The fourier series of a sine wave is 100% fundamental, 0% any harmonics.
When we do a Fourier transformation of a function we get the primary term which is the fundamental frequency and amplitude of the Fourier series. All the other terms, with higher frequencies and lower amplitudes, are the harmonics.
yes a discontinuous function can be developed in a fourier series
The Fourier series can be used to represent any periodic signal using a summation of sines and cosines of different frequencies and amplitudes. Since sines and cosines are periodic, they must form another periodic signal. Thus, the Fourier series is period in nature. The Fourier series is expanded then, to the complex plane, and can be applied to non-periodic signals. This gave rise to the Fourier transform, which represents a signal in the frequency-domain. See links.
Laplace = analogue signal Fourier = digital signal Notes on comparisons between Fourier and Laplace transforms: The Laplace transform of a function is just like the Fourier transform of the same function, except for two things. The term in the exponential of a Laplace transform is a complex number instead of just an imaginary number and the lower limit of integration doesn't need to start at -∞. The exponential factor has the effect of forcing the signals to converge. That is why the Laplace transform can be applied to a broader class of signals than the Fourier transform, including exponentially growing signals. In a Fourier transform, both the signal in time domain and its spectrum in frequency domain are a one-dimensional, complex function. However, the Laplace transform of the 1D signal is a complex function defined over a two-dimensional complex plane, called the s-plane, spanned by two variables, one for the horizontal real axis and one for the vertical imaginary axis. If this 2D function is evaluated along the imaginary axis, the Laplace transform simply becomes the Fourier transform.
The fourier series of a sine wave is 100% fundamental, 0% any harmonics.
When we do a Fourier transformation of a function we get the primary term which is the fundamental frequency and amplitude of the Fourier series. All the other terms, with higher frequencies and lower amplitudes, are the harmonics.
A Fourier series is a set of harmonics at frequencies f, 2f, 3f etc. that represents a repetitive function of time that has a period of 1/f. A Fourier transform is a continuous linear function. The spectrum of a signal is the Fourier transform of its waveform. The waveform and spectrum are a Fourier transform pair.
Fourier series and the Fourier transform
The Fourier series is important because it allows one to model periodic signals as a sum of distinct harmonic components. In other words, representing signals in this way allows one to see the harmonics in a signal distinctly, which makes it easy to see what frequencies the signal contains in order to filter/manipulate particular frequency components.
William Elwood Byerley has written: 'An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics'
what are the limitations of forier series over fourier transform
yes a discontinuous function can be developed in a fourier series
no
Yes. For example: A square wave has a Fourier series.
Fourier series is series which help us to solve certain physical equations effectively
Fourier series is the sum of sinusoids representing the given function which has to be analysed whereas discrete fourier transform is a function which we get when summation is done.