延安大学：《数字信号处理》课程PPT教学课件（DigitalSignal Processing，DSP）Chapter 7 FIR Digital Filter Design

7.1 Linear-phase FIR transfer function In the previous section, we pointed out why it is important to have a transfer function with a linear-phase property. In this section, we develop the forms of a linear-phase FIr transfer function H(z) with real impulse response h(n)

7.1 Linear-phase FIR transfer function In the previous section, we pointed out why it is important to have a transfer function with a linear-phase property. In this section, we develop the forms of a linear-phase FIR transfer function H(z) with real impulse response h(n)

7.1 Linear-phase FIR transfer function If H z is required to have a linear-phase, its frequency response must be of the form H(el)=Hg(o)e lo Where Ha(o) is called the amplitude response NOTE D Hg(o)is different from H(ejo)l a o(w) is a linear-phase function

7.1 Linear-phase FIR transfer function If H(z) is required to have a linear-phase, its frequency response must be of the form ( ) ( ) j j ( ) H e H e g    =  Where Hg(ω) is called the amplitude response. NOTE:  Hg(ω) is different from |H(ejω)|.  Θ(ω) is a linear-phase function

7.1 Linear-phase FIR transfer function Type 1 linear-phase: To T is a constant Type 2 linear-phase (o=8o-to r is a constant, 0 is a initial phase aa unify representation dElo

7.1 Linear-phase FIR transfer function Type 1 linear-phase:     ( ) = − is a constant Type 2 linear-phase:  A unify representation: ( ) 0 0       = − is a constant, is a initial phase d ( ) d     − =

7.1 Linear-phase FIR transfer function It can be proven that the transfer function have a linear-phase, if its impulse response h(n) is either symmetric h(n)=(N-1-n),0≤n≤N-1 or is antisymmetrIC h(n)=-h(N-1-n),0≤n≤N-1

7.1 Linear-phase FIR transfer function It can be proven that the transfer function have a linear-phase, if its impulse response h(n) is either symmetric h n h N n n N ( ) = − −   − ( 1 , 0 1 ) or is antisymmetric h n h N n n N ( ) = − − −   − ( 1 , 0 1 )

7.1 Linear-phase FIR transfer function Center of h(n) Type 2 Center of N=9 even-svmnetrv N=8 enen-svImmetry h(n) Type 3 Center of h(n) Type 4 Center of N9 dd-svmmetrv N=8 odd-symmetry n

7.1 Linear-phase FIR transfer function