# CHAPTER *3*

# Discrete-Time Simple Harmonic Oscillators

The goal of this chapter is to obtain a discrete-time version of the general solution to the motion of a simple harmonic oscillator. Equation 2.17. The complete solution, while not particularly complicated, does require significant effort to obtain. Section 3.1 therefore develops the first-order case, which is then used to solve the second-order problem in Section 3.3.

This chapter requires careful study for a full understanding. It is an introduction to some of the core subjects of digital signal processing and how those techniques can be used to mimic the real world. However, it is not necessary that you understand everything on the first reading. All the results from this chapter that are needed to proceed with the construction of the bell choir and organ of the next two chapters are repeated concisely at the beginning of Chapter 4.

## 3.1 FIRST-ORDER DISCRETE-TIME SOLUTIONS

A discrete-time version of Equation 2.17 is the goal of this chapter. This section shows how to solve the simpler first-order equation. Two such solutions are then combined to obtain the solution to the simple harmonic oscillator equation. The general first-order equation is

### 3.1.1 First-Order, Homogeneous

When *f* = 0, Equation 3.1 is called *homogeneous*, meaning simply that the equation is not forced. In that case, the continuous-time solution is trivially obtained: ...