Pulse wave

This article is about a pulse waveform. For a heart beat, see Pulse. For a Dirac pulse train, see Sampling function. For the aperiodic version, see Pulse function.

For other uses, see Pulse (disambiguation).

The shape of the pulse wave is defined by its duty cycle d, which is the ratio between the pulse duration (τ) and the period (T).

Duty cycles

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.

Pulse-width modulation refers to methods that vary the duty cycle of a pulse wave.

Frequency-domain representation

The Fourier series expansion for a rectangular pulse wave with period $T$, amplitude $A$ and pulse length $\tau$ is $$x(t)=A{\frac {\tau }{T}}+{\frac {2A}{\pi }}\sum _{n=1}^{\infty }\left({\frac {1}{n}}\sin \left(\pi n{\frac {\tau }{T}}\right)\cos \left(2\pi nft\right)\right)$$ where $f={\frac {1}{T}}$.

Equivalently, if duty cycle $d={\frac {\tau }{T}}$ is used, and $\omega =2\pi f$: $$x(t)=Ad+{\frac {2A}{\pi }}\sum _{n=1}^{\infty }\left({\frac {1}{n}}\sin \left(\pi nd\right)\cos \left(n\omega t\right)\right)$$

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

Alternatively, $x(t)$ can be written using the Sinc function, using the definition $\operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}$, as $$x(t)=A{\frac {\tau }{T}}\left(1+2\sum _{n=1}^{\infty }\left(\operatorname {sinc} \left(n{\frac {\tau }{T}}\right)\cos \left(2\pi nft\right)\right)\right)$$ or with $d={\frac {\tau }{T}}$ as $$x(t)=Ad\left(1+2\sum _{n=1}^{\infty }\left(\operatorname {sinc} \left(nd\right)\cos \left(2\pi nft\right)\right)\right)$$

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.

Fourier series of a 33.3% pulse wave, first fifty harmonics (summation in red)

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".

In digital electronics, a digital signal is a pulse train (a pulse amplitude modulated signal), a sequence of fixed-width square wave electrical pulses or light pulses, each occupying one of two discrete levels of amplitude. These electronic pulse trains are typically generated by metal–oxide–semiconductor field-effect transistor (MOSFET) devices due to their rapid on–off electronic switching behavior, in contrast to BJT transistors which slowly generate signals more closely resembling sine waves.

See also

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