Square Wave

The square wave is a periodic waveform consisting of instantaneous transitions between two levels which can be denoted $\pm 1$. The square wave is sometimes also called the Rademacher Function. Let the square wave have period $2L$. The square wave function is Odd, so the Fourier Series has $a_0=a_n=0$ and

$$b_n = {2\over L}\int_0^L\sin\left({n\pi x\over L}\right)\,dx$$

$$= {4\over n\pi} \sin^2({\textstyle{1\over 2}}n\pi) = {4\over n\pi} ... ...begin{array}{ll} 0 & \mbox{$n$\ even}\\ 1 & \mbox{$n$\ odd.}\end{array}\right.$$

The Fourier Series for the square wave is therefore

$$f(x)={4\over\pi}\sum_{n=1,3,5,\ldots}^\infty {1\over n}\sin\left({n\pi x\over L}\right).$$

See also Hadamard Matrix, Walsh Function

References

Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr. Interferometry and Synthesis in Radio Astronomy. New York: Wiley, p. 203, 1986.