7.2 Common discrete fourier series

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.

Deriving the coefficients

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

Fourier series approximation of a square wave

Real even signals

• $f\left(t\right)=f(-t)$ EVEN
• $f\left(t\right)=f$ * $\left(t\right)$ REAL
• therefore,
• $c_n=c_{-n}$ EVEN
• $c_n=c_n$ * REAL

Deriving the coefficients for other signals

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

Constant waveform

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.

Fourier series approximation of a constant wave

Sinusoid waveform

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.

Fourier series approximation of a sinusoid wave

Triangle waveform

• $f\left(t\right)=-f(-t)$ 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

Sawtooth waveform

Fourier series approximation of a sawtooth wave

Dft signal approximation

Conclusion

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