7.1 Linear-phase fIr transfer function Proof H(e)=∑h(n)em=∑h(m) cos on-j∑h(n) )sin on Ifh(n)is a real sequence, we have ∑h(n) sinan B(o)=arg tan-N-l ∑h(n) coson
7.1 Linear-phase FIR transfer function
7.1 Linear-phase FIR transfer function If this transfer function has a linear phase, such as 6(0)=-0 We obtain the following relationship ∑h(m) sin@n 8(o)=arg tanf-x-o ∑h(n) cos on Taking tan( on both sides of the above equation
7.1 Linear-phase FIR transfer function
7.1 Linear-phase FIR transfer function ) h(n)sinan 5111r tan to Cos 1O N-1 , ∑h(n) cos onsin to-∑h(n) ISin @2COS TO ∑h(n)sin[(x-n)]=0
7.1 Linear-phase FIR transfer function
7.1 Linear-phase FIR transfer function sin[(r-n)o is odd-symmetry on t=n Let T=(N-1)/2, thus equation holds if h(n) is even-symmetry on n=(N-1)/2 In other words,h(n)=h(N-1-n),0≤n≤N-1 Similarly, if 6(@=-T/2-to, we can arrive at h(m)=-h(N-1-n)0≤n≤N-1
7.1 Linear-phase FIR transfer function
7.1 Linear-phase FIR transfer function Since the length of the impulse response can be either even or odd. we can define four types of linear-phase FIR transfer functions For an antisymmetric FIR filter of odd length ie.,Nod:h{(N-1)2)}=0 e We examine next the each of the 4 cases
7.1 Linear-phase FIR transfer function