- Before a signal is applied to the input of a digital filter, the filter's internal “state” is assumed equal to zero.
- Digital filters are linear systems. One property of linear systems is that a sinusoidal input will produce a sinusoidal output of the same frequency.
- However, when a sinusoidal signal is first applied to the input of a digital filter, the output initially exhibits a region of transition (referred to as its transient response).
- For FIR filters, this transition region (or “warm-up” period) has a duration in samples equal to the filter order.
- For IIR filters, the length of the transition region is dependent on the filter order and the feedback coefficient values.
- Assuming a continued application of the sinusoidal input, the filter will subsequently settle into its steady-state region.
- If the input changes frequency or displays a discontinuity of any sort, another transient region will occur in the filter output.
- The frequency response of a digital filter is understood to represent its steady-state behavior.

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