Translation Table
The table below summarizes how to get one set of Fourier Series
coefficients from any other representation. Note that it is assumed
the function being represented is real - meaning $ \mathbb{X}[k]=\mathbb{X}^*[-k] $.
Also, $ n>0 $ in the table. The core equations at use in the
translation table are:
$ \begin{align} e^{j\theta}&=\cos(\theta)+j\sin(\theta)\\ \cos(\theta+\phi)&=\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)\\ \mbox{atan2}(b_n,a_n)&= \begin{cases} \tan^{-1}\left(\frac{b_n}{a_n}\right) & a_n>0\\ \tan^{-1}\left(\frac{b_n}{a_n}\right)-180^{\circ} & a_n<0 \end{cases} \end{align} $
$ \begin{array}{|c|c|c|c|} \hline \mbox{Find:} & \mbox{From trig} & \mbox{From cosine} & \mbox{From exponential} \\ \hline a_n & a_n & c_n\cos(\theta_n) & \mathbb{X}[n]+\mathbb{X}[-n]=2\Re\{\mathbb{X}[n]\}\\ \hline b_n & b_n & -c_n\sin(\theta_n) & j\left(\mathbb{X}[n]-\mathbb{X}[-n]\right)=-2\Im\{\mathbb{X}[n]\}\\ \hline a_0=c_0 & a_0 & c_0 & \mathbb{X}[0] \\ \hline c_n & \sqrt{a_n^2+b_n^2} & c_n & |\mathbb{X}[n]|+|\mathbb{X}[-n]|=2|\mathbb{X}[n]|\\ \hline \theta_n & -\mbox{atan2}(b_n,a_n) & \theta_n & \angle \mathbb{X}[n]\\ \hline \mathbb{X}[0] & a_0 & c_0 & \mathbb{X}[0] \\ \hline \mathbb{X}[n] & \frac{a_n}{2}+\frac{b_n}{2j}= \frac{a_n}{2}-j\frac{b_n}{2} & \frac{c_n}{2}\angle \theta_n & \mathbb{X}[n]\\ \hline \mathbb{X}[-n] & \frac{a_n}{2}-\frac{b_n}{2j}= \frac{a_n}{2}+j\frac{b_n}{2} & \frac{c_n}{2}\angle -\theta_n &\mathbb{X}[-n] \\ \hline \end{array} $
Anaylsis Equation Derivation
Assume you have some preiodic signal $ x(t) $ with fundamental frequency $ \omega_0 $ that has a Fourier Series representation of $ \mathbb{X}[k] $ such that:
$ \begin{align*} x(t)&=\sum_{k=-\infty}^{\infty}\mathbb{X}[k]e^{jk\omega_0t} \end{align*} $
Assume you have another (decidedly complex-valued) signal $ y(t) $ that is simply a complex exponential oscillating at a negative integer multiple $ -n $of $ \omega_0 $:
$ \begin{align*} y(t)&=e^{-jn\omega_0t} \end{align*} $
We will now look at the integral of the product of $ x(t)\,y(t) $ over one preiod. We will loko at the integral two different ways - one with the Fourier Series representation substitued in and one without:
$ \begin{align*} \int_Tx(\tau)\,y(\tau)\,d\tau&= \int_Tx(\tau)\,y(\tau)\,d\tau & ~&\mbox{Initial tautology}\\ \int_T\sum_{k=-\infty}^{\infty}\mathbb{X}[k]e^{jk\omega_0\tau}\,e^{-jn\omega_0\tau}\,d\tau&= \int_Tx(\tau)\,e^{-jn\omega_0\tau}\,d\tau & ~&\mbox{Substitute for }y(t)\mbox{ both places and for }x(t)\mbox{ on left only}\\ \sum_{k=-\infty}^{\infty}\mathbb{X}[k]\int_Te^{j(k-n)\omega_0\tau}\,d\tau&= \int_Tx(\tau)\,e^{-jn\omega_0\tau}\,d\tau & ~&\mbox{Swap order of sum and integral, pull functions of }k\mbox{ only out of integral} \end{align*} $
Let us focus for a moment on the integral:
$ \begin{align*} \int_Te^{j(k-n)\omega_0\tau}\,d\tau\end{align*} $
Using Euler's relationship, we can see that:
$ \begin{align*} e^{j(k-n)\omega_0\tau}=\cos((k-n)\omega_0\tau)+j\sin((k-n)\omega_0\tau)\end{align*} $
If $ k\neq n $, this represents a sinusoidal signal oscillating at some integer multiple $ n-k $ of the fundamental frequency $ \omega_0 $. And if we are integrating that over some period $ T $, the integral will be zero - the integral of a sinusoid over an integer number of periods is 0!
