Transient Response, Steady State, and Decay

Input
Signal
Filter Output Signal |

The terms *transient response* and *steady state response*
arise naturally in the context of sinewave analysis (*e.g.*,
§2.2). When the input sinewave is switched on, the filter
takes a while to ``settle down'' to a perfect sinewave at the same
frequency, as illustrated in Fig.5.7(b). The filter response during
this ``settling'' period is called the *transient response* of
the filter. The response of the filter *after* the transient
response, provided the filter is linear and time-invariant, is called
the *steady-state response*, and it consists of a pure sinewave
at the same frequency as the input sinewave, but with amplitude and
phase determined by the filter's *frequency response* at that
frequency. In other words, the steady-state response begins when the
LTI filter is fully ``warmed up'' by the input signal. More
precisely, the filter output is the same as if the input signal had
been applied since time minus infinity. Length
FIR filters
only ``remember''
samples into the past.
Thus, for length
FIR filters, the duration of the transient response is
samples.

To show this, (it may help to refer to the general FIR filter implementation in Fig.5.5), consider that a length (zero-order) FIR filter (a simple gain), has no state memory at all, and thus it is in ``steady state'' immediately when the input sinewave is switched on. A length FIR filter, on the other hand, reaches steady state one sample after the input sinewave is switched on, because it has one sample of delay. At the switch-on time instant, the length 2 FIR filter has a single sample of state that is still zero (instead of its steady-state value which is the previous input sinewave sample).

In general, a length
FIR filter is fully ``warmed up'' after
samples of input; that is, for an input starting at time
, by
time
, all internal state delays of the filter contain delayed
input samples instead of their initial zeros. When the input signal is
a unit step
times a sinusoid (or, by superposition, any linear
combination of sinusoids), we may say that the filter output reaches
*steady state* at time
.

- FIR Example
- IIR Example
- Transient and Steady-State Signals
- Decay Response, Initial Conditions Response
- Complete Response

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