Dsp00100-periodic motion and sinusoids

The individual terms

I'm not going to plot the individual sinusoidal terms in the triangular waveform. After the first couple of terms, they have such a small amplitude thatit is difficult to see them.

So what ?

By now, you are may be saying "So what?" What in the world does DSP have to do with bags of sand with holes in the bottom? The answer is everything.

Almost everything that we will discuss in the area of DSP is based on the premise that every time series, whether generated by sand leaking from a bagonto a moving carpet, or acoustic waves generated by your favorite rock band, can be decomposed into a large (possibly infinite) number of sine and cosine waves, each having its own amplitude and frequency.

A practical example

You have probably seen, the kind of stereo music component commonly known as an equalizer. An equalizer typically has about a dozen adjacent slider switchesthat can be moved up and down to cause the music that you hear to be more pleasing. This is a crude form of a frequency filter .

Many equalizers also have a set of vertical display lights that dance up and down as your music is playing. This is a crude form of a frequency spectrum analyzer .

The frequency filters

The purpose of each slider is to attenuate or amplify a band of adjacent frequencies (sine and cosine components, each having its own amplitude and frequency), before they make their way to the output amplifier and impinge on the system speakers. Thus, while you don't have the ability to attenuate oramplify each individual sine and cosine component, you do have the ability to attenuate or amplify them in groups.

In subsequent modules, we will learn how to use digital filters to attenuate or amplify the sine and cosine waves that make up a time series.

The spectrum analyzer

At an instant in time, the height of one of the vertical display lights is an indication of the combined power of the sine and cosine waves contained in asmall band of adjacent frequencies.

In subsequent modules, you will learn how to use Fourier analysis to perform spectral analysis on time series.

Summary

Many physical devices (and electronic circuits as well) exhibit a characteristic commonly referred to as periodic motion.

I used the example of a pendulum to introduce the concepts of periodic motion, harmonic motion, and sinusoids.

I introduced you to the concept of a time series.

I introduced you to sine and cosine functions and the Java methods that can be used to calculate their values.

I told you that almost everything we will discuss in this series on DSP is based on the premise that every time series can be decomposed into a largenumber of sinusoids, each having its own amplitude and frequency.

I introduced you to the concepts of period and frequency for sinusoids.

I introduced you to the concept of radians versus cycles.

I introduced you to the concept of decomposing a time series into a (possibly very large) set of sinusoids, each having its own frequency and amplitude. I told you that we will learn more about this later when we discuss frequencyspectrum analysis.

I introduced you to the concept of composition, where any time series can be created by adding together the correct (possibly very large) set of sinusoids, each having its own frequency and amplitude.

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

I identified the frequency equalizer in your audio system as an example of frequency filtering.

I identified the frequency display that may appear on your frequency equalizer as an example of real-time spectrum analysis

Miscellaneous

This section contains a variety of miscellaneous information.

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