Tham khảo tài liệu 'handbook of industrial automation - richard l. shell and ernest l. hall part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | h -t RCuf For a given periodic sampling period of Ts the resulting sampled impulse response is given by hd k Tsh kTfi or for k 0 hd fc ịfe kT IRC ak RC RC where a fi -RC it follows that H tzl 1 . - u RC 1 - az-1 RC z - a The frequency response of the impulse-invariant filter is given by RC e19 a which is periodic with a normalized period n. Bilinear Z-Transform Lowpass filters have known advantages as a signal interpolator see Chap. . In the continuous-time domain an integrator is a standard lowpass filter model. A continuous-time integrator interpolator is given by H i I 48 which has a common discrete-time Reimann model given by y k 1 y k I x k x k 1 49 which has a z-transform given by Y z z-1 Y z L z- A z X z 50 which results in the relationship 5 A z 1 51 5 Ts z - 1 or 2 T. 5 z ---- 52 Ts - s Equation 51 is called a bilinear z-transform. The advantage of the bilinear z-transform over the standard z-transform is that it eliminates aliasing errors introduced when an analog filter model with are arbitrarily long nonzero frequency response was mapped into the z-plane. The disadvantage in some applications is that the bilinear z-transform is not impulse invariant. As a result the bilinear z-transform is applied to designs which are specified in terms of frequency-domain attributes and ignore time-domain qualifiers. If impulse invariance is required the standard z-transform is used with an attendant loss of frequency-domain performance. Warping The frequency response of a classic analog filter denoted ỉ jữ Q e ro ro eventually needs to be interpreted as a digital filter denoted where rn e n n . The bilinear z-transform can map the analog frequency axis onto the digital frequency axis without introducing aliasing or leakage as was found with a standard z-transform. To demonstrate this claim consider evaluating Eq. 51 for a given analog frequency s j . Then 2e - 1 _ 2 j sin a 2 Tfi 1 Ts cos ờ 2 2 z _ - j tan ứ 2 53 which upon simplification reduces to
