Are there steps to find the Fourier series for transformed graphs?

[rforrevenge]

Hey everybody! I am an undergrad student ,in an ee department.Ive taken "signals and systems" and two ago we got into the Fourier part. The question i have is whether or not there are some steps someone can follow in order to find the Fourier series given some specific graph.Now,our professor told us that there are some graphs(square pulse,sawtooth etc) that we know their Fourier series function.How about some other graphs similar to them?How can we find their Fourier Series?Are there some steps someone can follow in order to do that?

P.S:When i say graphs similar to them i mean the graphs that we already know their Fourier series; but are a little "transformed" i.e a sawtooth with a bigger amplitude or a shifted square pulse.

 

[Proton Soup]

yes, you start by writing the function of your "graph"

 

[verdatel]

yes, we use the Fourier transforms of the shapes we already know (square, impulses, sines and cosines etc.). But then we combine them with the properties of the Fourier transform to get the Fourier transforms of more complicated signal waveforms.

For example, if we perform the convolution of two squares, we can produce a triangle. Convolution in the time domain is associated with multiplication in the Fourier domain. So when we perform the convolution of two squares, we multiply the sinc functions in the Fourier domain to get a sinc squared shape.

Another example, we can use the linearity property for shapes that involve the sum of two rectangles centred around the origin to give a 'box-upon-a-box' or 'staircase' kind of shape. We can divide the shape into the sum of two or how many ever rect functions and then treat each rect function separately.

Thats a crude way of putting it but I hope it helps.

 

[tabi]

what are the applications of Fourier series

 

[dstwofer]

The Fourier Transform can be used as a way of visualizing a signal composed of multiple frequencies, as individual frequency components. In other words It can be used to visualize Harmonic content of a signal. Fourier Transforms have many applications in digital & analogue signal processing.

 

[Phrak]

Umm. I think we're 'convoluting' transforms and sums.

 

[tabi]

what arre the physical significance of Hermites equation

 

[fizz_it]

Yes you can take any periodic wave shape and deconstruct it and then reconstruct it using Fourier analysis. If you scribble a wave form on a piece of graph paper and then use discrete analysis, you can get a close mathematical expression that represents the wave form. The higher the number of harmonics you choose to use, the closer the approximation.

I've done it before, it is very fun to do.

Look at this page
http://www.jhu.edu/~signals/phasorapplet/phasorappletindex.htm

If you choose any pulse shape but only one harmonic, you get a sine wave. Not surprising since the Fourier series is a sum of amplitude scaled sines and cosines, if you only include one sine, then you get a sine wave. As you include more sine waves at different frequencies and amplitudes then interesting things happen. Choose rectangular pulse and 2 harmonics.It looks more like a rectangle than a sine wave but it is still a lousy approximation. Increase the number of harmonics (or terms in the approximation) then the shape gets better and better

you see in the complex plane window all of the terms of the approximation rotate in a periodic fashion. And all of the terms are linked head to tail. Each term has its own length (this is the scaled part of the approximation) and each term rotates at it's own frequency. As the terms (vectors) rotates the last term holds a pen in it's hand and the path that it traces is the signal that comes out.

If you plot this path in the time domain instead of the complex domain you get a square pulseWhen you write out a Fourier series, each term is one of the rotating vectors.

You can scribble any (there are rules) periodic waveform on a piece of paper and then deconstruct it into an equation with Fourier analysis. then reconstruct it by plugging in values of time into the equation

Hope this helps