Relation To Fourier Analysis and Similar Methods
By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific, simple form, often the response becomes easier to compute,
For example, in Fourier analysis, the stimulus is written as the superposition of infinitely many sinusoids. Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different amplitude and phase.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses.
As another common example, in Green's function analysis, the stimulus is written as the superposition of infinitely many impulse functions, and the response is then a superposition of impulse responses.
Fourier analysis is particularly common for waves. For example, in electromagnetic theory, ordinary light is described as a superposition of plane waves (waves of fixed frequency, polarization, and direction). As long as the superposition principle holds (which is often but not always; see nonlinear optics), the behavior of any light wave can be understood as a superposition of the behavior of these simpler plane waves.
Read more about this topic: Superposition Principle
Famous quotes containing the words relation to, relation, analysis, similar and/or methods:
“We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
—D.H. (David Herbert)
“Only in a house where one has learnt to be lonely does one have this solicitude for things. One’s relation to them, the daily seeing or touching, begins to become love, and to lay one open to pain.”
—Elizabeth Bowen (1899–1973)
“... the big courageous acts of life are those one never hears of and only suspects from having been through like experience. It takes real courage to do battle in the unspectacular task. We always listen for the applause of our co-workers. He is courageous who plods on, unlettered and unknown.... In the last analysis it is this courage, developing between man and his limitations, that brings success.”
—Alice Foote MacDougall (1867–1945)
“Custom, then, is the great guide of human life. It is that principle alone, which renders our experience useful to us, and makes us expect, for the future, a similar train of events with those which have appeared in the past.”
—David Hume (1711–1776)
“A writer who writes, “I am alone” ... can be considered rather comical. It is comical for a man to recognize his solitude by addressing a reader and by using methods that prevent the individual from being alone. The word alone is just as general as the word bread. To pronounce it is to summon to oneself the presence of everything the word excludes.”
—Maurice Blanchot (b. 1907)