### Introduction

Discrete Fourier transform is one of the most important linear transformation for signal processing. Although I have learned discrete Fourier transformation in college and graduate school, it was not systematic at that time. Because I did not have too much chance to actually use it without having to be dependent on third-party library or software, I only remember it is a transformation between the time domain and the frequency domain without being able to tell too much mathematical details.

In this blog post, I would like to discuss some of the details of discrete Fourier transform and its relationship with Fourier transform.

### Fourier Transform

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes a function (often a function of time, or a signal) into its constituent frequencies, which is sometimes called frequency spectrum. The constituent frequencies could also be used to reconstruct the function back.

To decompose a function $f(x)$ to its frequency spectrum function $F(\xi)$ or $F(\omega)$, we use Fourier transform.

\[F(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i \xi x} dx\]where $\xi$ is any real-valued frequency.

Sometimes, if we define the radian-frequency $\omega = 2\pi \xi$,

\[F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i \omega x} dx\]To reconstruct the function $f(x)$ from its frequency spectrum function $F(\xi)$ or $F(\omega)$, we use inverse Fourier transform.

\[f(x) = \int_{-\infty}^{\infty} F(\xi) e^{2\pi i \xi x} d\xi\]or equivalently,

\[f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega x} d\omega\]### Discrete Fourier Transform

Discrete Fourier transform replaces the infinite integral in the Fourier transform with a finite sum. Given a signal sequence $\{f(x_0), f(x_1), \cdots, f(x_{n-1})\}$, where the sampling interval between $x_i$ and $x_{i+1}$ is a constant, we defined the discrete Fourier transform, and got a frequency sequence $\{F(\omega_0), F(\omega_1), \cdots, F(\omega_{n-1})\}$.

\[F(\omega_k) = \sum_{n = 0}^{N-1} f(x_n) e^{-i \omega_k x_n}\]where

$k = 0, 1, 2, \cdots, N-1$.

$N$ is the number of samples from $f$ and $F$.

$f(x_n)$ is the signal amplitude at $x_n$. It could be real or complex value.

$T$ is the sampling interval and $T = x_{i+1} - x_{i}$.

$x_n = nT$.

$F(\omega_k)$ is the spectrum of $f$ at radian-frequency $\omega_k$.

$f_0 = \frac{1}{NT}$ is the fundamental frequency, because we treat the $N$ signal examples as if they are from one period.

$\Omega = \frac{2\pi}{NT}$ is the radian-frequency sampling interval.

$\omega_k = k\Omega = \frac{2\pi k}{NT}$ is the $k$th radian-frequency sample.

$f_s = \frac{1}{T}$ is the sampling rate.

Similarly, the inverse discrete Fourier transform could be derived based on the definition of discrete Fourier transform.

\[f(x_n) = \frac{1}{N} \sum_{k = 0}^{N-1} F(\omega_k) e^{i \omega_k x_n}\]Sometimes, the sampling interval $T$ is not important and we are interested in the magnitude of the signal and frequency samples. Given a signal sequence $\{f[0], f[1], \cdots, f[N-1]\}$ and we got a frequency sequence $\{F[0], F[1], \cdots, F[N-1]\}$ after fourier transformation. We have the more general form of discrete Fourier transform.

\[\begin{align} F[k] &= \sum_{n = 0}^{N-1} f[n] e^{-i \omega_k x_n} \\ &= \sum_{n = 0}^{N-1} f[n] e^{-i \frac{2\pi k}{NT} nT} \\ &= \sum_{n = 0}^{N-1} f[n] e^{-i \frac{2\pi k n}{N}} \\ \end{align}\]Similarly, we have the inverse discrete Fourier transform.

\[\begin{align} f[n] &= \frac{1}{N} \sum_{k = 0}^{N-1} F[k] e^{i \omega_k x_n} \\ &= \frac{1}{N} \sum_{k = 0}^{N-1} F[k] e^{i \frac{2\pi k}{NT} nT} \\ &= \frac{1}{N} \sum_{k = 0}^{N-1} F[k] e^{i \frac{2\pi k n}{N}} \\ \end{align}\]The reason why we only need $N$ frequency samples becomes obvious here. Because

\[\begin{align} F[k+N] &= \sum_{n = 0}^{N-1} f[n] e^{-i \frac{2\pi (k+N) n}{N}} \\ &= \sum_{n = 0}^{N-1} f[n] e^{-i \frac{2\pi k n}{N}} e^{-i 2\pi n} \\ &= \sum_{n = 0}^{N-1} f[n] e^{-i \frac{2\pi k n}{N}} \times 1\\ &= \sum_{n = 0}^{N-1} f[n] e^{-i \frac{2\pi k n}{N}}\\ &= F[k] \\ \end{align}\]Because the frequency has a “period” of $N$, only $N$ frequency examples are required.

