Transient and Steady-State Signals

Loosely speaking, any sudden change in a signal is regarded as a transient, and transients in an input signal disturb the steady-state operation of a filter, resulting in a transient response at the filter output. This leads us to ask how do we define ``transient'' in a precise way? This turns out to be difficult in practice.

A mathematically convenient definition is as follows: A signal is said to contain a transient whenever its Fourier expansion [84] requires an infinite number of sinusoids. Conversely, any signal expressible as a finite number of sinusoids can be defined as a steady-state signal. Thus, waveform discontinuities are transients, as are discontinuities in the waveform slope, curvature, etc. Any fixed sum of sinusoids, on the other hand, is a steady-state signal.

In practical audio signal processing, defining transients is more difficult. In particular, since hearing is bandlimited, all audible signals are technically steady-state signals under the above definition. One way to pose the question is to ask which sounds should be ``stretched'' and which should be translated in time when a signal is ``slowed down''? In the case of speech, for example, short consonants would be considered transients, while vowels and sibilants such as ``ssss'' would be considered steady-state signals. Percussion hits are generally considered transients, as are the ``attacks'' of plucked and struck strings (such as piano). More generally, almost any ``attack'' is considered a transient, but a slow fade-in of a string section, e.g., might not be. In sum, musical discrimination between ``transient'' and ``steady state'' signals depends on our perception, and on our learned classifications of sounds. However, to first order, transient sounds can be defined practically as sudden ``wideband events'' in an otherwise steady-state signal. This is at least similar in spirit to the mathematical definition given above.

In summary, a filter transient response is caused by suddenly switching on a filter input signal, or otherwise disturbing a steady-state input signal away from its steady-state form. After the transient response has died out, we see the steady-state response, provided that the input signal itself is a steady-state signal (a fixed linear combination of sinusoids) and given that the filter is LTI.

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written by Julius Orion Smith III
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.