fourier transform

Table of Contents

The Discrete Fourier transform is a matrix multiplication operation with the Fourier matrix. It converts a signal from time domain to frequency domain and back.

Pointwise multiplication in the freq domain is equivalent to circular convolution in the time domain according to the convolution theorem.

The DFT is linear, so

\begin{equation} F\{a \cdot b\} = F\{a\} \cdot F\{b\} \end{equation}

This is an obvious result from the fact that the dft is a matrix multiplication, or a linear transformation.

1. DFT

\begin{equation} X[\omega_k] = \sum_{n = 0}^{N-1} x(t) e^{-j \omega t} \end{equation}

Here, \(\omega\) is the list of sample frequencies we want to evaluate the sequence at, i.e what the indices of \(X\) should mean. It is \(\omega = 2 \pi \frac{m}{N}\)

2. Continuous FT

In continuous time, the fourier transform of a function \(a\) is

\begin{equation} A(f) = \int_{-\infty}{\infty} a(t) e^{-j2\pi f t } dt \end{equation}

Note that the parameter changes from time to frequency.

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