A periodic function satisfies Dirichlet’s conditions. This implies that the function

Fourier Series:

• The main aim of the Fourier Series is that many frequency components are produced from a single period.
• This is a mathematical tool that allows the representation of any periodic signal as the sum of harmonically related sinusoids.
• If x(t) = x(t + T) then that is said to periodic

• A periodic signal can be expressed by its Fourier series if and only if the function follows Dirichlet Conditions.

Dirichlet Conditions:

1. If a signal is discontinuous, there should be a finite number of discontinuities in the period (T).

2. Signal should have a finite average value over a period (T).

3. Signal should have a finite number of positive and negative maxima in the period (T).

Note:

• It is applicable only when the signal is periodic.
• Non-sinusoidal signals can be approximated to sinusoidal form.

Notes:

Trigonometric Fourier Series:

g(t) = a₀ + a₁ cos ω₀t + a₂ cos 2ω₀t + …

+ b₁ sin ω₀t + b₂ sin 2ω₀t + …

\(g\left( t \right) = {a_0} + \mathop \sum \limits_{n = 1}^\infty {a_n}\cos n{\omega _0}t + {b_n}\sin n{\omega _0}t\)

\({a_0} = \frac{1}{T}\mathop \smallint \nolimits_T^1 g\left( t \right)dt\)

\({b_n} = \frac{2}{T}\mathop \smallint \nolimits_0^T g\left( t \right)\sin n{\omega _0}tdt\)

\({a_n} = \frac{2}{T}\mathop \smallint \nolimits_0^T g\left( t \right)\cos n{\omega _0}tdt\)

Polar Form:

\(g\left( t \right) = {d_0} + \mathop \sum \limits_{n = 1}^\infty {d_n}\cos \left( {n{\omega _0}t + {\theta _n}} \right)\)

d₀ = a₀

\(\left| {{d_n}} \right| = \sqrt {a_n^2 + b_n^2} \)

\({\theta _n} = {\tan ^{ - 1}}\left( {\frac{{ - {b_n}}}{{{a_n}}}} \right)\)

Assume a_n = d_n cos θ_n and b_n = -d­_n sin θ_n

Exponential Fourier series [complex F.S]

\(g\left( t \right) = {C_o} + \mathop \sum \limits_{n = 1}^\infty {C_n}{e^{jn{\omega _0}t}} + \mathop \sum \limits_{n = 1}^\infty {C_{ - n}}{e^{ - jn{\omega _0}t}}\)

\(g\left( t \right) = \mathop \sum \limits_{n = - \infty }^\infty {C_n}{e^{jn{\omega _0}t}}\)

\({C_n} = \frac{1}{T}\mathop \smallint \nolimits_0^T g\left( t \right){e^{ - jn{\omega _0}t}}dt\)

Conclusion: