Acoustic Signal Processing

Abstract

Signal processing refers to the acquisition, storage, display, and generation of signals – also to the extraction of information from signals and the re-encoding of information. As such, signal processing in some form is an essential element in the practice of all aspects of acoustics. Signal processing algorithms enable acousticians to separate signals from noise, to perform automatic speech recognition, or to compress information for more efficient storage or transmission. Signal processing concepts are the building blocks used to construct models of speech and hearing. Now, in the 21st century, all signal processing is effectively digital signal processing. Widespread access to high-speed processing, massive memory, and inexpensive software make signal processing procedures of enormous sophistication and power available to anyone who wants to use them. Because advanced signal processing is now accessible to everybody, there is a need for primers that introduce basic mathematical concepts that underlie the digital algorithms. The present handbook chapter is intended to serve such a purpose.

The chapter emphasizes careful definition of essential terms used in the description of signals per international standards. It introduces the Fourier series for signals that are periodic and the Fourier transform for signals that are not. Both begin with analog, continuous signals, appropriate for the real acoustical world. Emphasis is placed on the consequences of signal symmetry and on formal relationships. The autocorrelation function is related to the energy and power spectra for finite-duration and infinite-duration signals. The chapter provides careful definitions of statistical terms, moments, and single- and multi-variate distributions. The Hilbert transform is introduced, again in terms of continuous functions. It is applied both to the development of the analytic signal – envelope and phase, and to the dispersion relations for linear, time-invariant systems. The bare essentials of filtering are presented, mostly to provide real-world examples of fundamental concepts – asymptotic responses, group delay, phase delay, etc. This introduction is followed by more advanced ideas: matched filtering and time-reversal processing. Spectral estimation in the presence of noise is treated by several techniques: parametric models, autoregressive procedures, model-based signal processing implemented as Wiener and Kalman filters, and matched-field processing. There is a brief introduction to cepstrology, with emphasis on acoustical applications. The treatment of the mathematical properties of noise emphasizes the generation of different kinds of noise. Digital signal processing with sampled data is specifically introduced with emphasis on digital-to-analog conversion and analog-to-digital conversion. It continues with the discrete Fourier transform and with the z-transform, applied to both signals and linear, time-invariant systems. Digital signal processing continues with an introduction to maximum length sequences as used in acoustical measurements, with an emphasis on formal properties. The chapter ends with a section on information theory including developments of Shannon entropy and mutual information.