# Basics of Fourier Transform

There are several algorithms for transforming data from the time domain to the frequency domain as below.

## Fourier Series

A Fourier Series（傅里叶级数） is summarized as creating a complex waveform by summing pure sine waves with different amplitudes and frequencies, and to decompose a complex signal into a sum of sinusoids of different amplitudes and frequencies.

### Dirichlet Conditions

Dirichlet Conditions specify a set of conditions that must be met before a signal can be decomposed into a Fourier Series:

• The signal is a mathematical function, i.e., one and only one y-point corresponds to each x-point.
• The signal is periodic.
• The area bounded by the signal over one period is finite.

### Decompose into a Fourier Series

A complex signal that meets the Dirichlet Conditions can be represented by a sum of sinusoids:

$f(t)=a_0+A\{\sum_{n=1}^\infty[a_n cos(n \omega_1 t+\phi_n)+b_n sin(n \omega_1 t+\phi_n)]\}$

where:

• $$a_0$$ is the DC component.
• $$A$$ is an overall scale factor for all harmonic components.
• $$\omega_1$$ is the frequency of the fundamental.
• $$n$$ is an integer multiplier of the fundamental frequency for each harmonic term.

This proves that not only can we sum a series of sine and cosine waves to create any other wave, but also that the frequencies of the sinusoids are integer multiples (harmonics) of a single fundamental frequency.

## Discrete Fourier Transform (DFT)

Discrete Fourier Transform (DFT): takes amplitude versus time data, and then translates to amplitude versus frequency data.

Mathematically, the algorithm is a series summation of the product of each sample times a complex number:

$X(b)=\sum_{n=0}^{N-1}x[n](cos(2\pi nb/N)-jsin(2\pi nb/N))$

where:

• $$n$$ is one of $$N$$ samples.
• $$N$$ is total number of samples.
• $$b$$ is one of $$B$$ frequency bins (each bin represents a frequency range of $$F_s /N$$).
• $$j$$ is the imaginary operator.

The DFT algorithm uses each sample point in the summation from 0 to N-1 for each analyzed frequency. All N sample points contain information about all B frequencies, thus each of the B frequencies for which information is desired requires a summation of N time sample products. Because of the reasons above, processing a DFT is slow, because $$N^2$$ calculations are necessary. For example, a 2000 point DFT requires 4 million calculations, often floating point calculations, which are slower than integer calculations.

## Fast Fourier Transform (FFT)

Fast Fourier Transform (FFT) remedies the DFT speed problem by skipping over portions of the summations which produce redundant information. Rules for using FFT:

• The number of sample points must be a power of 2 ($$2^n$$).
• The number of additions and multiplications is: $$\frac{N}{2}\log_2 N$$.

## References & Acknowledgements

• Fundamentals of Testing Using ATE
• The-Fundamentals-of-Mixed-Signal-Testing_Brian-Lowe

Original: https://wiki-power.com/
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