Fourier Series Complex Form

Cuthbert Nyack

The general form of the Fourier Series is:-

Using the substitutions below in the general form:-

gives:-

Where

Positive frequencies correspond to anticlockwise rotating vectors and negative frequencies to clockwise rotating vectors. The coefficients d_n can be obtained from the equation:-

The applet below shows some features of sines, cosines and rotating vectors.
Changing Fn changes what is shown.
Fn = 0 shows how counterrotating vectors can produce a cosine.
Fn = 1 shows how counterrotating vectors can produce a sine.
Fn = 2 shows how counterrotating vectors can produce a cosine with variable phase.
Fn = 3 shows how counterrotating vectors can produce a sine with variable phase.
Fn = 4 shows how a sine and a cosine can produce an anticlockwise rotating vector.
Fn = 5 shows how a sine and a cosine can produce a clockwise rotating vector.
Fn = 6 shows how a signal consisting of 2 sinusoids can be derived from rotating vectors. The frequency and phase of the second sinusoid can be set by r and f.
Fn = 7 shows how an approximate triangular signal can be represented by rotating vectors.

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