Dsp00100-periodic motion and sinusoids

I will show you examples of using composition to create a square waveform and a triangular waveform.

I will identify some real-world examples of frequency filtering and real-time spectral analysis

Discussion and sample code

Periodic motion

The most difficult decision that I must make for this series is to decide where to begin. I need to begin at a sufficiently elementary level that you willunderstand everything that you read. So, before getting into the actual topic of digital signal processing, I'm going to take you back to some elementary physicsand mathematical concepts and discuss periodic motion .

Many physical devices (and electronic circuits as well) exhibit a characteristic commonly referred to as periodic motion. This includes pendulums,acoustic speakers, springs, human vocal cords, etc.

Motion of a pendulum

For example, consider the pendulum on a grandfather clock. Once you start the pendulum swinging, it will swing for a long time (actually, the spring in the clock gives it a little kick during each repetition, so it will continue toswing until the spring runs down).

The most important characteristic of the motion of the pendulum is that every repetition takes almost exactly the same amount of time. In other words, theperiod of time during which the pendulum swings from one side to the other is very constant. That is the source of the term periodic motion .

A bag of sand

Even without the spring to help it along, a pendulum swinging in a low-friction environment will swing for a long time. Visualize a pendulumconsisting of a bag of sand tied to the end of a long rope suspended from your ceiling (sheltered from the wind). Assume that the bag is filled with colored sand, and that it has a small hole in the bottom. Assume that you startit swinging in a back-and-forth motion (no circular motion).

The sand traces out a pattern

As the bag of sand swings back and forth, a little bit of sand will leak out of the hole and land on the floor below. The sand will trace out a line, whichis parallel to the direction of motion of the bag of sand.

Now assume that you carefully drag a carpet across the floor under the bag at a uniform rate, perpendicular to the direction of motion of the bag. This willcause the sand that leaks from the bag to trace out a zigzag pattern on the carpet very similar to the curved line shown in Figure 1 .

Figure 1. A sinusoidal function.

A sinusoidal function

The shape of the curve shown in Figure 1 is often referred to as a sinusoidal function.

In this assumed scenario, the bag is swinging back and forth along the vertical axis in Figure 1 , and the carpet is being dragged along the horizontal axis. The motion of the bag causes the sand to be leaked in a back-and-forthzigzag pattern. The motion of the carpet causes that pattern to be elongated along the horizontal axis.