The Fourier series to represent x-x2 for –π ≤ x ≤ π is given by \(x - {x^2} = \frac{{{a_0}}}{2} + \mathop \sum \limits_{n = 1}^\infty {a_n}cosnx + \mathop \sum \limits_{n = 1}^\infty {b_n}sinnx\)
Concept:
In the interval (-l, l) Fourier series is defined as:-
\(f\left( x \right) = \frac{{{a_0}}}{2} + \mathop \sum \limits_{n = 1}^\infty {a_n}\cos \frac{{n\pi x}}{l} + \mathop \sum \limits_{n = 1}^\infty {b_n}\sin \frac{{n\pi x}}{l}\)
Where, \({a_0} = \frac{1}{l}\mathop \smallint \nolimits_{ - l}^l f\left( x \right)dx\)
\({a_n} = \frac{1}{l}\mathop \smallint \nolimits_{ - l}^l f\left( x \right)\cos \frac{{n\pi x}}{l}dx\)
\({b_n} = \frac{1}{l}\mathop \smallint \nolimits_{ - l}^l f\left( x \right)\frac{{\sin n\pi x}}{l}dx\)
Euler Definition: In the interval (-π, π)
\(f\left( x \right) = \frac{{{a_0}}}{2} + \mathop \sum \limits_{n = 1}^\infty {a_n}\cos nx + \mathop \sum \limits_{n = 1}^\infty {b_n}\sin x\)
Where,
\({a_o} = \frac{1}{\pi }\mathop \smallint \nolimits_{ - \pi }^\pi f\left( x \right)dx\)
\({a_n} = \frac{1}{\pi }\mathop \smallint \nolimits_{ - \pi }^\pi f\left( x \right)\cos nx\;dx\)
\({b_n} = \frac{1}{\pi }\mathop \smallint \nolimits_{ - \pi }^\pi f\left( x \right)\sin nx\;dx\)
Calculation:
Given,
f(x) = (x – x2)
\(\therefore {a_0} = \frac{1}{l}\mathop \smallint \nolimits_{ - l}^l f\left( x \right)dx = \frac{1}{\pi }\mathop \smallint \nolimits_{ - \pi }^\pi \left( {x - {x^2}} \right)dx = \frac{1}{\pi }\left[ {\frac{{{x^2}}}{2} - \frac{{{x^3}}}{3}} \right]_{ - \pi }^\pi \)
= - 6.5797Important Questions from Fourier Series
If we use the Fourier transform ϕ(x, y) = \(\int {{{\rm{e}}^{{\rm{ikx}}}}} {ϕ _{\rm{k}}}\left( {\rm{y}} \right){\rm{dk}}\) to solve the partial differential equation \({\rm{ - }}\frac{{{\partial ^2}ϕ \left( {x,y} \right)}}{{\partial {y^2}}}\, - \,\frac{1}{{{y^2}}}\frac{{{\partial ^2}ϕ \left( {x,y} \right)}}{{\partial {x^2}}} + \frac{{{m^2}}}{{{y^2}}}ϕ \left( {x,y} \right) = 0\) in the half-plane {(x, y) : -∞ < x < ∞, 0 < y < ∞} the Fourier modes ϕ k(y) depend on y as y α and y β . The values of α and β are
When a time-domain signal is converted into its Fourier representation, which of the following is/are conserved?
I. Energy
II. Power
The trigonometric Fourier series of a periodic time function can have
The discrete-time Fourier series representation of a signal x[n] with period N is written as \(\rm x[n] = \sum_{k = 0}^{N - 1} a_k e^{j(2kn\pi/N)}\). A discrete-time periodic signal with period N = 3, has the non-zero Fourier series coefficients: a- 3 = 2 and a4 = 1. The signal is
let g(x) be a function defined by g(x) = x - [x], where [x] represents the integer part of x. (That is, it is the largest integer which is less than or equal to x). The value of the constant term in the Fourier series expansion of g(x) is ______.
