Trigonometric Fourier Series

>>  A non-sinusoidal periodic function can be expressed as an infinite sum of sinusoidal functions.

>>   A periodic function is a function that repeats its values in regular intervals or periods.

A function is said to be periodic if  >>  f (t) = f (t + nT)  (n is an integer) for all values of t in the domain. Where T is called the fundamental period or Basic period and T should have a nonzero positive value.

>>  According to the Fourier theorem, any periodic function of frequency ω can be expressed as an infinite sum of sine or cosine functions that are integral multiples of ω.

f (t) = a + a1 cos ωt + b1 sin ωt + a2 cos2ωt + bsin2ωt + ……….

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>>  Where ω is called the fundamental frequency in radian per second. The sinusiods sin (nωt) and cos (nωt) are called the nth harmonic of f(t). The constants a, an and bare called the fourier coefficients.

>>  A function that can be represented by a Fourier series must meet certain requirements, because the infinite series may or may not be converge.

Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

S_{n}=\sum _{k=1}^{n}a_{k}.

A series is convergent if the sequence of its partial sums (S1, S2, S…. ) tends to a limit ; that means that the partial sums become closer and closer to a given number when the number of their terms increases. (Any series that is not convergent is said to be divergent)

The reciprocals of the positive integers produce a divergent series (harmonic series):

{1 \over 1}+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+{1 \over 6}+\cdots \rightarrow \infty .

Alternating the signs of the reciprocals of positive integers produces a convergent series:

{1 \over 1}-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}\cdots =\ln(2)

The reciprocals of factorials produce a convergent series :

{\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{24}}+{\frac {1}{120}}+\cdots =e.

Dirichlet Conditions

>>  f(t) is single valued everywhere (Means f(t) is a function → A function can only have one output, f(t), for each unique input, t → Vertical line test).

>>  f(t) has a finite number of discontinuities in any one period (Then only an and bn will exist).fourier3

>> f(t) has a finite number of maxima and minima in any one period.

f(t) = Sin (1/t)

y ∈ (0,1] 

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A signal that does not satisfy this condition is x(t) = tan(t)

>>  These conditions are not necessary conditions, they are sufficient conditions for a Fourier series to exist.

Fourier Analysis

>> The process of determining the Fourier coefficients is called Fourier analysis.

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