2.4 Properties of fourier series

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Properties of the fourier series

The properties of the Fourier series are important in applying it to signal analysis and to interpreting it. The main properties are given here using thenotation that the Fourier series of a real valued function $x\left(t\right)$ over $\left\{0\le t\le T\right\}$ is given by $ℱ\left\{x\left(t\right)\right\}=c\left(k\right)$ and $\stackrel{˜}{x}\left(t\right)$ denotes the periodic extensions of $x\left(t\right)$ .

1. Linear: $ℱ\left\{x+y\right\}=ℱ\left\{x\right\}+ℱ\left\{y\right\}$ Idea of superposition. Also scalability: $ℱ\left\{ax\right\}=aℱ\left\{x\right\}$
2. Extensions of $x\left(t\right)$ : $\stackrel{˜}{x}\left(t\right)=\stackrel{˜}{x}\left(t+T\right)$ $\stackrel{˜}{x}\left(t\right)$ is periodic.
3. Even and Odd Parts: $x\left(t\right)=u\left(t\right)+jv\left(t\right)$ and $C\left(k\right)=A\left(k\right)+jB\left(k\right)=|C\left(k\right)|{e}^{j\theta \left(k\right)}$ $u$ $v$ $A$ $B$ $|C|$ $\theta$ even 0even 0even 0odd 00 oddeven 00 even0 eveneven $\pi /2$ 0 oddodd 0even $\pi /2$
4. Convolution: If continuous cyclic convolution is definedby $y\left(t\right)=h\left(t\right)\circ x\left(t\right)={\int }_{0}^{T}\stackrel{˜}{h}\left(t-\tau \right)\stackrel{˜}{x}\left(\tau \right)d\tau$ then $ℱ\left\{h\left(t\right)\circ x\left(t\right)\right\}=ℱ\left\{h\left(t\right)\right\}ℱ\left\{x\left(t\right)\right\}$
5. Multiplication: If discrete convolution is definedby $e\left(n\right)=d\left(n\right)*c\left(n\right)=\sum _{m=-\infty }^{\infty }d\left(m\right)c\left(n-m\right)$ then $ℱ\left\{h\left(t\right)x\left(t\right)\right\}=ℱ\left\{h\left(t\right)\right\}*ℱ\left\{x\left(t\right)\right\}$ This property is the inverse of property 4 and vice versa.
6. Parseval: $\frac{1}{T}{\int }_{0}^{T}{|x\left(t\right)|}^{2}dt=\sum _{k=-\infty }^{\infty }{|C\left(k\right)|}^{2}$ This property says the energy calculated in the time domain is the same as thatcalculated in the frequency (or Fourier) domain.
7. Shift: $ℱ\left\{\stackrel{˜}{x}\left(t-{t}_{0}\right)\right\}=C\left(k\right){e}^{-j2\pi {t}_{0}k/T}$ A shift in the time domain results in a linear phase shift in the frequencydomain.
8. Modulate: $ℱ\left\{x\left(t\right){e}^{j2\pi Kt/T}\right\}=C\left(k-K\right)$ Modulation in the time domain results in a shift in the frequency domain. This propertyis the inverse of property 7.
9. Orthogonality of basis functions:
${\int }_{0}^{T}{e}^{-j2\pi mt/T}{e}^{j2\pi nt/T}dt=T\delta \left(n-m\right)=\left\{\begin{array}{ll}T& {\text{if }}n=m\\ 0& {\text{if }}n\ne m\text{.}\end{array}$
Orthogonality allows the calculation of coefficients using inner products. It also allowsParseval's Theorem in property 6. A relaxed version of orthogonality is called "tight frames" and is important in over-specified systems, especially inwavelets.

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