If, however, $ k=n $, the signal becomes $ e^{j0\omega_0\tau}=1 $. So in the summation on the left above, the only term in the summation that is not zero will be when $ k=n $:
$ \begin{align*} \sum_{k=-\infty}^{\infty}\mathbb{X}[k]\int_Te^{j(k-n)\omega_0\tau}\,d\tau&= \int_Tx(\tau)\,e^{-jn\omega_0\tau}\,d\tau & ~&\mbox{Resume where we left things}\\ \mathbb{X}[n]\int_T\,d\tau&= \int_Tx(\tau)\,e^{-jn\omega_0\tau}\,d\tau & ~&\mbox{Pull out only }n^{th}\mbox{ summand}\\ \mathbb{X}[n]\,T&= \int_Tx(\tau)\,e^{-jn\omega_0\tau}\,d\tau & ~&\mbox{Integrate} \end{align*} $
meaning:
$ \begin{align*} \mathbb{X}[n]&=\frac{1}{T}\int_Tx(\tau)\,e^{-jn\omega_0\tau}\,d\tau\end{align*} $
Common Fourier Series Pairs and Properties
The next two subsections present tables of common Fourier series pairs
and Fourier series properties. The information in these tables has
been adapted from:
- Signals and Systems, 2nd ed. Simon Haykin and Barry Van Veen. John Wiley & Sons, Hoboken, NJ, 2005. pp. 774, 777.
- Signals and Systems, 2nd ed. Alan V. Oppenheim and Alan S. Willsky with S. Hamid Nawab. Prentice Hall, Upper Saddle River, NJ, 1997. p. 206.
Common Exponential Fourier Series Pairs
Note in the table below, the discrete form of the
Dirac delta function $ \delta[k] $ is
used. The definition of this function is:
$ \delta[k]= \begin{cases} 1, & k=0\\ 0, & k\neq 0 \end{cases} $
$ \begin{array}{l l l} \mbox{Name} & \mbox{Signal} & \mbox{Fourier Series} \\ \hline \mbox{Basic Signal} & x(t)\text{, Period } T & \mathbb{X}[k]\text{, } \omega_0=\frac{2\pi}{T}\\ \hline \mbox{Complex Exponential}& {\displaystyle x(t)=e^{jp\omega_0t}}& \mathbb{X}[k]=\delta[k-p]\\ \hline \mbox{Cosine}& {\displaystyle x(t)=\cos(p\omega_0t)}& {\displaystyle \mathbb{X}[k]=\frac{1}{2}\left(\delta[k-p]+\delta[k+p]\right) }\\ \hline \mbox{Sine}& {\displaystyle x(t)=\sin(p\omega_0t)}& {\displaystyle \mathbb{X}[k]=\frac{1}{j2}\left(\delta[k-p]-\delta[k+p]\right) }\\ \hline \mbox{Constant}& {\displaystyle x(t)=c}& \mathbb{X}[k]=c\delta[k]\\ \hline \mbox{Periodic Square Wave}& {\displaystyle \begin{array}{l} x(t)=\begin{cases} 1, & |t|<T_1\\ 0, & T_1<|t|\leq\frac{T}{2} \end{cases}\\ \text{and } x(t+T)=x(t) \end{array}}& {\displaystyle \mathbb{X}[k]=\frac{\sin(k\omega_0T_1)}{k\pi}}\\ \hline \mbox{Impulse Train}& {\displaystyle x(t)=\sum_{n=-\infty}^{\infty}\delta(t-nT)}& {\displaystyle \mathbb{X}[k]=\frac{1}{T}}\\ \hline \end{array} $
