Convolution, Fourier Analysis, Cross-Correlation and Their Interrelationship
The scope of this chapter is to derive and explain three fundamental concepts of acoustical signal analysis and synthesis: convolution, Fourier transformation, and cross-correlation. Convolution is an important process in acoustics to determine how a signal is transformed by an acoustical system that can be described through an impulse response, a room, for example. Fourier analysis enables us to analyze a signal's properties at different frequencies. This method is then extended to Fourier transformation to convert signals from the time domain to the frequency domain and vice versa. Further, the method of cross-correlation is introduced by extending the orthogonality relations for trigonometric functions that were used to derive Fourier analysis. The cross-correlation method is a fundamental concept to compare two signals. We will use this method to extract the impulse response of a room by comparing a signal measured with a microphone after being transformed by a room with the original, measurement signal emitted into the room using a loudspeaker. Based on this and other examples, the mathematical relationships between convolution, Fourier transformation, and correlation are explained to facilitate deeper understanding of these fundamental concepts.
discrete Fourier transformation
fast Fourier transform
I would like to thank my mentor Jens Blauert for teaching me the fundamentals of auditory signal processing. I learned the unique derivation of the convolution presented in this chapter when taking Jens' communication acoustics class in 1996. Rolf Walter at the University of Dortmund introduced me to the fundamentals of Fourier analysis and synthesis. Torben M. Pastore and Nikhil Deshpande were of great help proof-reading the manuscript.
- 14.1J. d’Alembert: Recherches sur Differens Points Importans du Systême du Monde (Researches on Different Important Points of the System of the World), Vol. 1. (Chez David l’aîné, Paris 1754)Google Scholar
- 14.2P.-S. Laplace: Mémoires de l’Académie Royale des Sciences de Paris in 1781 (Gautrier-Villars et Fils, Paris 1781)Google Scholar
- 14.6P.S. Single, D.S. McGrath: Implementation of a 32768-Tap FIR filter using real-time fast convolution. In: Proc. 87th Convent. Audio Eng. Soc. (1989), Preprint 2830Google Scholar
- 14.9A. Bravais: Analyse mathématique sur les probabilités des erreurs de situation d’un point (Mathematical analysis of the probabilities of errors in a point’s location), Mém. present. divers savants Acad. Sci. Inst. Fr., Sci. Math. Phys. 9, 255–332 (1846)Google Scholar
- 14.12J.M. Stanton: Galton, Pearson, and the Peas: A brief history of linear regression for statistics instructors, J. Stat. Educ. 9(3) (2001), online, last accessed 02/10/2016Google Scholar
- 14.14G.-B. Stan, J.-J. Embrechts, D. Archambeau: Comparison of different impulse response measurement techniques, J. Audio Eng. Soc. 50(4), 249–262 (2002)Google Scholar