Although the function to be expanded is defined only over a specific
finite region, the series converges to a function that is defined over thereal line and is periodic. It is equal to the original function over the
region of definition and is a periodic extension outside of the region.Indeed, one could artificially extend the given function at the outset and
then the expansion would converge everywhere.
A geometric view
It can be very helpful to develop a geometric view of the Fourier series
where
$x\left(t\right)$ is considered to be a vector and the basis functions are the
coordinate or basis vectors. The coefficients become the projections of
$x\left(t\right)$ on the coordinates. The ideas of a measure of distance, size, and
orthogonality are important and the definition of error is easy topicture. This is done in
[link] ,
[link] ,
[link] using Hilbert space
methods.
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 hereusing the notation that the Fourier series of a real valued function
$x\left(t\right)$ over
$\{0\le t\le T\}$ is given by
$\mathcal{F}\left\{x\right(t\left)\right\}=c\left(k\right)$ and
$\tilde{x}\left(t\right)$ denotes the periodic extensions of
$x\left(t\right)$ .
- Linear:
$\mathcal{F}\{x+y\}=\mathcal{F}\left\{x\right\}+\mathcal{F}\left\{y\right\}$
Idea of superposition. Also scalability:
$\mathcal{F}\left\{ax\right\}=a\mathcal{F}\left\{x\right\}$
- Extensions of
$x\left(t\right)$ :
$\tilde{x}\left(t\right)=\tilde{x}(t+T)$
$\tilde{x}\left(t\right)$ is periodic.
- Even and Odd Parts:
$x\left(t\right)=u\left(t\right)+jv\left(t\right)$ and
$C\left(k\right)=A\left(k\right)+jB\left(k\right)=\left|C\left(k\right)\right|\phantom{\rule{0.166667em}{0ex}}{e}^{j\theta \left(k\right)}$
$u$ |
$v$ |
$A$ |
$B$ |
$\left|C\right|$ |
$\theta $ |
even |
0 |
even |
0 |
even |
0 |
odd |
0 |
0 |
odd |
even |
0 |
0 |
even |
0 |
even |
even |
$\pi /2$ |
0 |
odd |
odd |
0 |
even |
$\pi /2$ |
- Convolution: If continuous cyclic convolution is defined by
$y\left(t\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}h\left(t\right)\circ x\left(t\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{\int}_{0}^{T}\tilde{h}(t-\tau )\phantom{\rule{0.166667em}{0ex}}\tilde{x}\left(\tau \right)\phantom{\rule{0.166667em}{0ex}}d\tau $
then
$\mathcal{F}\left\{h\right(t)\circ x(t\left)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathcal{F}\left\{h\right(t\left)\right\}\phantom{\rule{0.166667em}{0ex}}\mathcal{F}\left\{x\right(t\left)\right\}$
- Multiplication: If discrete convolution is defined by
$e\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}d\left(n\right)*c\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{\sum}_{m=-\infty}^{\infty}d\left(m\right)\phantom{\rule{0.166667em}{0ex}}c(n-m)$
then
$\mathcal{F}\left\{h\right(t\left)\phantom{\rule{0.166667em}{0ex}}x\right(t\left)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathcal{F}\left\{h\right(t\left)\right\}*\mathcal{F}\left\{x\right(t\left)\right\}$
This property is the inverse of
property 4 and vice versa.
- Parseval:
$\frac{1}{T}{\int}_{0}^{T}{\left|x\left(t\right)\right|}^{2}dt\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{\sum}_{k=-\infty}^{\infty}{\left|C\left(k\right)\right|}^{2}$
This property says the energy calculated in the time domain is the same as
that calculated in the frequency (or Fourier) domain.
- Shift:
$\mathcal{F}\left\{\tilde{x}(t-{t}_{0})\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}C\left(k\right)\phantom{\rule{0.166667em}{0ex}}{e}^{-j2\pi {t}_{0}k/T}$
A shift in the time domain results in a linear phase shift in the frequency domain.
- Modulate:
$\mathcal{F}\left\{x\left(t\right)\phantom{\rule{0.166667em}{0ex}}{e}^{j2\pi Kt/T}\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}C(k-K)$
Modulation in the time domain results in a shift in the frequency domain. This
property is the inverse of property 7.
- Orthogonality of basis functions:
$${\int}_{0}^{T}{e}^{-j2\pi mt/T}\phantom{\rule{0.166667em}{0ex}}{e}^{j2\pi nt/T}\phantom{\rule{0.166667em}{0ex}}dt\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}T\phantom{\rule{0.277778em}{0ex}}\delta (n-m)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\left\{\begin{array}{cc}T\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}n=m\hfill \\ 0\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}n\ne m.\hfill \end{array}\right)$$
Orthogonality allows the calculation of coefficients using inner products in
[link] and
[link] . It also allows Parseval's Theorem in
property 6 .
A relaxed version of orthogonality is called “tight frames" and is importantin over-specified systems, especially in wavelets.
Examples
- An example of the Fourier series is the expansion of a square wave
signal with period
$2\pi $ . The expansion is
$$x\left(t\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\frac{4}{\pi}[sin\left(t\right)+\frac{1}{3}sin\left(3t\right)+\frac{1}{5}sin\left(5t\right)\cdots ].$$
Because
$x\left(t\right)$ is odd, there are no cosine terms (all
$a\left(k\right)=0$ ) and,
because of its symmetries, there are no even harmonics (even
$k$ terms are
zero). The function is well defined and bounded; its derivative is not,therefore, the coefficients drop off as
$\frac{1}{k}$ .
- A second example is a triangle wave of period
$2\pi $ . This is a
continuous function where the square wave was not. The expansion of thetriangle wave is
$$x\left(t\right)=\frac{4}{\pi}[sin\left(t\right)-\frac{1}{{3}^{2}}sin\left(3t\right)+\frac{1}{{5}^{2}}sin\left(5t\right)+\cdots ].$$
Here the coefficients drop off as
$\frac{1}{{k}^{2}}$ since the function and
its first derivative exist and are bounded.