What is Fourier Series? Easily Explained with Examples

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What is Fourier Series

We always go for the best. Whatever be the problem, the solution should be accurate. We all are familiar with signals, and the representation of signals is an important task. The better analysis of a signal can be done in the frequency domain rather than in its original domain. That is for easy and accurate analysis and conversion of a time-domain signal into the frequency domain. Any periodic continuous-time / discrete-time signal can be represented exponentially or as the sum of sin or cos terms. This representation of a signal in the frequency domain is called Fourier series representation. Joseph Fourier describes this for the first time.

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The meaning of Fourier representation is only for periodic signals. A periodic signal is a signal that repeats its pattern at regular intervals. If f(x) is a periodic function with a period ‘2l’ in the closed interval (C, C+2l) satisfying the following condition.

  1. f(x) is piecewise continuous.
  2. f(x) should have a finite number of maxima and minima.

The Fourier series of a function is given by EULER’S FORMULA. That is a periodic function in the closed interval (C, C+2l) with period 2l can be expressed in the form of an infinite series of cosine & sin terms, which is given by

f(x) = (a₀÷2) + Σ [a cosnπx/l + b sinnπx/l]

Where a₀, a & b are called fourier constants given by the formula

a₀ = 1/l ∫f(x) dx

a = 1/l ∫[f(x) cosnπx/l]dx

b = 1/l ∫[f(x)sinnπx/l]dx       ( integrate over C to C+2l )

Existence of Fourier Series ( Dirichlet Condition)

The representation of all the periodic signals may not be in this series. The conditions under which a periodic signal can be represented by this series is called Dirichlet condition. They are…

In each period

  1. The function should have a finite number of maxima and minima.
  2. It should have a finite number of discontinuities.
  3. The function should be integral over one period.

ie, ∫ |r(t)dt < α ; ( limit 0 to T)

See also: How To Find Derivative Of A Graph | Easiest Method

Types of Fourier Series

Here are the different types of this Series.

Trigonometric Fourier Series

If a periodic signal is as the infinite sum of sin and cos terms, then it is called Trigonometric Series.

x(t) = a₀ + Σ[ acosnπx/l + bsinnπx/l ]

Polar & Cosine Fourier Series

If a periodic signal is in terms of cos terms, then it is called Cosine Series.

   x(t) = A₀ + Σ A cos(nω₀t + θn)

               A₀ = a₀

               A = √(a²+b²)

               a = 1/l ∫[f(x) cosnπx/l]dx

  b = 1/l ∫[f(x)sinnπx/l]dx

               θn = tan⁻(b/a)

Exponential Fourier Series

If a periodic signal is represented as an exponential function, then it is called Exponential Series.

 x(t) = Σ Cn eʲⁿʷᵒᵗ

where Cn = 1/T ∫x(t)e⁻ʲⁿʷᵒᵗdt

See Also: What is a Partial Derivative? Scope and Application

Properties of Fourier Series

Here are the properties of this Series

Linearity

Consider two signals x(t) & y(t) with period T. If Cn & Dn are the Fourier coefficient of x(t) & y(t) respectively, then the Fourier coefficient of their linear combination,

i.e. f. S {Ax(t) + By(t)} = Acₙ+BDₙ

Time Shifting Property

If this series of x(t) is Cn, then the series of x(t-tₒ) is given by

F.S{x(t-t₀)} = e⁻ʲⁿʷᵒᵗᵒCₙ

Any time shift of the signal does not change the magnitude of FS coefficients but change their phases.

Time Reversal

The series coefficients of the time-reversal of a signal are time-reversal of the FS coefficient of the corresponding signal.

  FS {x(t)} = Cₙ

  FS {x(-t)} = Cₙ-

if x(t) is an odd signal, it’s series coefficients are odd.

Time Scaling

If the period of x(t) is T, then period of x(at) is T/a, and the fundamental frequency become aω₀.    The series coefficients are the same as that of x(t), but the harmonics are ±aω₀, ±2aω₀, ±3aω₀…. etc

Multiplication

If x₁(t) and x₂(t) are both periodic with T and FS coefficients are Cₙ & Dₙ, respectively.

Then

  FS{x₁(t)*x₂(t)} = ΣCₗ(Dₙ_ₗ)

Convolution

If x₁(t) & x₂(t) are periodic with period T, then

FS{ X₁(t) * x₂(t) } = TCₙDₙ

Conjugate Property

If the series coefficient of x(t) is Cₙ, then the series coefficient of the complex conjugate of x(t) is the time-reversal of the complex conjugate of the series coefficient.

FS[x(t)]   = Cₙ

FS[x*(t)] = C-ₙ*

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Fundamental Theorem on Convergence

If f(x) is a periodic function with period 2l in C≤x≤C+2l, satisfying Dirichlet’s condition, then it’s Fourier converges at each point ‘x’ in the interval and the sum is by S(x)

S(x) = (a₀÷2) + Σ [a cos(nπx/l) + b sin(nπx/l)]

Also, it satisfies the following conditions

f(x) = S(x), if C≤x≤C+2l and f(x) is continuous at x

S(x) = [f(x-0)+f(x+0)]/2

If f(x) is continuous at x, then f(x-0) = f(x+0) = f(x)

Example

To find the Fourier series expansion of f(x) = x³ in -π≤x≤π ?

Solution:

Given f(x) = x³ , -π≤x≤π

as f(x) is an odd function, a₀=0 & aₙ=0

Therefore the series is

f(x)= Σ bₙsin(nx)

bₙ = Σ1/π∫x³sin(nx) ;{ lim -π to π }

bₙ = 2(-1)ⁿ/n³[6-n²π²]

Therefore f(x) = Σ 2(-1)/n³[6 – n²π²] sin(nx)

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