# Significance of Sinusoidal signals

Can anyone tell what are the significance of sinusoidal signals in the field of science and engineering? sorry for being noob,

• Because they are the eigenfunctions of LTI systems dsp.stackexchange.com/questions/12597/… – AlexTP Oct 4 '18 at 9:16
• might be best to say that $$x(t) = e^{j \Omega t}$$ or $$x[n] = e^{j \omega n}$$ are eigenfunctions of LTI systems and are complex sinusoids. – robert bristow-johnson Oct 4 '18 at 9:28

Significance of (pure) sinusoidals in engineering (summary):

1- Periodic signals: Pure sine-waves, represent a useful and simple tool in modeling various physical phenomena, such as standing waves, rotations, alternating currents etc.

2- Fourier analysis: Periodic signals of any engineering sort (not necessarily of pure sine waves) are conveniently decomposed into pure sine waves acording to Fourier series analysis.

3- Linear Systems: strictly speaking complex exponentials $$e^{j\omega t}$$, are eigenfunctions of LTI systems but sine waves are deeply related. More specifically, the frequency of the sinusoid is preseverd but the amplitude and phase are changed by LTI systems, according to the system's frequency response $$H(\omega)$$. For instance, for a zero phase system (a symmetric noncausal impulse response) where $$H(\omega)$$ is real, then the pure sinusoids will also display the eigenfunction property.

4- Nonperiodic Signals: By the extension of Fourier series analysis to integral analysis of continuum frequencies, almost any practical signal is analysed by periodic sine waves.

Three words: Simple harmonic motion.

Happens a lot in physics and engineering.

Fancier words: Solutions to homogenous second order linear differential equations having purely imaginary roots in their characteristic equation.

Sinusoidal signals are special because you can express any absolutely integrable signal as the sum of sinusoids with different amplitudes, frequencies and phases.

You will see the complex exponential used a lot to represent a sinusoid. It is a very convenient mathematical way of expressing them that allows for easy operations such as phase shifts and addition.