>> 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 ω∘t + b1 sin ω∘t + a2 cos2ω∘t + b2 sin2ω∘t + ……….
>> Where ω∘ is called the fundamental frequency in radian per second. The sinusiods sin (nω∘t) and cos (nω∘t) are called the nth harmonic of f(t). The constants a∘, an and bn are 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.
In mathematics, a series is the sum of the terms of an infinite sequence of numbers.
A series is convergent if the sequence of its partial sums (S1, S2, S3 …. ) 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):
Alternating the signs of the reciprocals of positive integers produces a convergent series:
The reciprocals of factorials produce a convergent series :
>> 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).
>> f(t) has a finite number of maxima and minima in any one period.
f(t) = Sin (1/t)
y ∈ (0,1]
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.
>> The process of determining the Fourier coefficients is called Fourier analysis.