14. 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.