The frequencies corresponding to the frequency samples $F[k]$ is $k f_0$, i.e., the frequencies are $0, \frac{1}{NT}, \frac{2}{NT}, \cdots, \frac{N-1}{NT}$ for $F[0], F[1], F[2], \cdots, F[n]$, respectively. Similarly the radient frequency corresponding to the frequency samples $F[k]$ is $2 \pi k f_0$.

### Unitary Discrete Fourier Transform

Because discrete Fourier transform is a linear transformation, we could express such transformation using matrix multiplications.

Surprisingly, the $e^{i \frac{2\pi k}{N}}$ for $k \in [0,N-1]$ in the inverse discrete Fourier transform are actually the $N$th root of unity, $\omega_N^{0}, \omega_N^{1}, \cdots, \omega_N^{N-1}$.

So we could write the inverse discrete Fourier transform using the following expression.

We define the inverse discrete Fourier transform Vandermonde matrix $\mathbf{DFT}^{\ast\dagger}$ as

\[\begin{align} \mathbf{DFT}^{\ast\dagger} &= \begin{bmatrix} \omega_N^{0 \cdot 0} & \omega_N^{0 \cdot 1} & \cdots & \omega_N^{0 \cdot N-1} \\ \omega_N^{1 \cdot 0} & \omega_N^{1 \cdot 1} & \cdots & \omega_N^{1 \cdot N-1} \\ \vdots & \vdots & \ddots & \vdots \\ \omega_N^{N-1 \cdot 0} & \omega_N^{N-1 \cdot 1} & \cdots & \omega_N^{N-1 \cdot N-1} \\ \end{bmatrix} \end{align}\]where $\omega_N^{i \cdot j} = (\omega_N^{i})^j = \omega_N^{ij}$.

We further arrange the signal and frequency samples as column vectors.

\[F_N = \begin{bmatrix} F[0] \\ F[1] \\ \vdots \\ F[N-1] \\ \end{bmatrix}\]and

\[f_N = \begin{bmatrix} f[0] \\ f[1] \\ \vdots \\ f[N-1] \\ \end{bmatrix}\]Therefore,

\[\begin{align} f_N &= \frac{1}{N} \mathbf{DFT}^{\ast\dagger} F_N \\ F_N &= \mathbf{DFT}^{\ast} f_N \\ \end{align}\]However, $\mathbf{DFT}^{\ast\dagger}$ is not unitary.

\[\mathbf{DFT}^{\ast\dagger}_{j,k} = \omega_N^{jk}\] \[\mathbf{DFT}^{\ast}_{j,k} = \omega_N^{-kj}\]Therefore,

\[\begin{align} (\mathbf{DFT}^{\ast} \mathbf{DFT}^{\ast\dagger})_{j,k} &= \sum_{i=0}^{N-1} \omega_N^{ji} \omega_N^{-ik} \\ &= \sum_{i=0}^{N-1} \omega_N^{-i(k-j)} \\ &= \begin{cases} \sum_{i=0}^{N-1} 1 & \text{when $k = j$}\\ \omega_N^{j-k} \sum_{i=0}^{N-1} \omega_N^{i} & \text{when $k \neq j$}\\ \end{cases} \\ &= \begin{cases} N & \text{when $k = j$}\\ \omega_N^{j-k} \frac{1 (1 - \omega_N^N)}{1-\omega_N^1} & \text{when $k \neq j$}\\ \end{cases} \\ &= \begin{cases} N & \text{when $k = j$}\\ \omega_N^{j-k} \frac{1 (1 - 1)}{1-\omega_N^1} & \text{when $k \neq j$}\\ \end{cases} \\ &= \begin{cases} N & \text{when $k = j$}\\ 0 & \text{when $k \neq j$}\\ \end{cases} \\ \end{align}\]It is not unitary, but we are close. We define a new matrix

\[\mathbf{DFT}^{\dagger} = \frac{1}{\sqrt{N}} \mathbf{DFT}^{\ast\dagger}\]This time, it is easy to show that this time $\mathbf{DFT}^{\dagger}$ is a unitary matrix.

To conduct discrete Fourier transform and inverse discrete Fourier transform, we have

\[\begin{align} f_N &= \frac{1}{\sqrt{N}} \mathbf{DFT}^{\dagger} F_N \\ F_N &= \sqrt{N} \mathbf{DFT}^{} f_N \\ \end{align}\]