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

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

Fourier series approximation of a constant wave

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.

$$x\left(t\right)=cos\left(2\pi t\right)$$

3

Fourier series approximation of a sinusoid wave

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

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

$$x\left(t\right)=t/2$$

5

Fourier series approximation of a sawtooth wave

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 $n\in \mathbb{Z}[0,N-1]$ | Frequency Domain Signal $k\in \mathbb{Z}[0,N-1]$ |

Constant Function | 1 | $\delta \left(k\right)$ |

Unit Impulse | $\delta \left(n\right)$ | $\frac{1}{N}$ |

Complex Exponential | ${e}^{j2\pi mn/N}$ | $\delta \left({(k-m)}_{N}\right)$ |

Sinusoid Waveform | $cos(j2\pi mn/N)$ | $\frac{1}{2}(\delta \left({(k-m)}_{N}\right)+\delta \left({(k+m)}_{N}\right))$ |

Box Waveform $(M<N/2)$ | $\delta \left(n\right)+{\sum}_{m=1}^{M}\delta \left({(n-m)}_{N}\right)+\delta \left({(n+m)}_{N}\right)$ | $\frac{sin\left(\right(2M+1)k\pi /N)}{Nsin(k\pi /N)}$ |

Dsinc Waveform $(M<N/2)$ | $\frac{sin\left(\right(2M+1)n\pi /N)}{sin(n\pi /N)}$ | $\delta \left(k\right)+{\sum}_{m=1}^{M}\delta \left({(k-m)}_{N}\right)+\delta \left({(k+m)}_{N}\right)$ |