Question:
use the principle of superposition to find the Fourier series for the output y(t)?
2009-05-09 22:33:10 UTC
"use the principle of superposition to find the Fourier series for the output y(t)"

This is a quote from a question i am working one. I was wondering if anyone knew what the "principle of super position" means in relation to finding a fourier series for a simple equation?
Three answers:
Bramble
2009-05-10 01:21:53 UTC
Superposition is simply adding things together (superimposing one upon the other), which you can do provided the prevailing physical conditions governing addition are linear.



If you have two Fourier series F1 and F2 which represent the (periodic) functions y1 and y2 respectively, then y1+y2 will be represented by F1 +F2.



End of message!
doug_donaghue
2009-05-10 05:45:24 UTC
It's 'superposition' (all one word). Yes, the principle of superposition simply states that if f(t) and g(t) are orthogonal functions which are solutions of some arbitrary ODE (Ordinary Differential Equation) and any attending boundry conditions, then any linear combination(s) of f(t) and g(t) are also solutions of that ODE. That is, the set of all solutions f(t), g(t), h(t)....... form the basis functions of a vector space over t in which any point is a solution of the ODE.

In the case of most 'linear transforms' (Fourier, LaPlace, Bessel, Walsh, Rademacher, etc.) the basis functions are complete over some interval (that is, there are an infinite number of them) so that the vector space that they span is also infinite-dimensional.



Doug
tabletennisrulez
2009-05-10 05:44:20 UTC
If we consider the trigonometric form of fourier series, the fourier series is the summation of a series of of sin and cos functions that give rise to the given expression.



e.g. y(t)=∑(ancos nx +bnsin nx)



summation for n=1 to n=infinity



an refers to a subscript n,

bn refers to b subscript n,



So, the principle of superposition refers to the superposition (addition) of many sin and cos functions to give rise to the actual equation.



Besides using trigo form, there's also the complex exponential form and amplitude-phase form. But essentially, principle of superposition refers to the approximation of the given eqn using the summation a series of other expressions.



Hope this helps.


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