This monograph addresses the fundamental concepts of frequency spectrum analysis using discrete Fourier transforms. Topics range from the discretization of measured physical phenomenon to the interpretation of periodogram spectral content in terms of the Fourier basis sinusoids. The text also discusses peculiarities that arise due to discretization, such as leakage and aliasing, and why de-trending and windows mitigate such effects. Included examples explain the details of implementation and interpretation, with MATLAB source code provided for a number of frequency-spectrum-analysis tools.