Can any periodic function be expanded into a Fourier series?

I understand from other related posts here that Fourier Expansion is possible only for Periodic Functions. , for all values of x in any interval c to c + 2π, of length 2π. The expansion of f(x) in the form of the above series is called Fourier Series”.

What are the conditions for expansion of a function in Fourier series?

a. f(x) sin nπ l x dx, which is the general form of Fourier series expansion for functions on any finite interval. Also note that this is applicable to the first case of our discussion, where we need to take a = −π, b = π, l = π and then everything becomes the same as in the previous section.

Why Fourier series is used for periodic signals?

Fourier series is just a means to represent a periodic signal as an infinite sum of sine wave components. A periodic signal is just a signal that repeats its pattern at some period. The primary reason that we use Fourier series is that we can better analyze a signal in another domain rather in the original domain.

What is periodic functions in Fourier series?

written in terms of a constant function, a function of period π or frequency 1 π (the “first harmonic”) and a function of period π 2 or frequency 2 π (the “second harmonic”). A constant term can therefore be expected to arise in the Fourier series of a function which has a non-zero average value.

What does it mean when a function is periodic?

A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of. radians, are periodic functions.

Which of the following Cannot be the Fourier series expansion of a periodic function?

Which of the following cannot be the Fourier series expansion of a periodic signal? Since x2(t) is not periodic, so it cannot be expanded in Fourier series.

How do Fourier series represent periodic signals?

Fourier Series Representation of Continuous Time Periodic Signals

1. x(t)=cosω0t (sinusoidal) &
2. x(t)=ejω0t (complex exponential)
3. These two signals are periodic with period T=2π/ω0.
4. A set of harmonically related complex exponentials can be represented as {ϕk(t)}
5. Where ak= Fourier coefficient = coefficient of approximation.

How do you know if a function is periodic?

1. A function f(x) is said to be periodic, if there exists a positive real number T such that f(x+T) = f(x).
2. The smallest value of T is called the period of the function.
3. Note: If the value of T is independent of x then f(x) is periodic, and if T is dependent, then f(x) is non-periodic.

How are periodic functions used in real life?

For example, high tides and low tides can be modeled and predicted using periodic functions because scientists can determine the height of the water at different times of the day (when the water level is low, the tide is low).

Is the function periodic if so what is its period?

From the graph perspective, if its graph repeats after an unchanging period, we call it a periodic function. From the algebraic perspective, if f(x)=f(x+T)(T is an unchanging period, x can be any real numbers), then this function is periodic.

How does Fourier series make it easier to represent periodic signal?

Explanation: Fourier series makes it easier to represent periodic signals as it is a mathematical tool that allows the representation of any periodic signals as the sum of harmonically related sinusoids.

How do you expand a function as a Fourier series?

2. State Dirichlet’s conditions for a function to be expanded as a Fourier series. Let a function f ( x) be defined in the interval c

How do you find the period of a Fourier series?

A function f (x)is said to have a period T if for all x , f (x +T )=f (x), where T is a positive constant. The least value of T >0 is called the period of f (x). Example : f (x)=sin x ; f (x +2p) sin= (x 2 +p) sin=x . 2. State Dirichlet’s conditions for a function to be expanded as a Fourier series.

What are the symmetry properties of Fourier series?

There are two symmetry properties of functions that will be useful in the study of Fourier series. Even and Odd Function A function f (x) is said to be even if f (−x) = f (x). The function f (x) is said to be odd if f (−x) = −f (x).

What is the difference between Laurent series and Fourier series?

What is the Fourier Series? A Fourier series is an expansion of a periodic function f (x) in terms of an infinite sum of sines and cosines. Fourier Series makes use of the orthogonality relationships of the sine and cosine functions. Laurent Series yield Fourier Series