**Search Introduction to Digital Filters**

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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

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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