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 ______.
Concept:
Greatest Integer Function: It is also known as floor function.
It is written as f(x) = [x]
The value of [x] is the largest integer that is less than or equal to x.
In Mathematical notation,
[x] = max{m ∈ z | m ≤ x}
Here notation "m ∈ z" means 'm' is an integer.
Fractional function:
x - [x] = {x} ⇒ fractional part of a integer
It's value always between 0 & 1 & {x} can also be zero.
0 ≤ {x} < 1
Also the constant term in Fourier analysis is defined as:
\(\rm a_0 = \frac{1}{T} \displaystyle\int _0^T g(x) dx\)
Calculation:
g(x) = x - [x] = {x}
It is periodic with a period (T) = 1
As we know,
∵ \(\rm a_0 = \frac{1}{T} \displaystyle\int _0^T g(x) dx\)
\(\rm a_0 = \frac{1}{1} \displaystyle\int _0^1 x dx\) (∵ x - [x] = x in (0, 1)
\(a_0 = \left. \left[ \frac{x^2}{2} \right] \right|_0^1\)
\(a_0 = \frac{1}{2}\)
∴ Constant term = 1/2
Important 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 Fourier series to represent x-x2 for –π ≤ x ≤ π is given by \(x - {x^2} = \frac{{{a_0}}}{2} + \mathop \sum \limits_{n = 1}^\infty {a_n}cosnx + \mathop \sum \limits_{n = 1}^\infty {b_n}sinnx\)
The value of a0 (round off to two decimal places), isThe 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
