# what is sinosoidal basis function in FFT or DFT?

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#### victoria_jitesh

##### Member level 2 what is sinosoidal basis function in FFT or DFT?
how it avoids interference in OFDM systems?

#### Dmitrij The basis function of DFT are harmonic functions (sinus and cosinus) of different frequencies. This basis, which comprises these harmonics, is complete and orthogonal - thus, satisfies the common requirements for being a basis. Also, these harmonics may be expressed in complex form through exponents with imaginary arguments, which also possess the mentioned basic properties:

DFT: f = sum[k=0,..,N-1] {f(k)*exp(-jwnk/N)}

{exp(-jnw)}, n=0,1,....

FFT is just the way of calculating the DFT which significantly increases the spped of computation and thus reduces the amount of time for it. So the basis functions for DFT and FFT are absolutely the same.

With respect,

Dmitrij basis functions -- what is you "unit" when you analyse your waveform. You can use any basis functions -- square Trangular ... but sinusoid is the preferred one for variety of reasons.

The Orthogonality of the sinusoids helps in recoversing the signals better nad hence used in OFDM

#### Dmitrij Sinusoids are really very desirable functions in many problems of Signal Processing, including yours. Due to the high importance of these properties, I'll proclaim them in brief:

1) Sinusoids (with different frequencies) are orthogonal (it may be proved both in time or frequency domain or with Parseval equation). It allows to evaluate decomposition coefficients much easier, than in the case of orthogonality absence (Remember the generalized Fourier series).
2) When analyzing the arbitrary linear chain with harmonic input signals we can use symbolic method (method of complex amplitudes)
3) Sinusoidal signal never changes its analytical form when passing through any linear system. It means, that the output reaction is sinusoidal as well, only amplitude and initial phase changes. They may be accurately computed by method of complex amplitudes.
4) Sinusoids are smooth, infinitely differentiated and easily tranformed in other trigonometric functions (remember formulas of double argument). Therefore they are suitable for approximating smooth, slowly-changing processes.
5) They have the best frequency localization (Fourier spectrum of sinus is just 1 harmonic on carrier fequency).

The main disatvantage is that they don't have time localization. They are global. Therefore in analyzing non-stationary processes they're usually neglected and wavelets, EMD and SSA are preferred.

With respect,

Dmitrij[/b]

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