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| Square wave | |
|---|---|
 | |
| General information | |
| General definition | |
| Fields of application | Electronics, synthesizers |
| Domain, codomain and image | |
| Domain | |
| Codomain | |
| Basic features | |
| Parity | Odd |
| Period | 1 |
| Antiderivative | Triangle wave |
| Fourier series | |
Asquare wave is anon-sinusoidal periodic waveform in which the amplitude alternates at a steadyfrequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions between minimum and maximum are instantaneous.
The square wave is a special case of apulse wave which allows arbitrary durations at minimum and maximum amplitudes. The ratio of the high period to the total period of a pulse wave is called theduty cycle. A true square wave has a 50% duty cycle (equal high and low periods).
Square waves are often encountered inelectronics andsignal processing, particularlydigital electronics anddigital signal processing. Itsstochastic counterpart is atwo-state trajectory.
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 are suitable for triggeringsynchronous logic circuits at precisely determined intervals. However, as the frequency-domain graph shows, square waves contain a wide range of harmonics; these can generateelectromagnetic radiation or pulses of current that interfere with other nearby circuits, causingnoise or errors. To avoid this problem in very sensitive circuits such as precisionanalog-to-digital converters,sine waves are used instead of square waves as timing references.
In musical terms, they are often described as sounding hollow, and are therefore used as the basis forwind instrument sounds created usingsubtractive synthesis. They also make up the "beeping" alerts used in many household, commercial, and industrial contexts. Additionally, the distortion effect used onelectric guitars clips the outermost regions of the waveform, causing it to increasingly resemble a square wave as more distortion is applied.
Simple two-levelRademacher functions are square waves.
The square wave in mathematics has many definitions, which are equivalent except at the discontinuities:
It can be defined as simply thesign function of a sinusoid:which will be 1 when the sinusoid is positive, −1 when the sinusoid is negative, and 0 at the discontinuities. Here,T is theperiod of the square wave andf is its frequency, which are related by the equationf = 1/T.
A square wave can also be defined with respect to theHeaviside step functionu(t) or therectangular function Π(t):
A square wave can also be generated using thefloor function directly:and indirectly:
Using the fourier series (below) one can show that the floor function may be written in trigonometric form[1]
UsingFourier expansion with cycle frequencyf over timet, an ideal square wave with an amplitude of 1 can be represented as an infinite sum of sinusoidal waves:
The ideal square wave contains only components of odd-integerharmonic frequencies (of the form2π(2k − 1)f).
A curiosity of the convergence of theFourier series representation of the square wave is theGibbs phenomenon.Ringing artifacts in non-ideal square waves can be shown to be related to this phenomenon. The Gibbs phenomenon can be prevented by the use ofσ-approximation, which uses theLanczos sigma factors to help the sequence converge more smoothly.
An ideal mathematical square wave changes between the high and the low state instantaneously, and without under- or over-shooting. This is impossible to achieve in physical systems, as it would require infinitebandwidth.
Square waves in physical systems have only finite bandwidth and often exhibitringing effects similar to those of the Gibbs phenomenon or ripple effects similar to those of the σ-approximation.
For a reasonable approximation to the square-wave shape, at least the fundamental and third harmonic need to be present, with the fifth harmonic being desirable. These bandwidth requirements are important in digital electronics, where finite-bandwidth analog approximations to square-wave-like waveforms are used. (The ringing transients are an important electronic consideration here, as they may go beyond the electrical rating limits of a circuit or cause a badly positioned threshold to be crossed multiple times.)
As already mentioned, an ideal square wave has instantaneous transitions between the high and low levels. In practice, this is never achieved because of physical limitations of the system that generates the waveform. The times taken for the signal to rise from the low level to the high level and back again are called therise time and thefall time respectively.
If the system isoverdamped, 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 these cases, the rise and fall times are measured between specified intermediate levels, such as 5% and 95%, or 10% and 90%. Thebandwidth of a system is related to the transition times of the waveform; there are formulas allowing one to be determined approximately from the other.