# Waveform

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Sine, square, triangle, and sawtooth waveforms
A sine, square, and sawtooth wave at 440 Hz
A composite waveform that is shaped like a teardrop.
A waveform generated by a synthesizer

A waveform is a variable that varies with time, usually representing a voltage or current.[1]

Waveforms are conventionally graphed with time on the horizontal axis.

In electronics, an oscilloscope can be used to visualize a waveform on a screen. A waveform can be depicted by a graph that shows the changes in a recorded signal's amplitude over the duration of recording.[2] The amplitude of the signal is measured on the ${\displaystyle y}$-axis (vertical), and time on the ${\displaystyle x}$-axis (horizontal).[2]

## Examples

Simple examples of periodic waveforms include the following, where ${\displaystyle t}$ is time, ${\displaystyle \lambda }$ is wavelength, ${\displaystyle a}$ is amplitude and ${\displaystyle \phi }$ is phase:

• Sine wave${\displaystyle (t,\lambda ,a,\phi )=a\sin {\frac {2\pi t-\phi }{\lambda }}}$. The amplitude of the waveform follows a trigonometric sine function with respect to time.
• Square wave${\displaystyle (t,\lambda ,a,\phi )={\begin{cases}a,&(t-\phi ){\bmod {\lambda }}<{\text{duty}}\\-a,&{\text{otherwise}}\end{cases}}}$. This waveform is commonly used to represent digital information. A square wave of constant period contains odd harmonics that decrease at −6 dB/octave.
• Triangle wave${\displaystyle (t,\lambda ,a,\phi )={\frac {2a}{\pi }}\arcsin \sin {\frac {2\pi t-\phi }{\lambda }}}$. It contains odd harmonics that decrease at −12 dB/octave.
• Sawtooth wave${\displaystyle (t,\lambda ,a,\phi )={\frac {2a}{\pi }}\arctan \tan {\frac {2\pi t-\phi }{2\lambda }}}$. This looks like the teeth of a saw. Found often in time bases for display scanning. It is used as the starting point for subtractive synthesis, as a sawtooth wave of constant period contains odd and even harmonics that decrease at −6 dB/octave.

The Fourier series describes the decomposition of periodic waveforms, such that any periodic waveform can be formed by the sum of a (possibly infinite) set of fundamental and harmonic components. Finite-energy non-periodic waveforms can be analyzed into sinusoids by the Fourier transform.

Other periodic waveforms are often called composite waveforms and can often be described as a combination of a number of sinusoidal waves or other basis functions added together.

## References

1. ^ David Crecraft, David Gorham, Electronics, 2nd ed., ISBN 0748770364, CRC Press, 2002, p. 62
2. ^ a b "Waveform Definition". techterms.com. Retrieved 2015-12-09.

## Further reading

• Yuchuan Wei, Qishan Zhang. Common Waveform Analysis: A New And Practical Generalization of Fourier Analysis. Springer US, Aug 31, 2000
• Hao He, Jian Li, and Petre Stoica. Waveform design for active sensing systems: a computational approach. Cambridge University Press, 2012.
• Solomon W. Golomb, and Guang Gong. Signal design for good correlation: for wireless communication, cryptography, and radar. Cambridge University Press, 2005.
• Jayant, Nuggehally S and Noll, Peter. Digital coding of waveforms: principles and applications to speech and video. Englewood Cliffs, NJ, 1984.
• M. Soltanalian. Signal Design for Active Sensing and Communications. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.
• Nadav Levanon, and Eli Mozeson. Radar signals. Wiley. com, 2004.
• Jian Li, and Petre Stoica, eds. Robust adaptive beamforming. New Jersey: John Wiley, 2006.
• Fulvio Gini, Antonio De Maio, and Lee Patton, eds. Waveform design and diversity for advanced radar systems. Institution of engineering and technology, 2012.
• John J. Benedetto, Ioannis Konstantinidis, and Muralidhar Rangaswamy. "Phase-coded waveforms and their design." IEEE Signal Processing Magazine, 26.1 (2009): 22-31.