 's. Now somebody asks me what frequencies correspond with a given \(k\). How should I respond? We need to know how many samples are taken per second. This is the frequency. Then, we can know how many cycles are completed per second. So why are we using an integer \(k\) in the first place? To make sure that an integer number of cycles are completed in the number of samples given.
- see here (wikipedia article)