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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Periodic rectangular waveform
A pulse wave's duty cycle D is the ratio between pulse duration 𝜏 and period T.
A pulse wave or pulse train or rectangular wave is a non-sinusoidal waveform that is the periodic version of the rectangular function . It is held high a percent each cycle (period ) called the duty cycle and for the remainder of each cycle is low. A duty cycle of 50% produces a square wave , a specific case of a rectangular wave. The average level of a rectangular wave is also given by the duty cycle.
The pulse wave is used as a basis for other waveforms that modulate an aspect of the pulse wave, for instance:
Frequency-domain representation [ edit ]
Fourier series of a 33.3 % pulse wave, first fifty harmonics (summation in red)
The Fourier series expansion for a rectangular pulse wave with period
T
{\displaystyle T}
, amplitude
A
{\displaystyle A}
and pulse length
τ
{\displaystyle \tau }
is [1]
x
(
t
)
=
A
τ
T
+
2
A
π
∑
n
=
1
∞
(
1
n
sin
(
π
n
τ
T
)
cos
(
2
π
n
f
t
)
)
{\displaystyle x(t )=A{\frac {\tau }{T}}+{\frac {2A}{\pi }}\sum _{n=1}^{\infty }\left({\frac {1}{n}}\sin \left(\pi n{\frac {\tau }{T}}\right)\cos \left(2\pi nft\right)\right)}
where
f
=
1
T
{\displaystyle f={\frac {1}{T}}}
.
Equivalently, if duty cycle
d
=
τ
T
{\displaystyle d={\frac {\tau }{T}}}
is used, and
ω
=
2
π
f
{\displaystyle \omega =2\pi f}
:
x
(
t
)
=
A
d
+
2
A
π
∑
n
=
1
∞
(
1
n
sin
(
π
n
d
)
cos
(
n
ω
t
)
)
{\displaystyle x(t )=Ad+{\frac {2A}{\pi }}\sum _{n=1}^{\infty }\left({\frac {1}{n}}\sin \left(\pi nd\right)\cos \left(n\omega t\right)\right)}
Note that, for symmetry, the starting time (
t
=
0
{\displaystyle t=0}
) in this expansion is halfway through the first pulse.
Alternatively,
x
(
t
)
{\displaystyle x(t )}
can be written using the Sinc function , using the definition
sinc
x
=
sin
π
x
π
x
{\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}}
, as
x
(
t
)
=
A
τ
T
(
1
+
2
∑
n
=
1
∞
(
sinc
(
n
τ
T
)
cos
(
2
π
n
f
t
)
)
)
{\displaystyle x(t )=A{\frac {\tau }{T}}\left(1+2\sum _{n=1}^{\infty }\left(\operatorname {sinc} \left(n{\frac {\tau }{T}}\right)\cos \left(2\pi nft\right)\right)\right)}
or with
d
=
τ
T
{\displaystyle d={\frac {\tau }{T}}}
as
x
(
t
)
=
A
d
(
1
+
2
∑
n
=
1
∞
(
sinc
(
n
d
)
cos
(
2
π
n
f
t
)
)
)
{\displaystyle x(t )=Ad\left(1+2\sum _{n=1}^{\infty }\left(\operatorname {sinc} \left(nd\right)\cos \left(2\pi nft\right)\right)\right)}
Generation [ edit ]
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.
Applications [ edit ]
The harmonic spectrum of a pulse wave is determined by the duty cycle.[2] [3] [4] [5] [6] [7] [8] [9] Acoustically, the rectangular wave has been described variously as having a narrow[10] /thin,[11] [3] [4] [12] [13] nasal[11] [3] [4] [10] /buzzy[13] /biting,[12] clear,[2] resonant,[2] rich,[3] [13] round[3] [13] and bright[13] sound . Pulse waves are used in many Steve Winwood songs, such as "While You See a Chance ".[10]
See also [ edit ]
References [ edit ]
^ Smith, Steven W. The Scientist & Engineer's Guide to Digital Signal Processing ISBN 978-0966017632
^ a b c Holmes, Thom (2015). Electronic and Experimental Music , p.230. Routledge. ISBN 9781317410232 .
^ a b c d e Souvignier, Todd (2003). Loops and Grooves , p.12. Hal Leonard. ISBN 9780634048135 .
^ a b c Cann, Simon (2011). How to Make a Noise , [unpaginated] . BookBaby. ISBN 9780955495540 .
^ 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). Handbook of Applications of Chaos Theory , [unpaginated] . CRC Press. ISBN 9781315356549 .
^ "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 .
^ a b c Kovarsky, Jerry (Jan 15, 2015). "Synth Soloing in the Style of Steve Winwood" . KeyboardMag.com . Retrieved May 4, 2018 .
^ a b Reid, Gordon (February 2000). "Synth Secrets: Modulation ", SoundOnSound.com . Retrieved May 4, 2018.
^ a b Aikin, Jim (2004). Power Tools for Synthesizer Programming , p.55-56. Hal Leonard. ISBN 9781617745089 .
^ a b c d e Hurtig, Brent (1988). Synthesizer Basics , p.23. Hal Leonard. ISBN 9780881887143 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Pulse_wave&oldid=1209163390 "
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● T h i s p a g e w a s l a s t e d i t e d o n 2 0 F e b r u a r y 2 0 2 4 , a t 1 6 : 3 4 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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