Fourier Harmonic Analysis

Harmonic Analysis Fourier Series
Even, Odd Functions » Heaviside » Mclaurins Series » Iterative Methods

What is Harmonic Analysis?

Harmonic Analysis is a numerical method used to approximate periodic functions based on a set of discrete data points, rather than a continuous function. It allows us to decompose a complex periodic signal into its constituent fundamental frequency and a series of harmonic components (sines and cosines).

This technique is invaluable when dealing with real-world data from experiments, sensors, or measurements where an explicit mathematical function might not be known or is too complex. It helps in understanding the underlying frequencies present in a signal, noise reduction, and signal reconstruction.

Formulas for Harmonic Analysis (Discrete Fourier Series)

Given $M$ equally spaced data points $y_0, y_1, \ldots, y_{M-1}$ over one period $T$, the Fourier series approximation is:

Where $N$ is the number of harmonic terms to calculate. The coefficients $a_0, a_n,$ and $b_n$ are calculated using the following summation formulas:

Here, $M$ is the number of data points, and $t_k$ represents the time (or independent variable) for the $k$-th data point. Assuming equally spaced points starting from $t=0$, we have $t_k = k \cdot \frac{T}{M}$.

Note: The accuracy of the approximation depends on the number of data points $M$ and the number of terms $N$. Generally, $N$ should be less than or equal to $M/2$ for meaningful results. For $N > M/2$, higher harmonics might not be accurately captured due to aliasing.

Perform Harmonic Analysis on Discrete Data