FPGA Based Digital World - Fourier Series(FS)

Fourier Series(FS)

Fourier series is a technique ofexpressing a periodic function in terms of sinusoids. Trigonometric Fourier series and exponential Fourier series are both popular.

According to Fourier theorem, any periodic function of frequency ω0 can be expressed as an infinite sum of sine or cosine functions with frequencies of integral multiples of ω0.

Fourier Series

where f(t) is period function with frequency ω0=2π/T.

f(t) = f(t+nT)

Dirichlet Conditions

The condition to yield a convergent Fourier series is Dirichlet conditions(sufficient but not necessary condition):

1) f(t) is single-valued everywhere;

2) f(t) has a finite number of finite discontinuites in any period;

3) f(t) has a finite number of maxima and minima in any period;

4) for any t0,

integral condition

Trigonometric Fourier Series

Equation (1) and (2) are trigonometric Fourier series, where

Trigonometric

Note: the coefficients are calcuated from one period.

Exponential Trigonometric Fourier Series

Equation (3) is complex or exponential trigonometric Fourier series, where

Exponential

or calcuated directly from f(t)

Cn direct calc

The plot of magnitude and phase of Cn versus nω0 are called complex amplitude spectrum and complex pahse spectrum of f(t).

Fourier Series of Typical Functions

1 Square wave

squarewave

For N=2, it could be taken as the combination of two phasors rotaging with ω0 and 3ω0, as shown below. It's similar for other Ns.

square wave

The curves of diffent N on influence of the final shape is below.

square wave

2 Rectangular Pulse Train -

rectangular pulse train

3 Sawtooth wave

sawtooth

4 Triangular wave

Triangular wave

5 Half-wave rectified sine

half rectified

6 Full-wave rectified sine

full wave rectified

7 Full-wave rectified sine

full rectified wave

Fourier Series in Spectrum Analyzer

Spectrums of f(t) are defined as

1) Amplitude spectrum and phase spectru: plot of the amplitude An and phase Фn versus nω0

2) Complex amplitude spectrum and complex pahse spectrum: plot of magnitude and phase of Cn versus nω0

3) Power spectrum: the plot of |Cn|*|Cn| versus nω0

Parameters An/Фn and Cn could also be acquried via Fourier tranform.

Sinusoidal Signal Properties

1) Linear operation of Sinusoidal signals with the same frequency: still sinusoidal with same freqeuncy

2) Linear operation of Sinusoidal signals with the different frequencies: non-sinusoidal


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