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Symmetry properties

Symmetry poperties of the Fourier series.

• If $f$ is a real function, then $c_{n}=c_{-n}^{*}$ (Hermitian symmetric) which implies:
  • $\Re {(c_{n})}=\Re {(c_{-n})}$ (real part is even symmetric)
  • $\Im {(c_{n})}=-\Im {(c_{-n})}$ (imaginary part is odd symmetric)
  • $|c_{n}|=|c_{-n}|$ (absolut value is even symmetric)
  • $\arg {(c_{n})}=\arg {(-c_{-n})}$ (argument is odd symmetric)
• If $f$ is a real and even function ($f(x)=f(-x)\forall x\in \mathbb {R}$), then all coefficients $c_{n}$ are real and $c_{n}=c_{-n}$ (even symmetric) which implies:
  • $b_{n}=0$ for all $n\geq 1$
• If $f$ is a real and odd function ($f(x)=-f(-x)\forall x\in \mathbb {R}$), then all coefficients $c_{n}$ are purely imaginary and $c_{n}=-c_{-n}$ (odd symmetric) which implies:
  • $a_{n}=0$ for all $n\geq 0$
• If $f$ is a purely imaginary function, then $c_{n}=-c_{-n}^{*}$ which implies:
  • $\Re {(c_{n})}=-\Re {(c_{-n})}$ (real part is odd symmetric)
  • $\Im {(c_{n})}=\Im {(c_{-n})}$ (imaginary part is even symmetric)
  • $|c_{n}|=|c_{-n}|$ (absolut value is even symmetric)
  • $\arg {(c_{n})}=\arg {(-c_{-n})}$ (argument is odd symmetric)
• If $f$ is a purely imaginary and even function ($f(x)=f(-x)\forall x\in \mathbb {R}$), then all coefficients $c_{n}$ are purely imaginary and $c_{n}=c_{-n}$ (even symmetric).
• If $f$ is a purely imaginary and odd function ($f(x)=-f(-x)\forall x\in \mathbb {R}$), then all coefficients $c_{n}$ are real and $c_{n}=-c_{-n}$ (odd symmetric).

Table of Fourier Series coefficients

Some common pairsof periodic functions and their Fourier Series coefficients are shown in the table below. The following notation applies:

• $f(x)$ designates a periodic function defined on $0\leq x<T$.
• $a_{k},b_{k},a_{0}$ designates a ...
• $c_{k}$ designates a ...

Time domain $f(x)$	Plot	Frequency domain (sine-cosine form) ${\begin{aligned}&a_{0}\\&a_{n}\quad {\text{for }}n\geq 1\\&b_{n}\quad {\text{for }}n\geq 1\end{aligned}}$	Remarks	Reference
$f(x)=A\left|\sin \left({\frac {2\pi }{T}}x\right)\right|\quad {\text{for }}0\leq x<T$		${\begin{aligned}a_{0}=&{\frac {4A}{\pi }}\\a_{n}=&{\begin{cases}{\frac {-4A}{\pi }}{\frac {1}{1-n^{2}}}&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\b_{n}=&0\\\end{aligned}}$	Full-wave rectified sine
$f(x)={\begin{cases}0\quad &{\text{for }}0\leq x<t_{1}\\A\left|\sin \left({\frac {2\pi }{T}}x\right)\right|\quad &{\text{for }}t_{1}\leq x<T/2\\0\quad &{\text{for }}T/2\leq x<T/2+t_{1}\\A\left|\sin \left({\frac {2\pi }{T}}x\right)\right|\quad &{\text{for }}T/2+t_{1}\leq x<T\end{cases}}$		${\begin{aligned}a_{0}=&{\frac {2A}{\pi }}\cos ^{2}\left({\frac {\alpha }{2}}\right)\\a_{n}=&{\begin{cases}{\frac {2A}{\pi (1-n^{2})}}\left(n\sin(\alpha )\cdot \sin(n\alpha )+1+\cos(\alpha )\cos(n\alpha )\right)&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\b_{n}=&{\begin{cases}{\frac {2A}{\pi (1-n^{2})}}\left(\cos(\alpha )\cdot \sin(n\alpha )-n\sin(\alpha )\cos(n\alpha )\right)&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\end{aligned}}$	Full-wave rectified sine cut by a phase-fired controller $\alpha ={\frac {2\pi t_{1}}{T}}$
$f(x)={\begin{cases}A\sin \left({\frac {2\pi }{T}}x\right)&\quad {\text{for }}0\leq x<T/2\\0&\quad {\text{for }}T/2\leq x<T\\\end{cases}}$		${\begin{aligned}a_{0}=&{\frac {2A}{\pi }}\\a_{n}=&{\begin{cases}{\frac {-2A}{\pi }}{\frac {1}{1-n^{2}}}&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\b_{n}=&{\begin{cases}{\frac {A}{2}}&\quad n=1\\0&\quad n>1\end{cases}}\\\end{aligned}}$	Half-wave rectified sine
$f(x)={\begin{cases}A&\quad {\text{for }}0\leq x<D\cdot T\\0&\quad {\text{for }}D\cdot T\leq x<T\\\end{cases}}$		${\begin{aligned}a_{0}=&2AD\\a_{n}=&{\frac {A}{n\pi }}\sin \left(2\pi nD\right)\\b_{n}=&{\frac {2A}{n\pi }}\left(\sin \left(\pi nD\right)\right)^{2}\\\end{aligned}}$	$0\leq D\leq 1$
$f(x)={\frac {Ax}{T}}\quad {\text{for }}0\leq x<T$		${\begin{aligned}a_{0}=&A\\a_{n}=&{\frac {-A}{n\pi }}\\b_{n}=&0\\\end{aligned}}$
$f(x)=A-{\frac {Ax}{T}}\quad {\text{for }}0\leq x<T$		${\begin{aligned}a_{0}=&A\\a_{n}=&{\frac {A}{n\pi }}\\b_{n}=&0\\\end{aligned}}$
$f(x)={\frac {4A}{T^{2}}}\left(x-{\frac {T}{2}}\right)^{2}\quad {\text{for }}0\leq x<T$		${\begin{aligned}a_{0}=&{\frac {2A}{3}}\\a_{n}=&{\frac {4A}{\pi ^{2}n^{2}}}\\b_{n}=&0\\\end{aligned}}$
$f(x)={\begin{cases}{\frac {A}{DT}}x&\quad {\text{for }}0\leq x<DT\\A{\frac {T-x}{T(1-D)}}&\quad {\text{for }}DT\leq x<T\end{cases}}$		${\begin{aligned}a_{0}=&A\\a_{n}=&\\b_{n}=&\\\end{aligned}}$	$0\leq D\leq 1$
$f(x)={\begin{cases}{\frac {A}{DT}}x&\quad {\text{for }}0\leq x<DT\\(x-T/2){\frac {-2A}{T(1-2D)}}&\quad {\text{for }}DT\leq x<(1-D)T\\(x-T){\frac {A}{DT}}&\quad {\text{for }}(1-D)T\leq x<T\end{cases}}$		${\begin{aligned}a_{0}=&0\\a_{n}=&0\\b_{n}=&{\frac {A}{D(1-2D)n^{2}\pi ^{2}}}\sin(2\pi nD)\\\end{aligned}}$	$0\leq D\leq 1/2$
				$\delta$ denotes the Dirac delta function.

