Fourier Series

  • Periodic Functions

A function f(x) is said to be periodic, when it repeats its values after a fixed interval . In other words, for a periodic function, \forall \ x in its domain, there exists a positive number T, such that

{f(x+T) = f(x)}

The least possible value of T is known as the primitive or fundamental period of f(x). If T is the fundamental period of f(x), then nT, \ n \in \mathbb N is also a period of f(x).

From the graph of a function, we can easily recognize whether a function is periodic and what its fundamental period is. Commonly known periodic functions are trigonometric functions.

Sine, cosine, cosecant and secant have fundamental period 2 \pi and tangent and cotangent have a fundamental period of \pi.

An aperiodic function can be made periodic by specifying the values of the function within its fundamental period. For example, g(x)= x is an aperiodic function. Consider the following function:

{f (x) = x, \ - \pi < x < \pi}

{\ f(x + 2 \pi) = f(x)}

When carefully observed, it can be seen that f(x) is periodic with fundamental period 2 \pi.

It may also be noted that f(x) is discontinuous at odd integral multiples of \pi. The jump is equal to 2 \pi. This is a finite jump. (See the graph)

fourier.jpg

The jump of a discontinuous function can be infinite, e.g. jump of tan (x) at odd multiples of \frac \pi 2.

  • Dirichlet’s Conditions

These are conditions required to be satisfied for a function to be expressed as a Fourier series.

Let f(x) be defined in A < x < A+2L, such that

I) f(x) is single valued in the interval and \int \limits_A^{A+2L} f(x)dx exists,

II) f(x) has (or may have) a finite number of finite discontinuities and

III) f(x) has (or may have) a finite number of maxima and minima in the given interval.

  • Fourier Series

When a function f(x) satisfies Dirichlet’s conditions, it can be written as a Fourier series. So, now consider such a function f(x) of period 2L and defined over (A, A+2L). It can be written as

{f(x) = \frac {a_0}{2}+ \sum \limits_{n=1}^{\infty} a_n cos \Big( \frac {n \pi x}{L} \Big) + b_n sin \Big( \frac {n \pi x}{L} \Big)}

The constants a_0, a_n and b_n are known as Fourier coefficients. They are calculated as:

{a_0 = \frac {1}{L} \int \limits_{A}^{A+2L} f(x)dx}

{a_n = \frac {1}{L} \int \limits_{A}^{A+2L} f(x) cos \Big( \frac {n \pi x}{L} \Big) dx}

{b_n = \frac {1}{L} \int \limits_{A}^{A+2L} f(x) sin \Big( \frac {n \pi x}{L} \Big) dx}

  • Important Points

I) To obtain a Fourier series of any function, identify its period. This will generally be upper limit – lower limit. The period is 2L in the nomenclature used above.

II) The term with no sine and cosine multiples is \frac {a_0}{2}.

III) To evaluate a_n and b_n, one has to evaluate the integrals with lower and upper limits as specified in the problem.

IV) All constants have a factor of \frac 1 L (half the period) to be multiplied with the corresponding integral.

V) Sine and cosine functions are orthogonal.

Eventually, one gets a better idea about these coefficients and their magnitudes. Notice that they act as weights of trigonometric functions and hence decide the contribution of each sine and cosine wave to the original function.

  • Fourier Series from Actual Data

In real life, we may not always have an analytical form of a function. Instead, we have numerical values. From the application, we may have an idea about the periodicity of the function. For example, in an I.C. engine, the torque generated is periodic and is a function of crank angle.

In such cases, the procedure of finding a_0, a_n, b_n is slightly different.

I) Get 2L, the period of the function.

II) If the period is split between N regular intervals in its period, then,

{a_0 = 2 \times \frac {\sum f(x)}{N}}

{a_n = 2 \times \frac {\sum f(x) \times cos \Big( \frac {n \pi x}{L} \Big)}{N}}

{b_n = 2 \times \frac {\sum f(x) \times sin \Big( \frac {n \pi x}{L} \Big)}{N}}

First harmonic is the sum of those 2 sine and cosine waves, where n=1.

mth harmonic is the sum of those 2 sine and cosine waves, where n=m.

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