1.2 Bounded Real Transfer Functions 1.2 Bounded Real Transfer Functions 1.2 Bounded Real Transfer Functions A causal stable real-coefficient transfer function A(z)is defined as a bounded real(BR) .Then the condition H)s1implies that Thus,for all finite-energy inputs,the output energy is less than or equal to the input energy transfer function if Y(e)sX(e) implying that a digital filter characterized by a H(e)s1 for all values of BR transfer function can be viewed as a ●Integrating the above from-rtoπ，and passive structure Let x(n)and y(n)denote,respectively.the applying Parseval's relation we get input and output of a digital filter characterized If H(e)=1,then the output energy is equal by a BR transfer function H(z)with X(e) 2bnfs2uaf to the input energy,and such a digital filter is and Y(ei)denoting their DTFTs therefore a lossless system 1.2 Bounded Real Transfer Functions 1.3 Allpass Transfer Function 1.3 Allpass Transfer Function Definition A causal stable real-coefficient transfer Hence,Az)can be written as function H(z)with H(e)=1 is thus called a An IIR transfer function A(z)with unity magnitude response for all frequencies,i.e., u()=Du) lossless bounded real(LBR)transfer function D.(z) The BR and LBR transfer functions are the (e)=1.for all o .Note from the above that if=zis a pole of a real coefficient allpass transfer function,then keys to the realization of digital filters with is called an allpass transfer function it has a zero at 2=1/ low coefficient sensitivity .An M-th order causal real-coefficient allpass The numerator of a real-coefficient allpass transfer function is of the form transfer function is said to be the mirror- 4v(回）=t+d++dEa+2"Dwe image polynomial of the denominator,and dtdE++dw2a+d-Dw闹 vice versa 113 1.2 Bounded Real Transfer Functions A causal stable real-coefficient transfer function H(z) is defined as a bounded real (BR) bounded real (BR) transfer function if Let x(n) and y(n) denote, respectively, the input and output of a digital filter characterized by a BR transfer function H(z) with and denoting their DTFTs ( ) 1 for all values of j H e  ( ) j X e ( ) j Y e 14 1.2 Bounded Real Transfer Functions Then the condition implies that Integrating the above from to , and applying Parseval’s relation we get ( )1 j H e  2 2 () () j j Ye Xe   2 2 () () n n y n xn      15 1.2 Bounded Real Transfer Functions Thus, for all finite-energy inputs, the output energy is less than or equal to the input energy implying that a digital filter characterized by a BR transfer function can be viewed as a passive structure passive structure If , then the output energy is equal to the input energy, and such a digital filter is therefore a lossless system ( )1 j H e  16 1.2 Bounded Real Transfer Functions A causal stable real-coefficient transfer function H(z) with is thus called a lossless bounded real (LBR) transfer function The BR and LBR transfer functions are the keys to the realization of digital filters with low coefficient sensitivity ( )1 j H e  17 1.3 Allpass Transfer Function Definition An IIR transfer function A(z) with unity magnitude response for all frequencies, i.e., is called an allpass allpass transfer function An M-th order causal real-coefficient allpass transfer function is of the form 2 ( ) 1, for all j A e  1 1 1 1 1 1 1 1 ( ) 1 M M M M M M M M M d d z dz z A z dz d z d z           ( ) DM z 1 ( ) M M z D z   18 1.3 Allpass Transfer Function Hence, AM(z) can be written as Note from the above that if z=z0 is a pole of a real coefficient allpass transfer function, then it has a zero at z=1/z0 The numerator of a real-coefficient allpass transfer function is said to be the mirror￾image polynomial image polynomial of the denominator, and vice versa 1 ( ) ( ) ( ) M M M M z D z A z D z