Properties

This table shows some mathematical operations in the time domain and the corresponding effects in the frequency domain.

• $x[n]^{*}$ is the complex conjugate of $x[n]$.
• $f(x),g(x)$ designate a $T$-periodic functions defined on $0<x\leq T$.
• ${\hat {f}}[n],{\hat {g}}[n]$ designates the Fourier series coefficients (exponential form) of $f$ and $g$ as defined in equation TODO!!!

Property	Time domain	Frequency domain (exponential form)	Remarks	Reference
Linearity	$a\cdot f(x)+b\cdot g(x)$	$a\cdot {\hat {f}}[n]+b\cdot {\hat {g}}[n]$	complex numbers $a,b$
Time reversal / Frequency reversal	$f(-x)$	${\hat {f}}[-n]$
Time conjugation	$f(x)^{*}$	${\hat {f}}[-n]^{*}$
Time reversal & conjugation	$f(-x)^{*}$	${\hat {f}}[n]^{*}$
Real part in time	$\Re {(f(x))}$	${\frac {1}{2}}({\hat {f}}[n]+{\hat {f}}[-n]^{*})$
Imaginary part in time	$\Im {(f(x))}$	${\frac {1}{2i}}({\hat {f}}[n]-{\hat {f}}[-n]^{*})$
Real part in frequency	${\frac {1}{2}}(f(x)+f(-x)^{*})$	$\Re {({\hat {f}}[n])}$
Imaginary part in frequency	${\frac {1}{2i}}(f(x)-f(-x)^{*})$	$\Im {({\hat {f}}[n])}$
Shift in time / Modulation in frequency	$f(x-x_{0})$	${\hat {f}}[n]\cdot e^{-i{\frac {2\pi x_{0}}{T}}n}$	real number $x_{0}$
Shift in frequency / Modulation in time	$f(x)\cdot e^{i{\frac {2\pi n_{0}}{T}}x}$	${\hat {f}}[n-n_{0}]\!$	integer $n_{0}$
Differencing in frequency
Summation in frequency
Derivative in time	${\frac {df(x)}{dx}}$	$j{\frac {2\pi n}{T}}{\hat {f}}[n]$
Derivative in time ($N$ times)	${\displaystyle }$	${\displaystyle }$
Integration in time	${\displaystyle }$	${\displaystyle }$
Convolution in time / Multiplication in frequency	$(f*g)(x)=\int _{0}^{T}f(\tau )g(x-\tau )d\tau$	${\frac {1}{T}}{\hat {f}}[n]\cdot {\hat {g}}[n]$	$*\!$ denotes continuous circular convolution.
Multiplication in time / Convolution in frequency	$f(x)\cdot g(x)$	$({\hat {f}}*{\hat {g}})[n]=\sum _{k=-\infty }^{\infty }{\hat {f}}[k]{\hat {g}}[n-k]$	$*\!$ denotes Discrete convolution.
Cross correlation	$\rho _{xy}[n]=x[-n]^{*}*y[n]\!$	$R_{xy}(\omega )=X_{2\pi }(\omega )^{*}\cdot Y_{2\pi }(\omega )\!$
Parseval's theorem	$E_{xy}=\sum _{n=-\infty }^{\infty }{x[n]\cdot y[n]^{*}}\!$	$E_{xy}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }{X_{2\pi }(\omega )\cdot Y_{2\pi }(\omega )^{*}d\omega }\!$