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Transient Response, Steady State, and Decay

Figure 5.7: Example transient, steady-state, and decay responses for an FIR ``running sum'' filter driven by a gated sinusoid.
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 $ N$ FIR filters only ``remember'' $ N-1$ samples into the past. Thus, for length $ N$ FIR filters, the duration of the transient response is $ N-1$ samples.

To show this, (it may help to refer to the general FIR filter implementation in Fig.5.5), consider that a length $ N=1$ (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 $ N = 2$ 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 $ N$ FIR filter is fully ``warmed up'' after $ N-1$ samples of input; that is, for an input starting at time $ n = 0$ , by time $ n=N-1$ , all internal state delays of the filter contain delayed input samples instead of their initial zeros. When the input signal is a unit step $ u(n)$ times a sinusoid (or, by superposition, any linear combination of sinusoids), we may say that the filter output reaches steady state at time $ n=N-1$ .

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University