What are Fourier Coefficients and how to calculate them?

February 14th, 2006

I will try to explain you what Fourier coefficients are in an as simple as possible way. It will require you to have some basic knowledge on how to solve integrals because we will use them! I will give my best to explain it to you, please notify me in case I made some mistakes or you don’t understand something. I would be happy if you leave some comments regarding this tutorial.
After we have learnt what Fourier series are it is essential to understand the Fourier coefficients because later on we will use them to break complicated signals into simple waves like sine and cosine. As you have noticed in the Fourier series formula, in figure 1, there are variables like a0, a1, b1, a2, b2 and so on. These variables are called Fourier coefficients.

Figure 1.

I am sure you asked yourself how we can get these values. Well there is a simple and straightforward procedure to get them.

To find an, you have to multiply the original signal by cos[nωt], find that area and then divide it by T/2.
To find bn, you have to multiply the original signal by sin[nωt], find that area and then divide it by T/2.

Why is this possible? Well, when we multiply a wave by another wave that is not the same then the area cancels itself and you get zero but if we multiply two same waves we get some area that is not equal to zero (the area we were looking for was just for one oscillation/period). The area we just got we divide by T/2 or multiply by 2/T.

Let me show you an example, on figure 2 there is a wave multiplied with another wave that is different (the frequency is different). As you can see for one period the area cancels itself and equals zero.

2 Figure 2
Figure 2.

On figure 3 we see two same waves multiplied by each other and we note there is no negative part (part bellow zero on y-axis) and we got some area. The area we just got we have to divide with T/2 and that’s it, that is our coefficient (in this case b1 because I used sine wave as an example).

2 Figure 3
Figure 3.

If it does not make sense to you why we multiplied it with the same wave try to take the integral of cos[ω]; where ω is from 0 to 2π (two pi) :-S. What will you get for the result? You will get 0 because the area from positive part (above zero on y-axis) and negative part (bellow zero on y-axis) cancel each other so that’s why we multiply it.

I think you understand it now . In figure 4 you have the formulas to calculate these coefficients.

2 Figure 4
Figure 4.

Simple! I know! I hope it will be easier for you to follow the upcoming tutorials, introductions and projects now. It would be nice if you leave me a comment so I know that it was helpful or useful to somebody. Thank you for reading it!

Entry Filed under: DSP, MATH

38 Comments Add your own

1. John Moore | | February 18th, 2006 at 6:01 pm

I am enjoying the review. I took courses in school a few years ago but have not had any opportunity to apply these techniques. It’s too easy to forget this stuff. Thank you.
2. Refik | | February 19th, 2006 at 10:40 am

Hey John,
Thanks for the comments. Oh, yeah, these things are easily forgetable, however just a reading can help… I am glad people like it and are enjoying the articles.
3. naveen | | February 23rd, 2006 at 1:04 pm

Its great. U made it easy for me. thankyou
4. James | | March 3rd, 2006 at 9:48 pm

Kudos on the article! You explained it in really understandable terms . Thx
5. MrBeefy | | March 5th, 2006 at 3:46 am

you did a much nicer job of explaining fourier transforms than any of my past math teachers. thanks!
6. refikh | | March 6th, 2006 at 5:31 pm

Dear MrBeefy,
thank you for the nice words. It took me a while to understand it as well. There was something about explaining in a DSP book I am currently reading: Books shouldn’t be written to impress your colleagues, they should be written to teach you something. I love it when something is explained in a simple manner
7. Steven Hepting | | March 29th, 2006 at 6:59 am

This is pretty awesome. I have a terrible time remembering things (I just tooks a Calculus V last semester but don’t remember any of it) so this is great. It’s also cool to see the waves being multiplied by one another. Not something I’d think of on my own. This is going right to del.icio.us to remember for later.
8. Stig | | April 5th, 2006 at 4:28 pm

I’m going through the methods of churning out the coefficients but its making my head go funny. Seeing a simple explanation helps because it assurs me I know the essence of what is going on. Onwards I muddle!
9. Alex Dong | | April 26th, 2006 at 9:14 am

Excellent article. In signal processing, the FFT has been treated as the fundamental concept but why is that? Is it only because via FFT, we could convert the time-domain signals into frequency domain and pick out only the signals we want?

