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Nov 29, 2017 Signal and System: Complex Exponential Fourier Series (Example-1) Topics Discussed: 1. Complex Exponential Fourier series solved problem 2. Calculation of Fo.

Sine, square, triangle, and sawtooth waveforms

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**square wave**is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. Although not realizable in physical systems, the transition between minimum and maximum is instantaneous for an ideal square wave.The square wave is a special case of a pulse wave which allows arbitrary durations at minimum and maximum. The ratio of the high period to the total period of a pulse wave is called the duty cycle. A true square wave has a 50% duty cycle (equal high and low periods).

Square waves are often encountered in electronics and signal processing. Its stochastic counterpart is a two-state trajectory.

## Origin and uses[edit]

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 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 radiation or pulses of current that interfere with other nearby circuits, causing noise or errors. To avoid this problem in very sensitive circuits such as precision analog-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 for wind instrument sounds created using subtractive synthesis. Additionally, the distortion effect used on electric guitars clips the outermost regions of the waveform, causing it to increasingly resemble a square wave as more distortion is applied.

Simple two-level Rademacher functions are square waves.

## Definitions[edit]

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 sinusoid:

- $\begin{array}{rl}x(t)& =\mathrm{sgn}\left(\mathrm{sin}\frac{2\pi t}{T}\right)=\mathrm{sgn}(\mathrm{sin}2\pi ft)\\ v(t)& =\mathrm{sgn}\left(\mathrm{cos}\frac{2\pi t}{T}\right)=\mathrm{sgn}(\mathrm{cos}2\pi ft),\end{array}$

which will be 1 when the sinusoid is positive, −1 when the sinusoid is negative, and 0 at the discontinuities. Here,

*T*is the period of the square wave, or equivalently,*f*is its frequency, where*f*= 1/*T*.A square wave can also be defined with respect to the Heaviside step function

*u*(*t*) or the rectangular function Π(*t*):- $\begin{array}{rl}x(t)& =2\left[\sum _{n=-\mathrm{\infty}}^{\mathrm{\infty}}\mathrm{\Pi}\left(\frac{2(t-nT)}{T}-\frac{1}{2}\right)\right]-1\\ =2\sum _{n=-\mathrm{\infty}}^{\mathrm{\infty}}\left[u\left(\frac{t}{T}-n\right)-u\left(\frac{t}{T}-n-\frac{1}{2}\right)\right]-1.\end{array}$

A square wave can also be generated using the floor function directly:

- $x(t)=2\left(2\lfloor ft\rfloor -\lfloor 2ft\rfloor \right)+1$

and indirectly:

- $x(t)={\left(-1\right)}^{\lfloor ft\rfloor}.$

## Fourier analysis [edit]

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 1000 Hz square wave

Graph showing the first 3 terms of the Fourier series of a square wave

Using Fourier expansion with cycle frequency

*f*over time*t*, an ideal square wave with an amplitude of 1 can be represented as an infinite sum of sinusoidal waves:- $\begin{array}{rl}x(t)& =\frac{4}{\pi}\sum _{k=1}^{\mathrm{\infty}}\frac{\mathrm{sin}\left(2\pi (2k-1)ft\right)}{2k-1}\\ =\frac{4}{\pi}\left(\mathrm{sin}(\omega t)+\frac{1}{3}\mathrm{sin}(3\omega t)+\frac{1}{5}\mathrm{sin}(5\omega t)+\dots \right),& \text{where}\omega =2\pi f\end{array}$

220 Hz square wave created by harmonics added every second over sine wave | |

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The ideal square wave contains only components of odd-integer harmonic frequencies (of the form 2π(2

*k*− 1)*f*). Sawtooth waves and real-world signals contain all integer harmonics.A curiosity of the convergence of the Fourier series representation of the square wave is the Gibbs 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 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 without under- or over-shooting. This is impossible to achieve in physical systems, as it would require infinite bandwidth.

Animation of the additive synthesis of a square wave with an increasing number of harmonics

Square waves in physical systems have only finite bandwidth and often exhibit ringing 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.)

5 seconds of square wave at 1 kHz | |

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## Characteristics of imperfect square waves[edit]

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 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 these cases, 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 formulas allowing one to be determined approximately from the other.

## See also[edit]

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

## External links[edit]

- Square Wave Approximated by Sines Interactive demo of square wave synthesis using sine waves.
- Flash applets Square wave.

Retrieved from 'https://en.wikipedia.org/w/index.php?title=Square_wave&oldid=901692714'

$begingroup$Here is a square-wave presented by Fourier series perspective:

Above coefficients shows that a square-wave is composed of only its odd harmonics.

But here below a square-wave is presented by Fourier transform perspective:

Above plot shows that a square-wave is composed of all frequencies not only harmonics, plot is continuous.

When I look at the FFT of a square-wave it looks like the Fourier transform which is continuous.

Series and transform gives different interpretation of a square wave. Why is that?

user16307user16307

$endgroup$## 1 Answer

$begingroup$The Fourier series expansion of a square wave is indeed the sum of sines with odd-integer multiplies of the fundamental frequency. So, responding to your comment, a 1 kHz square wave doest

**not**include a component at 999 Hz, but only odd harmonics of 1 kHz.The Fourier transform tells us what frequency components are present in a given signal. As the signal is periodic in this case, both the Fourier series and the Fourier transform can be calculated, and they should tell us the same information. The Fourier transform of a continuous periodic square wave is composed by impulses in every harmonic contained in the Fourier series expansion. Maybe this picture from Oppenheim's

*Signals and Systems*may help.The actual Fourier transform are only the impulses. The dotted-line is a sinc function that doesn't apply to this question, but gives the notion that this transform has something to do with the transform of a square pulse (i.e. a not periodic signal), which happens to be a sinc.

To put it mathematically:

- The Fourier series coefficients are $$frac{sin(komega_0 T)}{kpi}$$
- The Fourier transform is $$sumlimits_{k=-infty}^{infty}frac{2sin(komega_0 T)}{k}delta(omega - komega_0)$$

So the series coefficients and the Fourier transform are the same, except that there is a proportionality factor of $2pi$ and, in the first case, you plot bars (as the coefficients do not describe a function, they are just numbers), but in the second one you have impulses (because the Fourier transform is a function).

TenderoTendero

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