Fourier Analysis and Transforms
Spectrum analysis, also known as frequency domain analysis or spectral density estimation, is a crucial process for decomposing complex signals into simpler components. This article explores the fundamentals of Fourier analysis and transforms, which play a vital role in spectrum analysis and are widely used in hearing aid design and signal processing.
The Basics of Fourier Analysis:
Named after French mathematician Joseph Fourier, Fourier analysis breaks down repetitive waveforms into a series of sine waves with appropriate amplitudes and phases. Everyday sounds, such as speech sounds and music notes, consist of multiple frequency components. By combining pure tones with different frequencies and amplitudes, complex sounds can be created. Fourier analysis allows us to determine the sine wave components that constitute a given signal and express them as amplitude-frequency relationships.
Fourier Transform in Sound Analysis:
The Fourier transform provides a distinct energy or power spectrum for each sine function, enabling the description of sound as a series of energies at specific frequencies. This transformation maps a continuous time domain representation of a signal into its frequency domain representation, and vice versa. It decomposes a function into sinusoids of different frequencies, allowing for the analysis of a function’s components. In practical applications, the fast Fourier transform (FFT) is commonly used to efficiently compute the discrete Fourier transform (DFT) and approximate the spectrum of a signal.
Understanding FFT and its Application:
In practice, nearly all software and electronic devices that generate frequency spectra apply a fast Fourier transform (FFT), which is a specific mathematical approximation to the full integral solution. Formally stated, the FFT is a method for computing the discrete Fourier transform of a sampled signal. It is essentially an efficient algorithm to compute the Discrete Fourier Transform (DFT) and its inverse. Many distinct FFT algorithms exist, involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory. These are beyond the depth of this blog.
For certain signals, a Fourier transform can be performed analytically with calculus. For arbitrary signals, the signal must first be digitized, and a Discrete Fourier Transform (DFT) performed. The standard numerical algorithm used for the DFT is called the Fast Fourier Transform (FFT) or Discrete FFT (DFFT). Due to limitations inherent in digitization and numerical algorithms, the FFT will result in an approximation to the spectrum.
FFT algorithms are so commonly employed to compute DFTs that the term “FFT” is often used to mean “DFT” in colloquial settings. Formally, there is a clear distinction. DFT refers to a mathematical transformation or function, regardless of how it is computed, whereas FFT refers to a specific family of algorithms for computing DFTs. The FFT has been described as the most important numerical algorithm of our lifetime.
For those in the discipline of hearing, one important application of the FFT is for the analysis of sound. It is important to assess the frequency distribution of the power in a sound because the human ear exercises that capacity in the hearing process.
About the Author
Wayne Staab, PhD, is an internationally recognized authority in hearing aids. As President of Dr. Wayne J. Staab and Associates, he is engaged in consulting, research, development, manufacturing, education, and marketing projects related to hearing. His professional career has included University teaching, hearing clinic work, hearing aid company management and sales, and extensive work with engineering in developing and bringing new technology and products to the discipline of hearing. This varied background allows him to couple manufacturing and business with the science of acoustics to bring innovative developments and insights to our discipline. Dr. Staab has authored numerous books, chapters, and articles related to hearing aids and their fitting, and is an internationally-requested presenter. He is a past President and past Executive Director of the American Auditory Society and a retired Fellow of the International Collegium of Rehabilitative Audiology.
**this piece has been updated for clarity. It originally published on June 17, 2012