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Sine, square, triangle, and sawtooth waveforms

Square waveFrom Wikipedia, the free encyclopedia

A square wave is a non-sinusoidalperiodic waveform (which can berepresented as an infinite summation ofsinusoidal waves), in which theamplitude alternates at a steadyfrequency between fixed minimum andmaximum values, with the sameduration at minimum and maximum. Thetransition between minimum tomaximum is instantaneous for an idealsquare wave; this is not realizable inphysical systems. Square waves areoften encountered in electronics andsignal processing. Its stochasticcounterpart is a two-state trajectory. Asimilar but not necessarily symmetricalwave, with arbitrary durations atminimum and maximum, is called apulse wave (of which the square waveis a special case).

Contents

1 Origin and uses

2 Examining the square wave

3 Characteristics of imperfect

square waves

4 Other definitions

5 See also

6 External links

Origin and uses

Square waves are universally encountered in digital switching circuits and are naturally generated by binary(two-level) logic devices. They are used as timing references or "clock signals", because their fast transitions aresuitable for triggering synchronous logic circuits at precisely determined intervals. However, as the frequency-domain graph shows, square waves contain a wide range of harmonics; these can generate electromagnetic

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The six arrows represent the first six terms of the

Fourier series of a square wave. The two circles at

the bottom represent the exact square wave (blue)

and its Fourier-series approximation (purple).

(odd) harmonics of a square wave with 1000 Hz

Additive Square Demo

220Hz square wave created byharmonics added every second oversine wave.

Problems playing this file? See media help.

radiation or pulses of current that interfere with other nearby circuits, causing noise or errors. To avoid thisproblem in very sensitive circuits such as precision analog-to-digital converters, sine waves are used instead ofsquare waves as timing references.

In musical terms, they are often described as sounding hollow, and are therefore used as the basis for windinstrument sounds created using subtractive synthesis. Additionally, the distortion effect used on electric guitarsclips the outermost regions of the waveform, causing it to increasingly resemble a square wave as moredistortion is applied.

Simple two-level Rademacher functions are square waves.

Examining the square wave

Using Fourier expansion with cycle frequency f overtime t, we can represent an ideal square wave with anamplitude of 1 as an infinite series of the form

The ideal square wave contains only components ofodd-integer harmonic frequencies (of the form2π(2k-1)f). Sawtooth waves and real-world signalscontain all integer harmonics.

A curiosity of the convergence of the Fourier seriesrepresentation of the square wave is the Gibbsphenomenon. Ringing artifacts in non-ideal square waves

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Animation of the additive synthesis of a square wave with an

increasing number of harmonics

Square wave sound sample

5 seconds of square wave at 1 kHz

Problems playing this file? See media help.

can be shown to be related to this phenomenon. The Gibbs phenomenon can be prevented by the use of σ-approximation, which uses the Lanczos sigma factors to help the sequence converge more smoothly.

An ideal mathematical square wave changes between the high and the low state instantaneously, and withoutunder- or over-shooting. This is impossible to achieve in physical systems, as it would require infinite bandwidth.

Square-waves in physical systems have onlyfinite bandwidth, and often exhibit ringing effectssimilar to those of the Gibbs phenomenon, orripple effects similar to those of the σ-approximation.

For a reasonable approximation to the square-wave shape, at least the fundamental and thirdharmonic need to be present, with the fifthharmonic being desirable. These bandwidthrequirements are important in digital electronics,where finite-bandwidth analog approximationsto square-wave-like waveforms are used. (Theringing transients are an important electronicconsideration here, as they may go beyond theelectrical rating limits of a circuit or cause abadly positioned threshold to be crossedmultiple times.)

The ratio of the high period to the total period of any rectangular wave is called the duty cycle. A true squarewave has a 50% duty cycle - equal high and low periods. The average level of a rectangular wave is also givenby the duty cycle, so by varying the on and off periods and then averaging it is possible to represent any valuebetween the two limiting levels. This is the basis of pulse width modulation.

Characteristics of imperfectsquare waves

As already mentioned, an ideal square wave hasinstantaneous transitions between the high and low levels.In practice, this is never achieved because of physicallimitations of the system that generates the waveform. The times taken for the signal to rise from the low level tothe high level and back again are called the rise time and the fall time respectively.

If the system is overdamped, then the waveform may never actually reach the theoretical high and low levels,and if the system is underdamped, it will oscillate about the high and low levels before settling down. In thesecases, the rise and fall times are measured between specified intermediate levels, such as 5% and 95%, or 10%and 90%. The bandwidth of a system is related to the transition times of the waveform; there are formulasallowing one to be determined approximately from the other.

Other definitions

The square wave in mathematics has many definitions, which are equivalent except at the discontinuities:

It can be defined as simply the sign function of a periodic function, an example being a sinusoid:

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which will be 1 when the sinusoid is positive, −1 when the sinusoid is negative, and 0 at the discontinuities. Anyperiodic function can substitute the sinusoid in this definition.

A square wave can also be defined with respect to the Heaviside step function u(t) or the rectangular function⊓(t):

T is 2 for a 50% duty cycle. It can also be defined in a piecewise way:

when

In terms of sine and cosecant with period p and amplitude a:

A square wave can also be generated using the floor function in the following two ways:

Directly:

And indirectly:

where m is the magnitude and ν is the frequency.

See also

List of periodic functions

Rectangular functionPulse wave

Sine wave

Triangle wave

Sawtooth waveWaveform

Sound

2015/11/6 Square wave - Wikipedia, the free encyclopedia

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MultivibratorRonchi ruling, a square-wave stripe target used in imaging.

External links

Flash applets (http://www.electric1.es/armonicos/armonicosOC.html) Square wave.

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Categories: Waveforms Fourier series

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