Pulse wave

File:Pulse wave 33.33 percent Fourier series 50 harmonics.pngThe Fourier series expansion for a rectangular pulse wave with period $T$, amplitude $A$ and pulse length $\backslash tau$ isSmith, Steven W. The Scientist & Engineer's Guide to Digital Signal Processing {{ISBN|978-0966017632}}

$$x(t)\; =\; A\; \backslash frac\{\backslash tau\}\{T\}\; +\; \backslash frac\{2A\}\{\backslash pi\}\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash left(\backslash frac\{1\}\{n\}\; \backslash sin\backslash left(\backslash pi\; n\backslash frac\{\backslash tau\}\{T\}\backslash right)\; \backslash cos\backslash left(2\backslash pi\; nft\backslash right)\backslash right)$$

where $f\; =\; \backslash frac\{1\}\{T\}$.

Equivalently, if duty cycle $d\; =\; \backslash frac\{\backslash tau\}\{T\}$ is used, and $\backslash omega\; =\; 2\backslash pi\; f$:

$$x(t)\; =\; Ad\; +\; \backslash frac\{2A\}\{\backslash pi\}\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash left(\backslash frac\{1\}\{n\}\backslash sin\backslash left(\backslash pi\; n\; d\; \backslash right)\backslash cos\backslash left(n\; \backslash omega\; t\; \backslash right)\; \backslash right)$$

Note that, for symmetry, the starting time ($t=0$) in this expansion is halfway through the first pulse.

Alternatively, $x(t)$ can be written using the Sinc function, using the definition $\backslash operatorname\{sinc\}x\; =\; \backslash frac\{\backslash sin\; \backslash pi\; x\}\{\backslash pi\; x\}$, as

$$x(t)\; =\; A\; \backslash frac\{\backslash tau\}\{T\}\; \backslash left(1\; +\; 2\backslash sum\_\{n=1\}^\backslash infty\; \backslash left(\backslash operatorname\{sinc\}\backslash left(n\backslash frac\{\backslash tau\}\{T\}\; \backslash right)\backslash cos\backslash left(2\backslash pi\; n\; f\; t\backslash right)\; \backslash right)\; \backslash right)$$

or with $d\; =\; \backslash frac\{\backslash tau\}\{T\}$ as

$$x(t)\; =\; A\; d\; \backslash left(1\; +\; 2\backslash sum\_\{n=1\}^\backslash infty\; \backslash left(\backslash operatorname\{sinc\}\backslash left(n\; d\backslash right)\backslash cos\backslash left(2\backslash pi\; n\; f\; t\backslash right)\; \backslash right)\; \backslash right)$$

A pulse wave can be created by subtracting a sawtooth wave from a phase-shifted version of itself. If the sawtooth waves are bandlimited, the resulting pulse wave is bandlimited, too.

The harmonic spectrum of a pulse wave is determined by the duty cycle.Pejrolo, Andrea and Metcalfe, Scott B. (2017). Creating Sounds from Scratch, p.56. Oxford University Press. {{ISBN|9780199921881}}.Snoman, Rick (2013). Dance Music Manual, p.11. Taylor & Francis. {{ISBN|9781136115745}}.Skiadas, Christos H. and Skiadas, Charilaos; eds. (2017). [https://books.google.com/books?id=CAhEDwAAQBAJ&dq=%22rectangle+wave%22+harmonics&pg=PT440 Handbook of Applications of Chaos Theory], {{unpaginated}}. CRC Press. {{ISBN|9781315356549}}."[http://pages.uoregon.edu/emi/14.php Electronic Music Interactive: 14. Square and Rectangle Waves]", UOregon.edu.Hartmann, William M. (2004). Signals, Sound, and Sensation, p.109. Springer Science & Business Media. {{ISBN|9781563962837}}. Acoustically, the rectangular wave has been described variously as having a narrow/thin,Reid, Gordon (February 2000). "[http://www.soundonsound.com/sos/feb00/articles/synthsecrets.htm Synth Secrets: Modulation]", SoundOnSound.com. Retrieved May 4, 2018.Souvignier, Todd (2003). Loops and Grooves, p.12. Hal Leonard. {{ISBN|9780634048135}}.Cann, Simon (2011). [https://books.google.com/books?id=QTBVDQAAQBAJ&dq=pulse+wave+sawtooth+wave&pg=PT20 How to Make a Noise], {{unpaginated}}. BookBaby. {{ISBN|9780955495540}}.Aikin, Jim (2004). Power Tools for Synthesizer Programming, p.55-56. Hal Leonard. {{ISBN|9781617745089}}.Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. {{ISBN|9780881887143}}. nasal/buzzy/biting, clear,Holmes, Thom (2015). Electronic and Experimental Music, p.230. Routledge. {{ISBN|9781317410232}}. resonant, rich, round and bright sound. Pulse waves are used in many Steve Winwood songs, such as "While You See a Chance".{{cite web|url=http://www.keyboardmag.com/lessons/1251/synth-soloing-in-the-style-of-steve-winwood/50240|title=Synth Soloing in the Style of Steve Winwood |last=Kovarsky|first=Jerry|date=Jan 15, 2015|website=KeyboardMag.com|access-date=May 4, 2018}}

• Gibbs phenomenon
• Pulse shaping
• Sinc function
• Sine wave

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Category:Waves