# Very simple issue on Fourier series

by DWill
Tags: fourier, issue, series, simple
 P: 70 Hi all, I am just trying to prove to myself the Fourier series representation of a periodic rectangular pulse train. The pulses have some period T, and each pulse has magnitude equal to 1 over a duration of T/4, and 0 the rest of the cycle. Using trignometric Fourier series, I get the following: a0 = 1/4 - (dc value = 0.25, this makes sense) an = ∫1*cos(n2πft)dt over (0,T/4) = $\frac{sin(\frac{nπ}{2})}{nπ}$ bn = ∫1*sin(n2πft)dt over (0,T/4) = $\frac{1}{nπ}$(1-cos($\frac{nπ}{2})$ The bn term doesn't make sense to me, because it seems to contribute a dc term for each value of n. I double-checked the integration, although it's a very simple integral. This might just be the case of a really obvious mistake I'm making and I just can't seem to pinpoint it. Thanks for the clarification! EDIT: I recall that if I shift the reference axis so that t=0 lies at the midpoint of a pulse, the function has even symmetry and thus only the an terms will exist. However, I'm a bit confused why these extra dc terms result if I just shift the reference that I'm looking at a bit.