You will use the spectrogram to identify features in signals whose spectra vary with time, including a brief guitar solo and synthetic signal composed of a chirp and sinusoids. You will learn about the tradeoff between temporal resolution and resolution of features in the frequency domain by changing the DFT length used by the spectrogram.

# The DFT and Applications

## A Matrix Interpretation of the Discrete Fourier Transform

This lesson teaches you how to interpret the discrete Fourier transform as the inner product between a matrix and a vector and introduces a linear algebraic viewpoint. Linear algebra is an extremely powerful and widely used tool in signal processing. You will be introduced to these ideas in this lesson and build on them in future studies.

## A Matrix Interpretation of the Fast Fourier Transform Algorithm

This lesson shows you how to view the fast Fourier transform algorithm as a factorization of the discrete Fourier transform matrix into a product of matrices. The matrices in the product representation are very sparse which is why the fast Fourier transform algorithm is computationally efficient. As you see here, matrix factorization is a very useful and insightful tool in signal processing.