第9章-Z变换v1.0.1.ppt

§6.7 The Transfer Function Example - A causal LTI IIR digital filter is described by a constant coefficient difference equation given by y[n]=x[n-1]-1.2x[n-2]+x[n-3]+1.3y[n-1] -1.04y[n-2]+0.222y[n-3] Its transfer function is therefore given by §6.7 The Transfer Function Alternate forms: ROC: Note: Poles farthest from z=0 have a magnitude ★ §6.7.3 Frequency Response from Transfer Function If the ROC of the transfer function H(z) includes the unit circle, then the frequency response H(ej?) of the LTI digital filter can be obtained simply as follows: For a real coefficient transfer function H(z) it can be shown that ★ §6.7.3 Frequency Response from Transfer Function For a stable rational transfer function in the form the factored form of the frequency response is given by §6.7.3 Frequency Response from Transfer Function It is convenient to visualize the contributions of the zero factor (z-?k) and the pole factor (z-?k) from the factored form of the frequency response The magnitude function is given by §6.7.4 Frequency Response from Transfer Function The phase function is given by Rez jImz 1 -1 -j ?k ? §6.7.4 Geometric Interpretation of Frequency Response Computation The factored form of the frequency response is convenient to develop a geometric interpretation of the frequency response computation from the pole-zero plot as ω varies from 0 to 2π on the unit circle The geometric interpretation can be used to obtain a sketch of the response as a function of the frequency A typical factor in the factored form of the frequency response is given by §6.7.4 Geometric Interpretation of Frequency Response Computation where is a zero if it is zero factor or is a pole if it is a pole factor §6.7.4 Geometric Interpretation of Frequency Response Computation As shown below in the z-plane the factor represents a vector starting at the point and ending on the unit circle at