Next: Non-recirculating filter, second form Up: Elementary filters Previous: Elementary filters Contents Index

##
Elementary non-recirculating filter

The non-recirculating comb filter may be generalized to yield the design shown in Figure 8.7. This is the elementary non-recirculating filter, of the first form. Its single, complex-valued parameter controls the complex gain of the delayed signal subtracted from the original one.

Figure 8.7: A delay network with a single-sample delay and a complex gain . This is the non-recirculating elementary filter, first form. Compare the non-recirculating comb filter shown in Figure 7.3, which corresponds to choosing here.

To find its frequency response, as in Chapter 7 we feed the delay network a complex sinusoid whose frequency is . The th sample of the input is and that of the output is

so the transfer function is

This can be analyzed graphically as shown in Figure 8.8. The real numbers and are the magnitude and argument of the complex number :

The gain of the filter is the distance from the point to the point in the complex plane. Analytically we can see this because

Graphically, the number is just the number rotated backwards (clockwise) by the angular frequency of the incoming sinusoid. The value is the distance from to in the complex plane, which is equal to the distance from to .

Figure 8.8: Diagram for calculating the frequency response of the non-recirculating elementary filter (Figure 8.7). The frequency response is given by the length of the segment connecting to in the complex plane.

As the frequency of the input sweeps from 0 to , the point travels couterclockwise around the unit circle. At the point where , the distance is at a minimum, equal to . The maximum occurs which is at the opposite point of the circle. Figure 8.9 shows the transfer function for three different values of .

Figure 8.9: Frequency response of the elementary non-recirculating filter Figure 8.7. Three values of are used, all with the same argument (-2 radians), but with varying absolute value (magnitude) .

Next: Non-recirculating filter, second form Up: Elementary filters Previous: Elementary filters Contents Index