Fourier Series Properties

These are properties of Fourier series:

Linearity Property

If $ x(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{xn}$ & $ y(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{yn}$

then pnearity property states that

$ ext{a}, x(t) + ext{b}, y(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} ext{a}, f_{xn} + ext{b}, f_{yn}$

Time Shifting Property

If $ x(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{xn}$

then time shifting property states that

$x(t-t_0) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} e^{-jnomega_0 t_0}f_{xn} $

Frequency Shifting Property

If $ x(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{xn}$

then frequency shifting property states that

$e^{jnomega_0 t_0} . x(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{x(n-n_0)} $

Time Reversal Property

If $ x(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{xn}$

then time reversal property states that

If $ x(-t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{-xn}$

Time Scapng Property

If $ x(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{xn}$

then time scapng property states that

If $ x(at) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{xn}$

Time scapng property changes frequency components from $omega_0$ to $aomega_0$.

Differentiation and Integration Properties

If $ x(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{xn}$

then differentiation property states that

If $ {dx(t)over dt} xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} jnomega_0 . f_{xn}$

& integration property states that

If $ int x(t) dt xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} {f_{xn} over jnomega_0} $

Multippcation and Convolution Properties

If $ x(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{xn}$ & $ y(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{yn}$

then multippcation property states that

$ x(t) . y(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} T f_{xn} * f_{yn}$

& convolution property states that

$ x(t) * y(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} T f_{xn} . f_{yn}$

Conjugate and Conjugate Symmetry Properties

If $ x(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f_{xn}$

Then conjugate property states that

$ x*(t) xleftarrow[,]{fourier,series}xrightarrow[,]{coefficient} f*_{xn}$

Conjugate symmetry property for real valued time signal states that

$$f*_{xn} = f_{-xn}$$

& Conjugate symmetry property for imaginary valued time signal states that

$$f*_{xn} = -f_{-xn} $$