Legacy Documentation

Digital Image Processing (2000)

This is documentation for an obsolete product.
Current products and services
User's Guide / Filter Design /
Previous section-----Next section

9.2 Linear-Phase FIR Filters

Many image processing operations of practical interest may be formulated as an LSI filtering operation with an FIR filter. An FIR filter is completely specified by a finite set of coefficients. In most applications the data and filter are real valued. Recall from Section 5.2 that the filtering operation may then be described by the following convolution sum:
where h[n] is a FIR filter, x[n] is an input sequence, and y[n] the output. In image processing, FIR filters with linear phase are of particular interest. Such filters do not distort the phase of the input signal. It is well known that FIR filters can always be designed such that they exhibit this desirable characteristic. A FIR filter has linear phase if its impulse response satisfies either one of the following two symmetry conditions:
or
The FIR filter is called symmetric if it satisfies the former condition and antisymmetric in the latter. We now categorize linear phase filters according to their symmetry and length:
    Type 1 - symmetric, odd length
    Type 2 - symmetric, even length
    Type 3 - antisymmetric, odd length
    Type 4 - antisymmetric, even length
Examples of the four types of impulse response sequences are shown in Figure 9.2.1.

Figure 9.2.1. Illustration of four types of impulse response symmetry.

Note additionally that a Type 3 filter must have h [(N-1)/2 ] = 0. Each filter type has unique frequency domain characteristics. Here we summarize these characteristics to guide the users in matching the filter type with the application [Mit98, Opp89]. A Type 1 filter may be used to implement any desired bandpass frequency response. The symmetry condition of a Type 2 filter requires that H()=0, and therefore it may not be used to define a highpass filter. Otherwise it may be used instead of Type 1 in cases where an even-length filter is preferable. In contrast, antisymmetric filters (Type 3 and 4) cannot be used to implement any sort of bandpass filter. They are the only choice, however, for designing FIR differentiators, which are an important class of filters (recall the discussion of edge detectors).