- Source: Pulse wave
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:
Pulse-width modulation (PWM) refers to methods that encode information by varying the duty cycle of a pulse wave.
Pulse-amplitude modulation (PAM) refers to methods that encode information by varying the amplitude of a pulse wave.
Frequency-domain representation
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
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
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
The harmonic spectrum of a pulse wave is determined by the duty cycle. Acoustically, the rectangular wave has been described variously as having a narrow/thin, nasal/buzzy/biting, clear, resonant, rich, round and bright sound. Pulse waves are used in many Steve Winwood songs, such as "While You See a Chance".
See also
Gibbs phenomenon
Pulse shaping
Sinc function
Sine wave
References
Kata Kunci Pencarian:
- Power inverter
- Nami yo Kiitekure
- Glikasi
- Steve Lillywhite
- Musik pop
- Radar Doppler
- János Hebling
- Randolph Zaini
- Meta Platforms
- K-Beauty
- Pulse wave
- Pulse-width modulation
- Pulse wave velocity
- Square wave
- Pulse detonation engine
- Electromagnetic pulse
- Wave velocity
- Modulation
- Pulsus paradoxus
- Triangle wave
