Conclusion

An introduction to digital filters has been presented. The main utility of the analysis methods presented is in ascertaining how a given filter will affect the spectrum of a signal passing through it. Some of the concepts introduced were linearity, time-invariance, filter impulse response, difference equations, transient response, steady-state response, transfer functions, amplitude response, phase response, phase delay, group delay, linear phase, minimum phase, maximum phase, poles and zeros, filter stability, and the general use of complex numbers to represent signals, spectra, and filters. Additionally, practical filtering in matlab has been discussed.

However, this is still only the beginning. With these foundations there is an unlimited number of avenues of investigation into applications of digital filters. Some elementary examples are introduced in Appendix B, such as first- and second-order sections, the dc blocker, shelf equalizer, peaking equalizer, and time-varying resonators. A starting introduction to analog filters appears in Appendix E, and matrix formulations of digital filters are pursued in Appendices F and G. Some methods for digital filter design are discussed in Appendix I. Book III [86] of the music signal processing book series further discusses delay lines, comb filters, feedback delay networks, reverberator design, and computational physical modeling for sound synthesis and audio effects using digital filters, among other related topics. The fourth book [87] introduces FFT-based FIR filtering, and psychoacoustically motivated signal processing, with particular emphasis on time-varying spectral modifications in audio signal processing. Book IV also contains more about audio FIR filter design.

Given the immense range of naturally occurring filters in the domain of music, it is reasonable to expect that filter theory will continue to provide valuable tools for the analysis, synthesis, and manipulation of sound.

The following appendices provide elementary background material in support of the preceding chapters, as well as related and more advanced topics for further study.