# 7.2 Common discrete fourier series

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This module includes a table of common discrete fourier transforms.

## 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

$x\left(t\right)=\left\{\begin{array}{cc}1\hfill & t\le \frac{1}{2};\hfill \\ -1\hfill & t>\frac{1}{2}.\hfill \end{array}\right)$ Fourier series approximation to sq t . The number of terms in the Fourier sum is indicated in eachplot, and the square wave is shown as a dashed line over two periods.

## Real even signals

Given that the square wave is a real and even signal,
• $f\left(t\right)=f\left(-t\right)$ 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.

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

## 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.

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

## Triangle waveform

$x\left(t\right)=\left\{\begin{array}{cc}t& t\le 1/2\\ 1-t& t>1/2\end{array}\right)$
This is a more complex form of signal approximation to the square wave. Because of the Symmetry Properties of the Fourier Series, the triangle wave is a real and odd signal, as opposed to the real and even square wave signal. This means that
• $f\left(t\right)=-f\left(-t\right)$ ODD
• $f\left(t\right)=f$ * $\left(t\right)$ REAL
• therefore,
• ${c}_{n}=-{c}_{-n}$
• ${c}_{n}=-{c}_{n}$ * IMAGINARY

## Sawtooth waveform

$x\left(t\right)=t/2$
Because of the Symmetry Properties of the Fourier Series, the sawtooth wave can be defined as a real and odd signal, as opposed to the real and even square wave signal. This has important implications for the Fourier Coefficients.

## Conclusion

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 $n\in \mathbb{Z}\left[0,N-1\right]$ Frequency Domain Signal $k\in \mathbb{Z}\left[0,N-1\right]$ Constant Function 1 $\delta \left(k\right)$ Unit Impulse $\delta \left(n\right)$ $\frac{1}{N}$ Complex Exponential ${e}^{j2\pi mn/N}$ $\delta \left({\left(k-m\right)}_{N}\right)$ Sinusoid Waveform $cos\left(j2\pi mn/N\right)$ $\frac{1}{2}\left(\delta \left({\left(k-m\right)}_{N}\right)+\delta \left({\left(k+m\right)}_{N}\right)\right)$ Box Waveform $\left(M $\delta \left(n\right)+{\sum }_{m=1}^{M}\delta \left({\left(n-m\right)}_{N}\right)+\delta \left({\left(n+m\right)}_{N}\right)$ $\frac{sin\left(\left(2M+1\right)k\pi /N\right)}{Nsin\left(k\pi /N\right)}$ Dsinc Waveform $\left(M $\frac{sin\left(\left(2M+1\right)n\pi /N\right)}{sin\left(n\pi /N\right)}$ $\delta \left(k\right)+{\sum }_{m=1}^{M}\delta \left({\left(k-m\right)}_{N}\right)+\delta \left({\left(k+m\right)}_{N}\right)$

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Professor
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scanning tunneling microscope
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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