Tham khảo tài liệu 'fir filters -', công nghệ thông tin, kỹ thuật lập trình phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Chapter 15 HR Filters via the Bilinear Transformation A popular technique for the design of HR digital filters is the bilinear transformation method which offers several advantages over the other techniques presented in the previous chapter. Bilinear Transformation The bilinear transformation converts the transfer function for an analog filter into the system function for a digital filter by making the substitution 2 1 -z-1 S 7 l z 1 If the analog prototype filter is stable the bilinear transformation will result in a stable digital filter. Algorithm Bilinear transformation Step 1. Obtain the transfer function Ha s for the desired analog prototype filter. step 2. In the transfer function obtained in step 1 make the substitution 2 1-z-1 S T1 Z-1 where T is the sampling interval of the digital filter. Call the resulting digital system function H z . Step 3. The analog prototype filter s transfer function Ha s will in general be a ratio of polynomials in s. Therefore the system function H z obtained 287 288 Chapter Fifteen in step 2 will in general contain various powers of the ratio 1 z-1 1 z 1 in both the numerator and the denominator. Multiply both the numerator and denominator by the highest power of 1 z and collect terms to obtain H z as a ratio of polynomials in z 1 of the form H z VM h z k 2 k o kz 1 i akz k step 4. Use the ak and bk obtained in step 3 to realize the filter in any of the structures given in Sec. . Example Use the bilinear transform to obtain an IIR filter from a second-order Butterworth analog filter with a 3-dB cutoff frequency of 3 KHz. The sampling rate for the digital filter is 30 000 samples per second. solution The analog prototype filter s transfer function is given by io Ha s -------- --------- s2 -J2a cs w2 where wc 6000 . Making the substitution s 2 1 z T l z1 yields co2 ZV l-z- Y r 2 where T 1 30 000. After the appropriate algebraic simplifications and making use of the fact that 6000 r 7t 30 000 - 5 we obtain
