Filter Design and Implementation - Digital Filter Design Solutions - page 15

1.2 1.4

( a )

( b )

1 1.2

1

0.8

0.8

0.6

0.4 0.6

0.2 0.4

0 0.2

-0.2

0

0 20 40 0 20 40

Index, k Index, k

Figure 2.8 Step response for (a) linear phase filter and (b) minimum phase filter formed from the
coefficients of the linear phase.

In some applications, a minimum phase filter is required. The LPFCoef function
will return low-pass minimum phase filter coefficients when the input argument is
appended with the string ' min ' . If ' min ' is not present in the input argument, linear
phase coefficients are returned. In general, to convert a linear phase filter into a
minimum phase type, one may use the Matlab polystab function [7]:

hmin = polystab(h)*norm(h)/norm(polystab(h)) (2.9)

where hmin is the vector of minimum phase coefficients and h is the linear phase
input coefficients. Note that polystab stabilizes a polynomial with respect to the
unit circle by reflecting roots residing on the outside to inside the circle. It has
been observed that when the filter length approaches 90, and the cutoff frequency
is less than about 0.15, polystab itself may fail, and as such this approach is limited
to smaller filter lengths.
Figure 2.7 shows the impulse response for a linear phase filter and its
minimum phase counterpart using the LPFCoef function for a 55-tap filter with
cutoff at 0.4. Notice that (a) the minimum phase filter coefficients are not
symmetrical about its center like the linear phase, (b) the frequency responses of
both filters are identical, (c) the minimum phase impulse response may need to be
multiplied by -1 on occasions, and (d) the step response of the minimum phase is
asymmetrical whilst its counterpart is symmetrical (see Figure 2.8). In the next
section, we will consider how the filter is implemented.

2.5 FILTER IMPLEMENTATION

There are two ways of implementing the filter: (1) the direct, or time domain
method and, (2) the frequency domain approach. The method chosen is usually