2.3 Minimum-Phase and Maximum- 2.2 Linear-Phase Transfer Functions 2.2 Linear-Phase Transfer Functions Phase Transfer Functions Typical zero locations shown below Summary A causal stable transfer function with all .A Type 2 FIR filter cannot be used to design a zeros outside the unit circle has an excess highpass filter since it always has a zero1 phase compared to a causal transfer function .A Type 3 FIR filter has zeros at both==I and==-1, with identical magnitude but having all zeros and hence cannot be used to design either a lowpass inside the unit circle or a highpass or a bandstop filter zer or ploe vector A Type 4 FIR filter is not appropriate to design a △arg =2 inside the unit circle lowpass filter due to the presence of a zero at z=I .Type I FIR filter has no such restrictions and can be used to design almost any type of filter △arg rero or ploe vector outside the unit circle 2.3 Minimum-Phase and Maximum- 2.3 Minimum-Phase and Maximum- 2.3 Minimum-Phase and Maximum- Phase Transfer Functions Phase Transfer Functions Phase Transfer Functions It assumed that a causal stable transfer A causal stable transfer function with all 。Questions function has Mzeros (with m inside the UC zeros inside the unit circle (m=0)is called a An LTI system is said to be minimum-phase if the and mo outside the UC)and Npoles(with n minimum-phase transfer function system and its inverse are causal and stable. inside the UC and no outside the UC) A causal stable transfer function with all (A(z)B(z=1) AargHe =2πm-2πm,+2a(N-M) zeros outside the unit circle(m=M)is called .Is a causal stable allpass filter minimum or a maximum-phase transfer function maximum phase? =2π(6-mg） A transfer function with zeros inside and .What is case for a linear phase FIR filter? Since N=n and i.e.,no=0,we have outside the unit circle is called a mixed-phase Aarg[m(m transfer function49 Typical zero locations shown below 2.2 Linear-Phase Transfer Functions Re z jIm z Re z jIm z Re z jIm z Re z jIm z 1 1 1 1 j e j e j0 e j0 e Type 1 N odd Type 4 N even Type 3 N odd Type 2 N even 50 Summary A Type 2 FIR filter cannot be used to design a highpass highpass filter since it always has a zero z=ˉ1 A Type 3 FIR filter has zeros at both z = 1 and z=ˉ1, and hence cannot be used to design either a lowpass lowpass or a highpass highpass or a bandstop filter A Type 4 FIR filter is not appropriate to design a lowpass lowpass filter due to the presence of a zero at z = 1 Type 1 FIR filter has no such restrictions and can be used to design almost any type of filter 2.2 Linear-Phase Transfer Functions 51 2.3 Minimum-Phase and Maximum￾Phase Transfer Functions A causal stable transfer function with all zeros outside outside the unit circle has an excess phase compared to a causal transfer function with identical magnitude but having all zeros inside the unit circle jIm z Re z  2 0 zero or ploe vector inside the unit circle arg 2 !   2 0 zero or ploe vector outside the unit circle arg 0 !  unit circle 52 2.3 Minimum-Phase and Maximum￾Phase Transfer Functions It assumed that a causal stable transfer function has M zeros (with mi inside the UC and m0 outside the UC) and N poles (with ni inside the UC and n0 outside the UC) Since N= ni and i.e., n0=0 , we have " # 2 0 0 0 arg ( ) 2 2 2 ( ) 2 j He m n N M i i n m !          2 0 0 arg ( ) 2 j He m !      53 2.3 Minimum-Phase and Maximum￾Phase Transfer Functions A causal stable transfer function with all zeros inside the unit circle (mo=0) is called a minimum-phase transfer function A causal stable transfer function with all zeros outside the unit circle (mo=M) is called a maximum-phase transfer function A transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer transfer function 54 2.3 Minimum-Phase and Maximum￾Phase Transfer Functions Questions An LTI system is said to be minimum-phase if the system and its inverse are causal and stable. (A(z)B(z)=1) Is a causal stable allpass filter minimum or maximum phase? What is case for a linear phase FIR filter?