You will find that a square wave can be made from an infinite number of odd harmonic sine waves of progressively reducing amplitude. So, for a Hz square wave, you would need to add together a Hz sine wave as well as Hz, Hz, Hz etc.
You need an infinite number of harmonics to produce a perfect square wave and quite a lot of harmonics to get close to a square wave. The example above shows what happens if you start with Hz sine wave red trace then add in the right proportion of Hz green trace , then add Hz as well blue trace and finally Hz cyan trace. While the waveform is now looking a lot less like a sine wave it is a long way from a perfect square wave.
Even with 32 frequencies i. However, this is a mathematical exercise. Dym, H. Fourier Series and Integrals. New York: Academic Press, Folland, G.
Fourier Analysis and Its Applications. Groemer, H. New York: Cambridge University Press, Fourier Analysis. Cambridge, England: Cambridge University Press, Exercises for Fourier Analysis.
Krantz, S. Lighthill, M. Introduction to Fourier Analysis and Generalised Functions. Morrison, N. Introduction to Fourier Analysis. New York: Wiley, Sansone, G. English ed. New York: Dover, pp. Weisstein, E. Whittaker, E. Weisstein, Eric W. SE, and I thought that people on this site might find some of it worthwhile, and might even want to participate.
Fourier series date at least as far back as Ptolemy's epicyclic astronomy. Adding more eccentrics and epicycles, akin to adding more terms to a Fourier series, one can account for any continuous motion of an object in the sky. Add a comment. Active Oldest Votes. History: To understand the importance of the Fourier transform, it is important to step back a little and appreciate the power of the Fourier series put forth by Joseph Fourier.
The Fourier transform: The Fourier transform can be viewed as an extension of the above Fourier series to non-periodic functions. Thus, the transform is energy preserving. While this falls under the elementary property category, this is a widely used property in practice, especially in imaging and tomography applications, Example: When a wave travels through a heterogenous medium, it slows down and speeds up according to changes in the speed of wave propagation in the medium.
Digital signal processing DSP vs. Analog signal processing ASP The theory of Fourier transforms is applicable irrespective of whether the signal is continuous or discrete, as long as it is "nice" and absolutely integrable.
Improve this answer. Lorem Ipsum Lorem Ipsum 5, 3 3 gold badges 31 31 silver badges 41 41 bronze badges. Note that a Taylor series is not an expansion in terms of the constituent frequencies. For e. The latter is the correct frequency representation, so I'm not sure if any comparisons with Taylor series is apt here. Signals can be represented with respect to many different bases.
Sines and cosines are special because they are the eigenfunctions of LTI systems. Show 4 more comments. Community Bot 1. Peter K. I think that following the philosophy of choice on the academic correctness over "popularity" of an answer, your answer should be be integrated into the above answer provideded by Lorem Ipsum, which despite being selected as the answer with 96 points by the users, lacks this very important point of view.
Edit: Since people asked me to write why the FFT is fast It's because it cleverly avoids doing extra work. It has everything to do with discrete Fourier transforms.
Without the FFT, your Fourier transforms would take time proportional to the square of the input size, which would make them a lot less useful.
But with the FFT, they take time proportional the size of the input times its logarithm , which makes them much more useful, and which speeds up a lot of calculations. Also this might be an interesting read. Where its fast and why do we care that its fast? It's fast because the algorithm is fast It should be paraphrased - "Beside all the nice characteristics explained in other peoples answer, FFT allows it to become a feasible approach in real-time applications". Show 3 more comments.
I want to calculate the TWA osha. We can use the formal mathematical definition too. Scott Scott 4 4 silver badges 7 7 bronze badges. David David 61 1 1 silver badge 1 1 bronze badge.
The Overflow Blog. Does ES6 make JavaScript frameworks obsolete? Podcast Do polyglots have an edge when it comes to mastering programming Featured on Meta. Why are Fourier series important? Are there any real life applications of Fourier series? February 09, A Fourier series is a way of representing a periodic function as a possibly infinite sum of sine and cosine functions.
It is analogous to a Taylor series, which represents functions as possibly infinite sums of monomial terms. A sawtooth wave represented by a successively larger sum of trigonometric terms. For functions that are not periodic, the Fourier series is replaced by the Fourier transform.
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