Fourier Series of Piecewise Function, , PWL approximation to a sine.

Cuthbert Nyack

The spectrum of signals with jumps in their values tend to converge slowly. Continuous functions tend to have faster converging series. The function opposite is continuous although it has jumps in its first derivative and its spectrum can be seen in the applet below.

The Fourier series of a piecewise continuous function with 8 segments and no discontinuities can be found from the above applet with Fn = 1. The segments are set by the parameters 'a' to 'h'.
Fn = 2 shows the special case of the segments approximating a sine. The spectrum contains only terms with b_n.
Fn = 3 shows Fn = 2 with the spectrum amplified.
Note that with M segments per period, the dominant contributions to the spectrum after n = 1 occur at (nM ± 1) w ie 7,9; 15,17 etc.
Fn = 4 shows the special case of the segments approximating a cosine. The spectrum contains only terms with a_n.
Fn = 5 shows Fn = 4 with the spectrum amplified.
Fn = 4 shows the special case of the segments approximating a cosine delayed by p/4. The spectrum contains equal odd and even components.
Fn = 7 shows a half wave symmetric wave with n segments per quarter period set by 'a' to 'm'. The number of segments is set by the scrollbar labelled 'n'.
Fn = 8 shows the spectrum of an 8 segment PWL approximation to one period of a sine.
Fn = 9 shows the spectrum of a 12 segment PWL approximation to one period of a sine.
Fn = 10 shows the spectrum of a 16 segment PWL approximation to one period of a sine.
Fn = 11 shows the spectrum of a 20 segment PWL approximation to one period of a sine.
Fn = 12 shows the spectrum of a 24 segment PWL approximation to one period of a sine.
Fn = 13 shows the spectrum of a 28 segment PWL approximation to one period of a sine.
Fn = 14 shows the spectrum of a 32 segment PWL approximation to one period of a sine.
Fn = 15 shows the spectrum of a 36 segment PWL approximation to one period of a sine.
Fn = 16 shows the spectrum of a 40 segment PWL approximation to one period of a sine.

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