This chapter thoroughly examines linear systems, with an emphasis on linear time-invariant (LTI) systems, essential in signal processing and systems analysis. It begins by introducing system fundamentals, including linearity, translation (shifting), and the effects of cascading systems. The concept of impulse response is then presented as a key method to predict system output based on given inputs. LTI systems are explored in detail, highlighting their importance due to their analytical simplicity and reliability. The chapter further discusses eigenfunctions and their role in defining system responses, along with the Fourier transform, which is used to analyze and design LTI systems. Essential system characteristics like causality and stability are examined, providing insights into system behavior over time. Finally, matched filters are introduced as a method to improve signal detection by aligning filters with specific signal characteristics.