Square wave

The square wave signal or the square wave designates a periodic signal that switches back and forth between two values and shows a square curve over time in a diagram. It can be unipolar or bipolar.

Comparison of the elementary forms of oscillation

The square-wave signal is one of the basic forms of sound generation in synthesizers and has a "hollow" sound character, because only odd whole-number multiples of a fundamental frequency are generated, which is why it is often used as a basis for imitating flutes and brass instruments.

Sound example for square wave signal with 220 Hz

Signals with an ideally rectangular course only exist theoretically. In reality, the flanks cannot rise vertically and thus make an infinitely steep jump; the instead real jump is described by the rise and fall times . Owing to the capacitive and inductive behavior of the transmission lines, among other things, a square-wave signal often shows undershoot and overshoot .

If a square wave is used as the clock signal , the exact value of the duty cycle is irrelevant in many cases; however, it influences the equivalence and the proportion of the harmonics.

generation

A square wave signal is generated either with an astable multivibrator , generally with a square wave generator , or from another signal form by means of a Schmitt trigger .

An additional temporally symmetrical signal is obtained with a stable frequency by halving the frequency . When transmitting as alternating voltage (i.e. without a direct component), a time-symmetrical square-wave voltage is also symmetrical in terms of voltage level.

Also crystal oscillators usually give off a square wave, for example, as a clock signal for a microprocessor is used. However, the quartz oscillator itself performs a sinusoidal oscillation.

Forms that deviate from this (e.g. for measurement purposes) are now generated with function generators using direct digital synthesis (DDS) .

properties

Square-wave signals are characterized by the following properties:

Frequency or period
Duty cycle: With a symmetrical square wave it is 50% and can otherwise be in the range between 0 and 100%.
Rise and fall time: square waves with a high slope contain a particularly large number of harmonics (see Fourier analysis below )
Low and high level (for example unipolar with 0 and 5 volts in TTL circuits)

Another property in digital technology is jitter . H. the time discrepancies between the pulses or the frequency fluctuations.

use

Square wave signals are the basis of digital signal processing. Square waves (i.e., periodic square waves) may occur. a. on:

as a clock signal for digital processors and controllers
as a pulse-width modulated signal for sensors, digital-to-analog and analog-to-digital converters , switching regulators and switching power supplies as well as class D audio amplifiers
as a test signal on oscilloscopes to adjust the frequency compensation of the connected measuring tips
at the output of pulse and function generators for laboratory purposes
in synthesizers as one of the fundamental waveforms, often with adjustable pulse width modulation
as a simple, digitally generated sound signal (e.g. signal tones for devices, children's toys)

Spectral observation

The square wave with a single frequency can also be viewed as the sum of an infinite number of individual sine waves with discrete frequencies.

Fourier analysis

The Fourier analysis made possible by use of mathematical methods, the decomposition of a signal into sine and cosine functions. Assuming an ideal and symmetrical square wave signal without a constant component, the following Fourier series results :

{\ displaystyle f (t) = {\ frac {4h} {\ pi}} \ left [\ sin (\ omega t) + {\ frac {1} {3}} \ sin (3 \ omega t) + { \ frac {1} {5}} \ sin (5 \ omega t) + \ ldots \ right] = {\ frac {4h} {\ pi}} \ sum _ {k = 1} ^ {\ infty} {\ frac {\ sin ((2k-1) \ omega t)} {2k-1}}}

with the peak value of the square wave, its fundamental frequency or fundamental angular frequency and the time . The formula shows that the frequency spectrum of a symmetrical square wave signal consists exclusively of odd harmonics. These discrete frequencies and their amplitudes are accordingly also displayed on a spectrum analyzer when a square wave is applied. The amplitudes of the harmonics decrease with increasing frequency. The steeper and sharper the square wave, the higher the harmonics on the frequency scale. ${\ displaystyle h}$ ${\ displaystyle f}$ ${\ displaystyle \ omega = 2 \ pi f}$ ${\ displaystyle t}$

Fourier synthesis

Gibbs phenomenon for a square wave

If the process is reversed and a Fourier synthesis is carried out, the result is of course not an ideal square-wave signal, because such signals cannot be achieved in practice. Due to the lack of the high frequency components that cannot be transmitted in practice, one would expect rounding at the jump points. But there is no square-wave signal with rounded corners - the Fourier series development rather leads to a signal form in which the signal dips under the lower (imaginary) impulse roof in front of and behind the jump points and shoots out over the upper (imaginary) impulse roof and fades out in a dampened oscillation. Even if very high frequency components are used, the overshoot does not disappear - the maximum oscillation amplitude of the overshoot remains the same.

This phenomenon is referred to as Gibbs' phenomenon and must not be confused with the under- and overshoot already mentioned, but is nevertheless often referred to in the same way.

(That is to reduce the Gibbs phenomenon essential) to approach in the synthesis of harmonics to a real square wave signal without overshooting and with finite flanks and rounded corners, in the finite sequence of the Fourier development instead of the last link of a so-called Lanczos sigma factor are introduced

Demonstration of the generation of a square wave by superimposing sine waves (Fourier synthesis)

literature

Michael Dickreiter: Handbook of the recording studio technology . Volume 1. 6., improved edition. KG Saur, Munich et al. 1997, ISBN 3-598-11320-X .
Curt Rint (ed.): Handbook for high frequency and electrical technicians . Volume 3. 12th, supplemented and completely revised edition. Hüthig and Pflaum, Munich et al. 1979, ISBN 3-8101-0044-7 .
Dieter Zastrow: Electronics. Textbook and workbook. Introduction to analog technology, digital technology, power electronics, programmable logic controllers . 2nd, revised edition. Vieweg, Braunschweig et al. 1984, ISBN 3-528-14210-3 .

Web links

Commons : Square Wave - Collection of pictures, videos and audio files

Calculating the 15 overtones from the fundamental frequency (harmonics)

Individual evidence

↑ http://mathworld.wolfram.com/LanczosSigmaFactor.html Lanczos Sigma Factor on the website of Wolfram Research

[1] ttp://mathworld.wolfram.com/LanczosSigmaFactor.html Lanczos Sigma Factor on the website of Wolfram Research