Common Exponential Fourier Series Properties
$ \begin{array}{l l l} \mbox{Property} & \mbox{Periodic Signal} & \mbox{Fourier Series}\\ \hline \mbox{Basic Signals} & x(t), y(t), z(t);~T_x=T_y=T & \mathbb{X}[k], \mathbb{Y}[k], \mathbb{Z}[k];~\omega_0=\frac{2\pi}{T}\\ \hline \mbox{Linearity} & z(t)=Ax(t)+By(t) & \mathbb{Z}[k]=A\mathbb{X}[k]+B\mathbb{Y}[k]\\ \hline \mbox{Time Shifting} & z(t)=x\left(t-t_0\right) & \mathbb{Z}[k]=\mathbb{X}[k]e^{-jk\omega_0t_0}\\ \hline \mbox{Frequency Shifting} & z(t)=e^{jk_0\omega_0t}x(t) & \mathbb{Z}[k]=\mathbb{X}[k-k_0]\\ \hline \mbox{Conjugation} & z(t)=x^*(t) & \mathbb{Z}[k]=\mathbb{X}^*[-k]\\ \hline \mbox{Time Reversal} & z(t)=x(-t) & \mathbb{Z}[k]=\mathbb{X}[-k]\\ \hline \mbox{Time Scaling} & z(t)=x(\alpha t), \alpha>0 & \mathbb{Z}[k]=\mathbb{X}[k], T_z=\frac{T_x}{\alpha}\\ \hline \mbox{Periodic Convolution} & z(t)={\displaystyle \int_{T}x(\tau)y(t-\tau)d\tau} & \mathbb{Z}[k]=T\mathbb{X}[k]\mathbb{Y}[k]\\ \hline \mbox{Multiplication} & z(t)=x(t)y(t) & {\displaystyle \mathbb{Z}[k]=\sum_{l=-\infty}^{\infty}\mathbb{X}[l]\mathbb{Y}[k-l]}\\ \hline \mbox{Differentiation} & z(t)=\frac{dx(t)}{dt} & \mathbb{Z}[k]=jk\omega_x\mathbb{X}[k]\\ \hline \mbox{Integration} & {\displaystyle z(t)=\int_{-\infty}^{t}x(\tau)~d\tau}, \mathbb{X}[0]=0& \mathbb{Z}[k]=\left(\frac{1}{jk\omega_x}\right)\mathbb{X}[k]\\ \hline \mbox{Properties of Real Signals} & z(t)\mbox{ real} & \left\{ \begin{array}{l} \mathbb{Z}[k]=\mathbb{Z}^*[-k]\\ \Re\{\mathbb{Z}[k]\}=\Re\{\mathbb{Z}[-k]\}\\ \Im\{\mathbb{Z}[k]\}=-\Im\{\mathbb{Z}[-k]\}\\ |\mathbb{Z}[k]|=|\mathbb{Z}[-k]|\\ \measuredangle \mathbb{Z}[k]=-\measuredangle \mathbb{Z}[-k] \end{array} \right.\\ \hline \mbox{Properties of Real, Even Signals} & z(t)\mbox{ real and even}&\mathbb{Z}[k]\mbox{ real and even}\\ \hline \mbox{Properties of Real, Odd Signals} & z(t)\mbox{ real and odd}&\mathbb{Z}[k]\mbox{ imaginary and odd}\\ \hline \mbox{Isolation of Even Part} & z(t)=x_e(t)\mbox{ with x(t) real}& \mathbb{Z}[k]=\Re\{\mathbb{X}[k]\} \\ \hline \mbox{Isolation of Odd Part} & z(t)=x_o(t)\mbox{ with x(t) real}& \mathbb{Z}[k]=j\Im\{\mathbb{X}[k]\} \\ \hline \mbox{Parseval's Relation (Power)} & {\displaystyle P_{ave}=\frac{1}{T}\int_{T}|z(t)|^2~dt}& {\displaystyle P_{ave}=\sum_{k=-\infty}^{\infty}|\mathbb{Z}[k]|^2} \end{array} $
