The Harmonic Spectrum

🎶 Signal Decomposition

Fourier analysis is the study of how complex functions may be represented or approximated by sums of simpler trigonometric functions. It originated from the study of Fourier series, which Joseph Fourier used to simplify the analysis of heat transfer.

The core idea is to break down a signal (like sound waves or electrical signals) into its fundamental frequency components.

🔄 Analysis and Synthesis

In science and engineering, this decomposition process is often called Fourier analysis, while reconstructing the signal from these components is known as Fourier synthesis. For instance, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis.

🧮 Mathematical Framework

The process of decomposition itself is called a Fourier transformation. Its output, the Fourier transform, provides a representation of the function in the frequency domain. This field has evolved significantly over time, encompassing more abstract and general situations, and is often known as harmonic analysis.

〰️ Continuous Fourier Transform

Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, producing a continuous function of frequency. This operation is reversible.

When the domain is time (t), the transform yields the frequency-domain function S(f), conveying amplitude and phase information for each frequency.

The Fourier Transform:

${\displaystyle S(f)=\int _{-\infty }^{\infty }s(t)\cdot e^{-i2\pi ft}\,dt}$

The Inverse Fourier Transform:

${\displaystyle s(t)=\int _{-\infty }^{\infty }S(f)\cdot e^{i2\pi ft}\,df}$

〰️ Continuous Fourier Transform

Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, producing a continuous function of frequency. This operation is reversible.

When the domain is time (t), the transform yields the frequency-domain function S(f), conveying amplitude and phase information for each frequency.

The Fourier Transform:

${\displaystyle S(f)=\int _{-\infty }^{\infty }s(t)\cdot e^{-i2\pi ft}\,dt}$

The Inverse Fourier Transform:

${\displaystyle s(t)=\int _{-\infty }^{\infty }S(f)\cdot e^{i2\pi ft}\,df}$

🔢 Discrete-Time Fourier Transform (DTFT)

This transform is the dual of the time-domain Fourier series. It relates a periodic frequency-domain function (a Dirac comb) to a discrete-time sequence.

The DTFT of a sequence s[n] is the Fourier transform of a modulated Dirac comb.

View Relationship ⬇️

The relationship between a sampled function and its DTFT:

${\displaystyle S_{\tfrac {1}{T}}(f)\ \triangleq \ \underbrace {\sum _{k=-\infty }^{\infty }S\left(f-{\frac {k}{T}}\right)\equiv \overbrace {\sum _{n=-\infty }^{\infty }s[n]\cdot e^{-i2\pi fnT}} ^{\text{Fourier series (DTFT)}}} _{\text{Poisson summation formula}}={\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }s[n]\ \delta (t-nT)\right\},\,}$

💻 Discrete Fourier Transform (DFT)

The DFT is crucial for practical computation, especially with the Fast Fourier Transform (FFT) algorithm. It transforms a finite sequence of samples into a finite sequence of frequency components.

It's derived from the DTFT by sampling the frequency domain, effectively representing periodic data.

View Formulas ⬇️

The DFT coefficients S[k] from a sequence s[n]:

${\displaystyle S[k]=\sum _{n}s_{_{N}}[n]\cdot e^{-i2\pi {k}{n}/N},\quad k\in \mathbb {Z} ,}$

The Inverse DFT:

${\displaystyle s_{_{N}}[n]={\frac {1}{N}}\sum _{k}S[k]\cdot e^{i2\pi {kn/N}}}$

🧩 Decomposing Functions

Complex functions can be decomposed into their even and odd parts. Fourier analysis reveals a direct mapping between these components in the time and frequency domains.

For example, a real-valued function in the time domain corresponds to a conjugate symmetric function in the frequency domain.

The relationship between time-domain and frequency-domain components:

${\displaystyle {\begin{array}{rccccccccc}{\text{Time domain}}&s&=&s_{_{\text{RE}}}&+&s_{_{\text{RO}}}&+&is_{_{\text{IE}}}&+&\underbrace {i\ s_{_{\text{IO}}}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {{\Bigg \Updownarrow }{\mathcal {F}}}&&\ \ {{\Bigg \Updownarrow }{\mathcal {F}}}&&\ \ {{\Bigg \Updownarrow }{\mathcal {F}}}\\{\text{Frequency domain}}&S&=&S_{\text{RE}}&+&\overbrace {\,i\ S_{\text{IO}}\\,} &+&iS_{\\text{IE}}&+&S_{\\text{RO}}\\end{array}}}$

📜 Ancient Roots to Modern Tools

The concept of harmonic series traces back to Babylonian mathematics for astronomical calculations. The Ptolemaic system used epicycles, which are related to trigonometric series.

In modern times, precursors to the DFT were used by Alexis Clairaut (1754) for orbital mechanics and Joseph Louis Lagrange (1759) for analyzing vibrating strings.

💡 Gauss and Fourier

Carl Friedrich Gauss (1805) utilized a true DFT for trigonometric interpolation of asteroid orbits. However, Joseph Fourier's seminal work in the early 19th century on heat transfer, demonstrating the power of representing functions as sums of trigonometric terms, truly established Fourier analysis as a distinct field.

📜 Discover other topics to study!

📜 Important Notice

This page was generated by an Artificial Intelligence and is intended for informational and educational purposes only. The content is based on a snapshot of publicly available data from Wikipedia and may not be entirely accurate, complete, or up-to-date.

This is not professional mathematical or scientific advice. The information provided on this website is not a substitute for professional consultation or rigorous academic study. Always refer to authoritative textbooks and peer-reviewed sources for definitive understanding and application.

The creators of this page are not responsible for any errors or omissions, or for any actions taken based on the information provided herein.