The convolution integral is the fundamental expression relating the input and output of an LTI system. However, it has three shortcomings:

- It can be tedious to calculate.
- It offers only limited physical interpretation of what the system is actually doing.
- It gives little insight on how to design systems to accomplish certain tasks.

In fact, by making our notions of signals and linear systems more mathematically abstract, we will be able to draw enlightening parallels between signals and systems and linear algebra.