The Inverse Discrete Fourier Transform

Complex sinusoids act as building blocks for all signals. This is similar to the $x$, $y$, and $z$ dimensions that define any point in 3-D space and the red, green, and blue colors that combine to form any color.

The DFT

We derive the DFT that takes a time-domain signal into the frequency domain. For a signal of $N$ length, $x[n]$, we have:

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi\frac{k}{N}n}, \qquad k =0,1,\cdots,N-1$$ ...