How can the fractional part function be expanded into Fourier series?
If you plot it for positive and negative x, you will see it is a sawtooth which is antisymmetrical i.e. odd. Trouble is it is not strictly periodic and therefore doesn't have a Fourier series unless you ignore the negative part. You could replace the negative half with a mirror image of the positive half. It would then be periodic, even and have a Fourier series with just cosine terms. The coefficients you could work out using the usual integrals (see Wikipedia).
MATLAB HELP!!!! HOW TO PLOT PIECEWISE FUNCTION AND FOURIER SERIES?
Question is not clear. From my understanding, you need to plot outcomes of two functions for same data range with a specified interval in same plot with common X& Y axis. one is the square wave and onother one is "g(x)=(4/pi)(sin(x)/1+sin(3x)/3+ sin(5x)/5)+sin(7x)/7" you can do this by discreatizing continues functions with some interval generate datasets (x1,y1) for "f(x)={1 0
Can a non-periodic function be expanded into Fourier series?
I find it peculiar that so many people, here, claim that a non-periodic function cannot be expanded into a Fourier series. It can! In fact, a well known exercise demonstration of this is calculating the Fourier expansion of the function, f(x) = x. However, there are very important caveats.The most important of these is that the Fourier Series will only be valid in the interval used for the Fourier series. In fact, the result of doing this will yield a function that is periodic; it will be a sequence of “copies” of the function you are expanding within the interval. Remember the Fourier series of f(x) = x?The expansion gives you this:As you can see, this is only valid in the interval, [math](-\pi, \pi)[/math].I would like to point out, here, that an expansion being invalid outside of a neighborhood is nothing new. Most Taylor expansions have radii of convergence, which means that outside of a radius around a point, the Taylor series is no longer valid.This is all assuming that you can compute a meaningful Fourier series in the interval you are interested in, of course, and for that to be the case, it would have to be integrable in that interval.The other answers, here, are confusing Fourier Transforms with Fourier coefficients. They are related, but they are not the same thing.The Fourier Transform is what you get, when take the limit of the Fourier Coefficients, as you expand your interval out to [math](-\infty, \infty)[/math].And, even then, your transform will be correct, if the function being expanded is Square Integrable, and the expansion will no longer be a series, but an integral.Fourier Transform
Is fourier transform applicable to periodic functions?
First came Fourier series, which consists of decomposing a given continuous time function into harmonics separated by a fundamental frequency f0. Fourier series is applicable only to periodic signals of period T. By setting T as very very large, we can turn a periodic signal to an aperiodic signal. This is Fourier Transform. So FT is for aperiodic signals.Fourier series coefficients at f0, 2f0, 3f0, etc. are always discrete depending on the fundamental frequency chosen. But if fundamental frequency (the space between the harmonics) goes to zero, then Fourier Transform becomes continuous.This is of course for continuous time functions. Discrete signals have additional issues. We have CTFT for continuous signals and DTFT for discrete signals. We DFT for finite length discrete signals.
What is the exact difference between continuous fourier transform, discrete Time Fourier Transform(DTFT), Discrete Fourier Transform (DFT) Fourier series and Discrete Fourier Series (DFS) ? In which cases is which one used?
Fourier series decomposes a given signal into discrete harmonic sinusoids. Assumption is that the signal presented is periodic. Hence the Fourier series coefficients, called FSC, are discrete. We say that that the spectrum developed by the Fourier series is discrete, with each frequency being separated by f0, the lowest frequency in the signal.The Continuous time Fourier Transform (CTFT), is a modification of the Fourier series such that the period of the signal is assumed to extend to infinity as shown in Figure 4.2 above. Hence, the frequency resolution now become very very small. The harmonics are now so very close to each other that the spectrum become continuous as we see in Figure below. However, if you compute the CTFT of a periodic signal, which has a well defined period, the CTFT is discrete just like the FSC. (Very confusing!)DTFTHere the signal is discrete. But just as in CTFT, here we also let the period go to infinity as in figure below.And now its spectrum is also continuous but complicate matters, repeats! Note that CTFT is an integral and DTFT is a summation.If you want to deeply understand this topic, which is of fundamental importance in DSP and many other sciences, you should read my book (these figures come from my book.)This is not a topic that is well taught in universities as it should be.Figures contained here are from “Intuitive Guide to Fourier Analysis and Spectral Estimation”The Intuitive Guide to Fourier Analysis and Spectral Estimation: with Matlab 1, Charan Langton, Victor Levin, Richard G. Lyons - Amazon.com
What is a Fourier series?
A Fourier series is way of approximating a periodic waveform as a weighted sum of harmonically related sine/cosine waves. For example, a square wave may be approximated as the following sum: f(x) = sin x + 1/3 sin 3x + 1/5 sin 5x + 1/7 sin 7x etc + 1/9 sin 9x + .... 1/n sin nx. I have put together a set of common Fourier waveforms using the fabulous Desmos online graphing calculator. Interactive Fourier Series. Try moving the slider about which changes the number of terms. As you add more terms the approximation improves.
Why we take Fourier series for periodic signal and Fourier transform form for non-periodic signal?
Fourier series is the decomposition of a periodic signal into infinite sum series of sinusoidal harmonics. It is essentially a synthesis equation. You synthesize a signal from multiple smaller signals. Why was this method developed? In essence, we needed a representative signal : One signal to represent all kinds of periodic signals. The exponential signal was chosen since it is somewhat a robust signal ie. it retains it’s form after addition, multiplication differentiation, integration etc. Now, we all know that both periodic and aperiodic signals exist in the world. We can think of an aperiodic signal as a signal whose period is infinity. Now, if we attempt to extend the Fourier series to an aperiodic signal, we take the limit of the frequency as tending to zero(since period -> infinity). The summation of the Fourier series becomes an integration and the resulting equation would be the inverse Fourier transform. Then what is the Fourier transform? The Fourier transform of aperiodic signals is actually the counterpart of the Fourier coefficients in periodic signals.
What's is a real world application of complex numbers?
Yea, So we have this project for Algebra Class. We have to research different areas and fields complex numbers are used. We have to summarizze our findings into a one page summary. It sounds preety basic, but it's hard finding stuff on the web. Then we have to create a poster qith picture and equations on it that relate to this field. Hopefully someone can help me out. Thanks.
What is the relationship between the Fourier transform and Fourier series representation of a continuous function?
“How do I convert a distribution over the continuous basis set, into a sum of discrete basis functions?”Just sample the Fourier transform of the non-periodic signal at the frequency of the periodic wave you want to create. The resulting coefficients will then form the Fourier series coefficients of the periodic waveform that can be created by repeating the non-periodic signal, on the time axis. This relation is formally called the Poisson summation formula .