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Linear Time-Invariant Digital Filters

In this chapter, the important concepts of linearity and time-invariance (LTI) are discussed. Only LTI filters can be subjected to frequency-domain analysis as illustrated in the preceding chapters. After studying this chapter, you should be able to classify any filter as linear or nonlinear, and time-invariant or time-varying.

The great majority of audio filters are LTI, for several reasons: First, no new spectral components are introduced by LTI filters. Time-varying filters, on the other hand, can generate audible sideband images of the frequencies present in the input signal (when the filter changes at audio rates). Time-invariance is not overly restrictive, however, because the static analysis holds very well for filters that change slowly with time. (One rule of thumb is that the coefficients of a quasi-time-invariant filter should be substantially constant over its impulse-response duration.) Nonlinear filters generally create new sinusoidal components at all sums and differences of the frequencies present in the input signal.5.1This includes both harmonic distortion (when the input signal is periodic) and intermodulation distortion (when at least two inharmonically related tones are present). A truly linear filter does not cause harmonic or intermodulation distortion.

All the examples of filters mentioned in Chapter 1 were LTI, or approximately LTI. In addition, the $ z$ transform and all forms of the Fourier transform are linear operators, and these operators can be viewed as LTI filter banks, or as a single LTI filter having multiple outputs.

In the following sections, linearity and time-invariance will be formally introduced, together with some elementary mathematical aspects of signals.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University