One problem with most signal processing book is that they simply scared me away by showing tons of fomula without telling me what the overall techniques are most useful for. Your articles fill this gap pretty well. Good job and keep it up.
10. radbert | | May 26th, 2006 at 2:54 pm

Great job guys…
11. Le Béru | | May 31st, 2006 at 1:06 pm

Just perfect. Thanks a lot!
12. Kayun | | June 7th, 2006 at 9:15 am

Excellent!!! Thank you very much.
13. nurisya | | June 19th, 2006 at 5:59 am

the writing was indeed help me to understand better about Fourier Series. I look forward for your other article
14. Vieux_Machin | | June 29th, 2006 at 6:17 pm

Really easy to understand, I feel really better with Fourier now. Thank you very much
15. Atul | | July 21st, 2006 at 3:45 pm

Great job done to explain fourier coefficients, Simply the DSP books makes u scary by making the whole book full of cumbersome formulas without telling what they want to prove and how to implement it. Good Job but I would like to see this tuorial to go further in the same easy way………. and explain all these complex things like convolutions (linear circular ) where they are used typical applications and most importanly topics covering IFFT and FFT filters and how they use these complex mathematical computations in filtering and modulation of signals.

Looking forward for really cool good tutorials….

“DSP is real fun, if tought practically and properly”
16. Jorhabib | | September 8th, 2006 at 1:28 pm

Hey… I liked the way I found this information… well distributed and simple… However I wondered: What happens when Coefficients are the same. My teacher told us about about a Compact Representation.
Thanks
17. niba | | September 15th, 2006 at 11:47 pm

rocking!! Great stuff!!
18. deepa | | September 18th, 2006 at 10:15 pm

excellent description
19. Mike | | October 6th, 2006 at 12:53 am

Please, this might be a stupid question. But I was wondering, what is “T” ==> Period for the whole wave or one oscillaton?
20. Bill | | October 22nd, 2006 at 1:59 pm

Just a quick one….

When you are determining the coefficient “a0″
should the value not also be 2/T as opposed to 1/T???
21. Alif Din | | October 26th, 2006 at 10:58 pm

Great job, Now I realy come to know it.
22. suba | | November 27th, 2006 at 7:32 am

this is very excellent one.really i apriciated u.
23. onwordi | | November 29th, 2006 at 4:16 pm

FS is a very tech topic.you are really a good teacher.
24. Gavin | | December 20th, 2006 at 9:41 am

Thanks for these Fourier tutorials. I’m looking to get a grasp of them so I can show my students one reason sine and cosine waves are important. These articles helped me in that quest.
25. Yuli Vlasdimirsky | | February 1st, 2007 at 6:20 pm

Thanks for this short, but comprehensive tutorial.
Regards,
Yuli
26. James | | February 9th, 2007 at 3:33 pm

Very usefiul tutorial. Like the laymans language too. How would i analyze a specific signal using this fourier series? Please put up an example if you can
27. Jim Herzing | | March 20th, 2007 at 2:34 pm

Very useful tutorial. I am involved in tire uniformity work where we measure the forces in a tire. (i.e. radial and lateral force). We apply a known load to a tire on a testing machine and receive a signal from load cells to tell us how the tire strength varies. (we call “highs” and “low” around the tire. “Highs” are strong points and “Low” are weak points. For example, say we collected the following points (positive numbers are highs and negative numbers are “lows”.) How would you calculate the Fourier series?

Point
-7
-4
1
7
8
9
10
10
13
13
14
14
13
13
8
10
6
3
0
-1
-3
-5
-7
-8
-7
-6
-7
-9
-7
-9
-7
-6
-8
-12
-11
-12
-10
-7
28. Muhammad Awais | | March 21st, 2007 at 1:08 pm

i am studying image processing.
your page is the best in the internet to understand the role of forier series/coeffecients used to represent the periodic pattern.
e.g when we want to represent the complex signal/pattern constructing the shape of some polygon then different numbers of signals appropriatly cancelling and reinforcing each other to draw the required shape nicely handeled by the coeffecients with them.
29. Mark | | March 29th, 2007 at 9:24 am

Thank you for a good tutorial.
30. Najib | | March 29th, 2007 at 9:35 am

great job, her is some applets if it has interest. follow the link
http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=33

Regards
Najib Lugar
31. iserver | | April 7th, 2007 at 10:40 am

thank’s for the tut
i need to know how to determinate the length of the frequency?
32. Juan G | | April 23rd, 2007 at 5:44 am

Thanks for taking the time to do this. You’re awesome!
33. robab | | May 25th, 2007 at 2:17 pm

thank you, but i need to know about furrier coefficients of functions that has been defined in T(T is the unit circle), if you help me, i will be happy.
34. Radha | | May 29th, 2007 at 3:21 pm

really really great and very useful.
god bless you.

Radha.
35. vijay cvr college of engg | | July 26th, 2007 at 2:45 pm

thank’s.very good tutorial
36. Rinjo Jose | | July 30th, 2007 at 8:26 am

could you please help me for creating a traingular wave in 8051 with out DAC or in C or c++
37. MAyank GUpta | | September 10th, 2007 at 3:21 pm

Sir, could u please help me in Fractional Fourier Transform and Gabor Transform
38. shweta | | October 2nd, 2007 at 9:34 pm

sir it ws really gud…………………..
u made it easy 4 me