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 they vary 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 a filter's coefficients should be substantially constant over its impulse-response duration.) Nonlinear filters 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.^5.2

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