The output V(z)of the k-th branch is a linear combination of intermediate signals rkn] Xk(z),0≤k≤R-1: R-1 Vi(z)=>Xp(=)Hkp(),Hkp(z)=e Hop(e 2). p=0 Observe that xo[n],x1[n],...,R-n]are the polyphase components of the input sequence x[n]. Because they are common to each parallel branch of the MFB,we can share them at the input to obtain the following amalgamated structure: MIMO system R-inputs N-outputs tn IR coln] voln] IR n +h[n] H IR xaln] +2[n] IR ZR-1[n] UN-1[n] Polyphase implementation of the MFB In this system: Filter bank output rate N No.of filter bank channels =N. Filter bank input rate R The transfer function matrix H satisfies Xo(z) 6(z) X1(z) (z） XR-1(2) VN-1(2) Hp()=eop(ej特z). Such a polyphase implementation is preferred over the direct implementation of the MFB because it is more efficient in terms of computational complexity.To see why this is true, assume our system operates with complex arithmetic.In the original system: 3� The output Vk(z) of the k-th branch is a linear combination of intermediate signals xk[n] Xk(z),0 ≤ k ≤ R − 1: ⇔ R−1 p=0 Observe that x0[n], x1[n], . . . , xR−1[n] are the polyphase components of the input sequence x[n]. Because they are common to each parallel branch of the MFB, we can share them at the input to obtain the following amalgamated structure: Polyphase implementation of the MFB In this system: Filter bank output rate = N No. of filter bank channels = N, . Filter bank input rate R The transfer function matrix H satisfies Hkp(z) = e−j 2πkp H0p(e−j 2πk V N N k(z) = Xp(z)Hkp(z) , z) . ⎡ ⎢ ⎢ ⎢ ⎢ ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎢ ⎢ ⎤ ⎥ ⎥ ⎥ ⎥ X0(z) X1(z) ⎡ ⎢ ⎢ ⎢ ⎢ ⎤ ⎥ ⎥ ⎥ ⎥ V0(z) V1(z) ⎤ ⎥ ⎥ ⎥ Hkp ⎥ N (z) . = . . . ⎣ ⎣⎦ . ⎦ ⎣ . ⎦ XR−1(z) VN−1(z) Hkp(z) = e−j 2πkp H0p(e−j 2πk N N z) . Such a polyphase implementation is preferred over the direct implementation of the MFB because it is more efficient in terms of computational complexity. To see why this is true, assume our system operates with complex arithmetic. In the original system: 3