### 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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