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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.
missing image

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.
missing image

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

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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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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