Information Technology Reference
In-Depth Information
Figure 3.7:
In discrete time convolution, the mirrored impulse response is stepped through the input one sample
period at a time. At each step, the sum of the cross-products is used to form an output value. As the input in this
example is a constant height pulse, the output is simply proportional to the sum of the coincident impulse response
samples. This figure should be compared with
Figure 3.6
.
In digital-to-analog convertors, the impulse response of the reconstruction filter is convolved with the sample pulses
to create a continuous output waveform. In a wavelet decoder the output is reconstructed by convolving the
wavelet function with a pulse whose amplitude is determined by a coefficient.
3.4 FIR and IIR filters
Filters can be described in two main classes, as shown in
Figure 3.8
, according to the nature of the impulse
response. Finite-impulse response (FIR) filters are always stable and, as their name suggests, respond to an
impulse once, as they have only a forward path. In the temporal domain, the time for which the filter responds to an
input is finite, fixed and readily established. The same is therefore true about the distance over which a FIR filter
responds in the spatial domain. FIR filters can be made perfectly phase linear if a significant processing delay is
accepted. Most filters used for image processing, sampling rate conversion and over- sampling fall into this
category.
Figure 3.8:
An FIR filter (a) responds only once to an input, whereas the output of an IIR filter (b) continues
indefinitely rather like a decaying echo.
