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# 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)

Boxcar Function, Hadamard Matrix, Heaviside Step Function, Rectangle Function, Sawtooth Wave, Triangle Wave, Walsh Function

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## References

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

Square Wave

## Cite this as:

Weisstein, Eric W. "Square Wave." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SquareWave.html