Editing Linear system (section)

===Examples===
A [[simple harmonic oscillator]] obeys the differential equation:
<math display="block">m \frac{d^2(x)}{dt^2} = -kx.</math>

If <math display="block">H(x(t)) = m \frac{d^2(x(t))}{dt^2} + kx(t),</math>
then {{math|''H''}} is a linear operator.  Letting {{nowrap|{{math|1=''y''(''t'') = 0}},}} we can rewrite the differential equation as {{nowrap|{{math|1=''H''(''x''(''t'')) = ''y''(''t'')}},}} which shows that a simple harmonic oscillator is a linear system.

Other examples of linear systems include those described by <math>y(t) = k \, x(t)</math>, <math>y(t) = k \, \frac{\mathrm dx(t)}{\mathrm dt}</math>, <math>y(t) = k \, \int_{-\infty}^{t}x(\tau) \mathrm d\tau</math>, and any system described by ordinary linear differential equations.<ref name="Nahvi_2014" /> Systems described by <math>y(t) = k</math>, <math>y(t) = k \, x(t) + k_0</math>, <math>y(t) = \sin{[x(t)]}</math>, <math>y(t) = \cos{[x(t)]}</math>, <math>y(t) = x^2(t)</math>, <math display="inline">y(t) = \sqrt{x(t)}</math>, <math>y(t) = |x(t)|</math>, and a system with odd-symmetry output consisting of a linear region and a saturation (constant) region, are non-linear because they don't always satisfy the superposition principle.<ref name="DeerghaRao_2018">{{cite book | title = Signals and Systems | first = K. | last = Deergha Rao | publisher = Springer | year = 2018 | pages = 43–44 | isbn = 978-3-319-68674-5}}</ref><ref name="Chen_2004">{{cite book | title = Signals and systems | edition = 3 | first = Chi-Tsong | last = Chen | publisher = Oxford University Press | year = 2004 | pages = 55–57 | isbn = 0-19-515661-7}}</ref><ref name="ElAliKarim_2008">{{cite book | title = Continuous Signals and Systems with MATLAB | edition = 2 | first1 = Taan S. | last1 = ElAli | first2 = Mohammad A. | last2 = Karim | publisher = CRC Press | year = 2008 | page = 53 | isbn = 978-1-4200-5475-0}}</ref><ref name="Apte_2016">{{cite book | title = Signals and Systems: Principles and Applications | first = Shaila Dinkar | last = Apte | publisher = Cambridge University Press | year = 2016 | page = 187 | isbn = 978-1-107-14624-2}}</ref>

The output versus input graph of a linear system need not be a straight line through the origin. For example, consider a system described by <math>y(t) = k \, \frac{\mathrm dx(t)}{\mathrm dt}</math> (such as a constant-capacitance [[capacitor]] or a constant-inductance [[inductor]]). It is linear because it satisfies the superposition principle. However, when the input is a sinusoid, the output is also a sinusoid, and so its output-input plot is an ellipse centered at the origin rather than a straight line passing through the origin.

Also, the output of a linear system can contain [[Harmonic analysis|harmonics]] (and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by <math>y(t) = (1.5 + \cos{(t)}) \, x(t)</math>. It is linear because it satisfies the superposition principle. However, when the input is a sinusoid of the form <math>x(t) = \cos{(3t)}</math>, using [[List of trigonometric identities#Product-to-sum and sum-to-product identities|product-to-sum trigonometric identities]] it can be easily shown that the output is <math>y(t) = 1.5 \cos{(3t)} + 0.5 \cos{(2t)} + 0.5 \cos{(4t)}</math>, that is, the output doesn't consist only of sinusoids of same frequency as the input ({{nowrap|3 rad/s}}), but instead also of sinusoids of frequencies {{nowrap|2 rad/s}} and {{nowrap|4 rad/s}}; furthermore, taking the [[least common multiple]] of the fundamental period of the sinusoids of the output, it can be shown the fundamental angular frequency of the output is {{nowrap|1 rad/s}}, which is different than that of the input.