Introduction

Once one has obtained a solid understanding of the fundamentals of Fourier series analysis and the General Derivation of the Fourier Coefficients, it is useful to have an understanding of the common signals used in Fourier Series Signal Approximation.

Consider a square wave f(x) of length 1. Over the range [0,1), this can be written as

$$x\left(t\right)=\left\{\begin{array}{cc}1\hfill & t\le \frac{1}{2};\hfill \\ -1\hfill & t>\frac{1}{2}.\hfill \end{array}\right.$$

1

Fourier series approximation of a square wave

Real Even Signals

Given that the square wave is a real and even signal,

- $f\left(t\right)=f(-t)$ EVEN
- $f\left(t\right)=f$*$\left(t\right)$ REAL
- therefore,
- ${c}_{n}={c}_{-n}$ EVEN
- ${c}_{n}={c}_{n}$* REAL

Consider this mathematical question intuitively: Can a discontinuous function, like the square wave, be expressed as a sum, even an infinite one, of continuous signals? One should at least be suspicious, and in fact, it can't be thus expressed.

The extraneous peaks in the square wave's Fourier series
**never** disappear; they are termed
Gibb's phenomenon after the American physicist
Josiah Willard Gibbs. They occur whenever the signal is
discontinuous, and will always be present whenever the signal
has jumps.

The Square wave is the standard example, but other important signals are also useful to analyze, and these are included here.

This signal is relatively self-explanatory: the time-varying portion of the Fourier Coefficient is taken out, and we are left simply with a constant function over all time.

$$x\left(t\right)=1$$

2

With this signal, only a specific frequency of time-varying Coefficient is chosen (given that the Fourier Series equation includes a sine wave, this is intuitive), and all others are filtered out, and this single time-varying coefficient will exactly match the desired signal.

$$x\left(t\right)=sin\left(\pi t\right)$$

3

$$x\left(t\right)=\left\{\begin{array}{cc}t& t\le 1/4\\ 2-\mathrm{4t}& 1/4\le t\le 3/4\\ -7/4+\mathrm{4t}& 3/4\le t\le 1\end{array}\right.$$

4

- $f\left(t\right)=-f(-t)$ ODD
- $f\left(t\right)=f$*$\left(t\right)$ REAL
- therefore,
- ${c}_{n}=-{c}_{-n}$
- ${c}_{n}=-{c}_{n}$* IMAGINARY

Fourier series approximation of a triangle wave

$$x\left(t\right)=t-Floor\left(t\right)$$

5

Fourier series approximation of a sawtooth wave

link: http://yoder-3.institute.rose-hulman.edu/visible3/chapters/03spect/demosLV/fseries/index.htm

To summarize, a great deal of variety exists among the common Fourier Transforms. A summary table is provided here with the essential information.

Description | Time Domain Signal for $t\in [0,1)$ | Frequency Domain Signal |

Constant Waveform | $x\left(t\right)=1$ | ${c}_{k}=\left\{\begin{array}{cc}1& k=0\\ 0& k\ne 0\end{array}\right.$ |

Sinusoid Waveform | $x\left(t\right)=sin\left(\pi t\right)$ | ${c}_{k}=\left\{\begin{array}{cc}1/2& k=\pm 1\\ 0& k\ne \pm 1\end{array}\right.$ |

Square Waveform | $x\left(t\right)=\left\{\begin{array}{cc}1& t\le 1/2\\ -1& t>1/2\end{array}\right.$ | ${c}_{k}=\left\{\begin{array}{cc}4/\pi k& \mathrm{k}\mathrm{odd}\\ 0& \mathrm{k}\mathrm{even}\end{array}\right.$ |

Triangle Waveform | $x\left(t\right)=\left\{\begin{array}{cc}t& t\le 1/2\\ 1-t& t>1/2\end{array}\right.$ | ${c}_{k}=\left\{\begin{array}{cc}-8\mathrm{Sin(k\pi )/2)}/{\left(\pi k\right)}^{2}& \mathrm{k}\mathrm{odd}\\ 0& \mathrm{k}\mathrm{even}\end{array}\right.$ |

Sawtooth Waveform | $x\left(t\right)=t/2$ | ${c}_{k}=\left\{\begin{array}{cc}0.5& k=0\\ \mathrm{-1}/\pi k& k\ne 0\end{array}\right.$ |