Square Wave

The square wave, also called a pulse train, or pulse wave, is a periodic waveform consisting of instantaneous transitions between two levels. The square wave is sometimes also called the Rademacher function. The square wave illustrated above has period 2 and levels  and 1/2. Other common levels for square waves include  and  (digital signals).

Analytic formulas for the square wave  with half-amplitude , period , and offset  include

			(1)
			(2)
			(3)

where  is the floor function,  is the sign function, and  is the inverse hyperbolic tangent.

The square wave is implemented in the Wolfram Language as SquareWave[x].

Let the square wave have period . The square wave function is odd, so the Fourier series has  and

			(4)
			(5)
			(6)
			(7)

The Fourier series for the square wave with period , phase offset 0, and half-amplitude 1 is therefore

(8)

REFERENCES:

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