The trigonometric Fourier series of a periodic time function can have
DC, cosine & Sine terms
Understanding Trigonometric Fourier Series Components
The trigonometric Fourier series is a powerful mathematical tool used to represent a periodic function as an infinite sum of simpler sinusoidal components. This representation helps in analyzing the frequency content of the signal.
Components of the Trigonometric Fourier Series
A periodic function, say $f(t)$, with a fundamental period $T$ and fundamental angular frequency $\omega_0 = \frac{2\pi}{T}$, can be represented by its trigonometric Fourier series as follows:
$$ f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right) $$
Let's break down the components in this series:
- DC Component ($a_0$): This is the constant term, also known as the average value or the zero-frequency component of the function $f(t)$. It's calculated as:
$$ a_0 = \frac{1}{T} \int_{0}^{T} f(t) \, dt $$
- Cosine Terms ($\sum a_n \cos(n\omega_0 t)$): These terms represent the even-ordered harmonics (including the fundamental frequency when $n=1$) of the function. The coefficients $a_n$ (for $n \ge 1$) are calculated as:
$$ a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos(n\omega_0 t) \, dt $$
- Sine Terms ($\sum b_n \sin(n\omega_0 t)$): These terms represent the odd-ordered harmonics (including the fundamental frequency when $n=1$) of the function. The coefficients $b_n$ are calculated as:
$$ b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin(n\omega_0 t) \, dt $$
Conclusion on Series Components
Based on the general form of the trigonometric Fourier series, it includes a constant term (DC component), cosine terms (for even harmonics), and sine terms (for odd harmonics). Therefore, a trigonometric Fourier series representation of a periodic time function can have all three: DC, cosine, and sine terms.
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 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
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 ______.
