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.^{5.1}This 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
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.

- Definition of a Signal
- Definition of a Filter
- Examples of Digital Filters
- Linear Filters

- Time-Invariant Filters
- Showing Linearity and Time Invariance
- Dynamic Range Compression

- A Musical Time-Varying Filter Example
- Analysis of Nonlinear Filters
- Conclusions
- Linearity and Time-Invariance Problems

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University