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- Fourier Series Properties
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- Fourier Transforms Properties
- Distortion Less Transmission
- Hilbert Transform
- Convolution and Correlation
- Signals Sampling Theorem
- Signals Sampling Techniques
- Laplace Transforms
- Laplace Transforms Properties
- Region of Convergence
- Z-Transforms (ZT)
- Z-Transforms Properties
- Signals and Systems Resources
- Signals and Systems - Resources
- Signals and Systems - Discussion
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
$$f*_{xn} = -f_{-xn} $$