TABLE OF CONTENTS Here is guarantee: if you can understand the contents of this chapter, then you will be able to understand the rest of the book. Here we introduce linear systems and provide an account of the basic mathematical equipment that we need for DSP, together with how they are applied in certain practical circumstances. If you are already schooled in these areas, or are familiar with DSP, then the entire chapter can be skipped. If, however, you have forgotten how to differentiate, or have never used complex numbers before, then what follows is for you. These subjects are important because if you can master them then you will have little difficulty in coding DSP algorithms that can do some truly astonishing things. As the chapter title suggests, these are essential tools. We will not go into the proofs as to why they are true, but we will see why they are used, and when they are appropriate. Let us start with a consideration of linear systems.
Imagine one day climbing the bell tower of a church and, having gained the belfry, you softly tap the lip of a large bell with a small hammer you have been carrying in your backpack. The bell emits a deep, resonant clang and you leave, satisfied with your work. The next day, you repeat the exercise and find that the bell performs in exactly the same way. The day after that, you hit the bell slightly harder.
