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 linearity property states that

$ \text{a}\, x(t) + \text{b}\, y(t) \xleftarrow[\,]{fourier\,series}\xrightarrow[\,]{coefficient} \text{a}\, f_{xn} + \text{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^{-jn\omega_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^{jn\omega_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 Scaling Property

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

then time scaling property states that

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

Time scaling property changes frequency components from $\omega_0$ to $a\omega_0$.

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} jn\omega_0 . f_{xn}$

& integration property states that

If $ \int x(t) dt \xleftarrow[\,]{fourier\,series}\xrightarrow[\,]{coefficient} {f_{xn} \over jn\omega_0} $

Multiplication and Convolution Properties

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

then multiplication 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

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