Fourier Series
- Decomposes periodic signals into constituent sine and cosine components for easier analysis.
- Enables manipulation of specific signal characteristics (e.g., pitch, timbre, contrast).
- Applicable to one- and two-dimensional signals such as sound waveforms and images.
Definition
Section titled “Definition”The Fourier series is a mathematical tool used to represent periodic functions as the sum of a series of sines and cosines. This allows for the analysis and manipulation of complex periodic signals, such as those found in sound and electrical signals.
Explanation
Section titled “Explanation”By expressing a periodic function as a sum of sine and cosine terms, the Fourier series breaks a complex waveform into individual frequency components. Each component is described by its frequency, amplitude, and phase, which together reveal the underlying structure of the original signal and make it possible to analyze or modify specific aspects of that signal.
Examples
Section titled “Examples”A musical note can be represented as a periodic function with a specific frequency, amplitude, and phase. Using the Fourier series, the complex waveform of a musical note can be broken down into its individual sine and cosine components, allowing for the manipulation of specific aspects of the sound, such as its pitch or timbre.
Image processing
Section titled “Image processing”An image can be thought of as a two-dimensional function, with the intensity of each pixel representing the amplitude of the function at a specific location. Using the Fourier series, the complex spatial information of an image can be broken down into a series of sines and cosines, allowing for the manipulation of specific aspects of the image, such as its contrast or sharpness.
Use cases
Section titled “Use cases”- Analysis and manipulation of complex periodic signals found in sound and electrical signals.
- Breaking down musical notes into frequency components to modify pitch or timbre.
- Decomposing images into spatial frequency components to adjust contrast or sharpness.
Related terms
Section titled “Related terms”- Sines and cosines
- Periodic function
- Waveform
- Frequency
- Amplitude
- Phase
- Timbre
- Image processing