# Dsp00100-periodic motion and sinusoids  (Page 5/7)

 Page 5 / 7

## The argument to the sine and cosine functions

If you think of x as being a measurement of time in seconds, and 1/n being a measurement of frequency in cycles/second, the arguments to the sine and cosine functions can be viewed as:

`2*pi(radians/cycle)*(x sec)*(1/n)( cycle/sec)`

If you cancel out like terms, this reduces to:

`2*pi(radians)*(x )*(1/n )`

Thus, with a fixed value of n , for each value of x , the argument represents an angle in radians, which is what is required for use withthe functions of the Java Math library.

## Where does sin(arg) equal zero ?

The value of the sine of an angle goes through zero at every integer multiple of pi radians. This explanation will probably make more sense if you refer back to Figure 3 .

The curve in Figure 3 was calculated and plotted for n equal to 50. The sine curve has a zero crossing for every value of x such that x is a multiple of n/2, or 25.

## Where are the peaks in the cosine function ?

Similarly, the peaks in the cosine curve in Figure 3 occur for every value of x such that x is a multiple of n/2, or 25.

## Composition and decomposition

In theory, it is possible to decompose any time series into a number (quite possibly a very large number) of sine and cosine functions each having its own amplitude and frequency. (In a future module, we will learn how this is possible using a Fourier series or a Fourier transform.)

Conversely, it is theoretically possible to create any time series by adding together just the right combination of sine and cosine functions, each havingits own amplitude and frequency.

## An approximate square waveform

As an example of composition, suppose that I need to create a time series that approximates a square waveform, as shown at the bottom of Figure 6 .

Figure 5. An approximate square waveform.

I can create such a waveform by adding together the correct combination of sinusoids, each having its own frequency and amplitude.

Figure 6. An improved approximate square waveform.

## Successive approximations

The ten curves plotted in Figure 5 and Figure 6 show successive approximations to the creation of the desired square waveform. The bottom curvein Figure 6 is a plot of the following sinusoidal expression containing the algebraic sum of ten sinusoidal terms.

```cos(2*pi*x/50) - cos(2*pi*x*3/50)/3+ cos(2*pi*x*5/50)/5 - cos(2*pi*x*7/50)/7+ cos(2*pi*x*9/50)/9 - cos(2*pi*x*11/50)/11+ cos(2*pi*x*13/50)/13 - cos(2*pi*x*15/50)/15+ cos(2*pi*x*17/50)/17 - cos(2*pi*x*19/50)/19```

## Each curve contains more sinusoidal terms

The top curve in Figure 5 is a plot of only the first sinusoidal term shown above. It is a pure cosine curve.

Each successive plot, moving down the page in Figure 5 and Figure 6 adds another term to the expression being plotted, until all ten terms are includedin the bottom curve in Figure 6 .

## Reasonably good approximation

As you can see, the bottom curve in Figure 6 is a reasonably good approximation to a square wave, but it is not perfect.

(A perfect square wave would have square corners, a flat top, no ripple, and perfectly vertical sides.)

## Each term improves the approximation

If you start at the top of Figure 5 and examine the successive curves, you will see that the approximation to a square wave improves as each new sinusoidalterm is added.

#### Questions & Answers

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Eke Reply
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Joseph
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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da
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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da
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RAW Reply
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Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Brian Reply
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Rafiq
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Damian
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LITNING Reply
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LITNING Reply
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LITNING Reply
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LITNING
scanning tunneling microscope
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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Rafiq
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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Bob Reply
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Bob
The nanotechnology is as new science, to scale nanometric
brayan
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Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
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