Fourier Analysis of a Periodic Function

This applet is designed to demonstrate features of the real and imaginary
components in the Fourier series of a periodic function. The Fourier expansion of a real function ƒ(t) may be written:

.

Each component is specified by its amplitude, A_{n}, angular frequency ω_{n} and phase φ_{n}.
If ƒ(t) has period T then (for n = 0, ω_{0}=0,
φ_{0}=0, A_{0}=ƒ(t)).
The display shows the coefficients of the e^{iωnt} and e^{-iωnt} terms namely:

C _{+ωn}= A_{n}e^{iφn}= A_{n}cos(φ_{n})+isin(φ_{n})and C _{-ωn}= A_{n}e^{-iφn}= A_{n}cos(φ_{n})-isin(φ_{n})

The 'real' Fourier series graph shows the real parts of the coefficients and the 'imaginary' Fourier series graph shows
the imaginary parts. The 'real' graph is even since the coefficients of e^{iωnt} and
e^{-iωnt} are the same (Acos(φ)). The 'imaginary' graph is odd since the
coefficients of e^{iωnt} and e^{-iωnt} have opposite signs.

The interactive capability shows the effect on the Fourier spectrum of changing the position of the time origin and also of varying ƒ(t) (by moving the t-axis vertically).

Adjusting the phase angles (φ_{n}): This is done by dragging the wave horizontally with the left mouse button

Adjusting the 'DC' term (ƒ(t)):The phase angles φ

_{n}depend on where the origin is located. If it is taken in the centre of a pulse (for the square wave) or a trough, ƒ(t) is then an even function and all the φ_{n}are zero; so the Fourier components are purely real.

If the time origin is on a rising or falling edge for a duty cycle of 50% then ƒ(t) is an odd function and the Fourier components are purely imaginary.

Dragging the origin makes no difference to the amplitudes A_{n}, as seen in the amplitude spectrum.

The term for ω=0 is purely real and equal to ƒ(t). Dragging the t-axis vertically with the right mouse button affects only the value of the Fourier component at t=0.

As in the previous applet the duty cycle and number of terms displayed can be altered on the panel.