Waveform
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

The term "waveform" refers to the shape of any [graph] of a primary variable plotted on the vertical axis against another secondary variable (commonly time) plotted on the horizontal axis. The axes are sometimes rotated through 90 degrees. An instrument called an oscilloscope can be used pictorially to represent a wave as an image 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.[1] The amplitude of the signal is measured on the ${\displaystyle y}$-axis (vertical), and time on the ${\displaystyle x}$-axis (horizontal).[1]

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 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. ^ a b "Waveform Definition". techterms.com. Retrieved .

• 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.
• 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.

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