TAILIEUCHUNG - APPLICATIONS OF MATLAB IN SCIENCE AND ENGINEERING - PART 6

thiết kế phương pháp nội suy Thay vì giảm thiểu một chức năng thông báo lỗi, các hệ số FDF được tính toán từ thực hiện chức năng lỗi tối đa bằng phẳng = 0. Điều này có nghĩa là các dẫn xuất của một chức năng thông báo lỗi bằng số không vào thời điểm này tần số: | 254 Applications of MATLAB in Science and Engineering a n Fig. 8. FDF Frequency responses using minimax method for D to with Nfd 20 and a . Interpolation design approach Instead of minimizing an error function the FDF coefficients are computed from making the error function maximally-flat at 0 0. This means that the derivatives of an error function are equal to zero at this frequency point õnec o 80 0 n 0 1 2 .NFD -1 0 0 17 the complex error function is defined as ec o HFD H - Hid H 18 where HFD a i is the designed FDF frequency response and Hid a i is the ideal FDF frequency response given by equation 6 . The solution of this approximation is the classical Lagrange interpolation formula where the FDF coefficients are computed with the closed form equation . NFD D-k hL n nD-J n 0 1 2 .Nfd k n 19 where NFD is the FDF length and the desired delay D _NFD 2 J id. We can note that the filter length is the unique design parameter for this method. The FDF frequency responses designed with Lagrange interpolation with a length of 10 are shown in Fig. 9. As expected a flat magnitude response at low frequencies is presented a narrow bandwidth is also obtained. Fractional Delay Digital Filters 255 Magnitude Responses Phase Responses ffl rc Fig. 9. FDF Frequency responses using Lagrange interpolation for D to with Nfd 10. The use of this design method has three main advantages Laakson et al. 1994 1 the ease to compute the FDF coefficients from one closed form equation 2 the FDF magnitude frequency response at low frequencies is completely flat 3 a FDF with polynomial-defined coefficients allows the use of an efficient implementation structure called Farrow structure which will be described in section . On the other hand there are some disadvantages to be taken into account when a Lagrange interpolation is used in FDF design 1 the achieved bandwidth is narrow 2 the design is made in time-domain and then any frequency information of the processed signal

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