5.4.3 Detailed Example

1. 5.4.3 Detailed Example The following example is intended to illustrate the process of multirate complementary filtering, and show the actual frequency response of the individual subfilters.

2. Example 5.4: A band-separating filter with a very narrow band is to be designed. Its low-pass part is to have a pass-band cutoff frequency ?p = 9?/16 and stop- band cutoff frequency of ?s=17?/30. For the high-pass part, these values are to be interchanged. The relative transition bandwidth is

5. Fig. 5.20 shows the structure of the chosen multirate complementary filter, which consists of three cascaded multirate stages. The first two stages involve the taking of complements, while the third does not. The corresponding frequency scheme follows from Fig. 5.21The required band-separating filter is formed from the two filters HLP1(z) and HHP1(z), which are complementary to each other, as can be seen from Figs. 5.21a and 5.21b. In the structure in Fig 5.20, the two corresponding outputs are shown, before and after the complement is taken

7. The high-pass HHP1(z) is formed from the high-pass filters HDI1(z) and the kernel filter HLP2(z), refer to Figs. 5.21c and 5.21d. The kernel filter is derived from the high-pass filter by taking the complement. This latter filter is itself realised using the high-pass filters HHP2(z) and the kernel filter HLP3(z), as shown in the Figs. 5.21e-g. Since the cutoff frequencies of the required kernel filter HLP3(z) are below ?/2, the complement is not taken in the final stage, and low-pass decimation and interpolation filters HDI3(z) are used, as in Fig. 5.21h. The kernel filter HK(z) in the third stage is then implemented directly, see Fig. 5.21i.

8. The first step when designing the filter is to transpose the cutoff frequenices of the slope that is to be realised onto those of the kernel filter HK(z). Taking into account the reduction of the sampling rate by a factor of 8, the kernel filter HK(z) must have cutoff frequencies of ??p = ? /2 in the pass-band and ?s= 8 ? /15 in the stop-band. The cutoff frequencies of the decimation are determined by the stop-band cutoff frequencies of the subsequent low-pass filters, see Fig. 5.21.

14. The decimation and interpolation filters are designed as half-band filters using the Parks-McClellan method. The filter HDI1(z) has 21 nonzero coefficients, the filter HDI2(z) five and the filter HDI3(z) seven. The kernel filter HK(z) is not a half-band filter, and requires 121 coefficients. The required complexity is thus

15. This is just 6.8% of the complexity required by the direct implementation, namely 960�fs . It is worth noting that the kernel filter HK(z), which has by far the largest number of coefficients, accounts for just 23% of the complexity, while the decimation and interpolation filter HDI1(z) in the first stage requires 64%. Fig. 5.22 to 5.27 show, in dB, the actual frequency responses of the subfilters of the multirate complementary filter in Fig 5.20. They can best be understood by comparing them with Fig. 5.20 and 5.21.

17. The above example shows that the slope of the resulting frequency response is predominantly determined by the kernel filter HK(z). Despite this, it is responsible for only a small part of the overall computational complexity . This is for two reasons. Firstly, the relative bandwidth in the transition band is larger than with the direct implementation, 1/60 instead of 1/480 in the above example. Secondly, the kernel filter is operated at different sampling rate, f0/8 instead of f0.

18. However, the complexity is increased by the fact that the ripples of subfilters must be less than the overall ripple of the desired band-separation filter.The first decimation and interpolation filter HDI1(z) requires a relatively large complexity. This is partly because this filter is operated at the higher sampling rate, and partly because the cutoff frequencies of the filter HLP2(z) are close to ?, see Figs. 5.25 and 5.26. The design method that has been illustrated here always fails when the cutoff frequencies are either side of ? or very close to it. In such a case, a modified scaling factor can be used. For instance, instead of M=L=2, a factor 3/2 could be chosen.

19. 5.3.5 Estimation of Computational Complexity With multirate complementary filters, the filter slope can be made arbitrarily steep without increasing the computational complexity. This can be shown as follows. Let nmax be the number of coefficients of the largest decimation and interpolation filter.

20. The required complexity of the first stage is then not more than 2nmaxf0 filter filter operations per second. The complexity of the second stage is not more than 2nmaxf0 /2, that of the third stage 2nmaxf0/4 , etc. The total complexity for S stages is then

21. Where nk is the number of coefficients of the kernel filter. When S becomes large, the term for the kernel filter become negligible, and the total complexity can be roughly approximated as 4f0nmax filter operations per second. Even with a high number S of filter stages, the computational complexity remains limited. However, the propagation time through the filter, and the number of delay elements grow exponentially. From this, the length of the cascade is restricted in practical applications.