Last updated: October 2023 | Fact-checked for accuracy.
Easily compute the trigonometric components of periodic functions with our Fourier Expansion Calculator. Whether you are analyzing square waves, sawtooth signals, or triangle waves, this tool provides precise coefficients and series expansions for your engineering and physics projects.
Fourier Expansion Calculator
Fourier Expansion Calculator Formula:
Formula Source: Wolfram MathWorld – Fourier Series | Wikipedia: Fourier Analysis
Variables:
- Waveform Type: The shape of the periodic signal (Square, Sawtooth, or Triangle).
- Amplitude (A): The peak value or height of the wave from the center.
- Period (T): The time duration for one full cycle of the wave.
- Number of Terms (n): The upper limit of the summation (harmonics).
Related Calculators:
- Root Mean Square (RMS) Calculator
- Harmonic Distortion Calculator
- Signal Frequency Calculator
- Waveform Wavelength Calculator
What is a Fourier Expansion Calculator?
A Fourier expansion calculator is a mathematical tool used to decompose a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines. This process is fundamental in signal processing, acoustics, and heat transfer.
By calculating the Fourier coefficients ($a_0$, $a_n$, and $b_n$), this calculator helps visualize how complex waveforms are constructed from fundamental frequencies and their harmonics.
How to Calculate Fourier Expansion (Example):
Let’s calculate the expansion for a Square Wave with Amplitude A=1 and Period T=2π for 3 terms:
- Identify the function symmetry. A square wave is an “odd” function, so $a_0 = 0$ and $a_n = 0$.
- Apply the formula for $b_n$: $b_n = (4A / n\pi)$ for odd $n$.
- Calculate for $n=1$: $b_1 = 4(1)/\pi \approx 1.273$.
- Calculate for $n=3$: $b_3 = 4(1)/3\pi \approx 0.424$.
- The result is: $f(x) \approx 1.273 \sin(x) + 0.424 \sin(3x) + \dots$
Frequently Asked Questions (FAQ):
What is the purpose of the a0 term? It represents the DC offset or the average value of the function over one period.
Why does increasing ‘n’ make the result more accurate? As the number of terms increases, the series better approximates the sharp edges and details of the waveform (Gibbs phenomenon).
Can any function be expanded? No, only periodic functions that satisfy the Dirichlet conditions can be fully expanded into a Fourier series.
Is this the same as a Fast Fourier Transform (FFT)? No, this expansion provides the continuous analytical series, while FFT is a discrete numerical algorithm used for digital data.