 4.45. In analogy with the N = 2 kernel described above, the first N = 2 of the N kernel encoded Given a m X n matrix A and a p X q matrix B, the Kronecker product C=AQ B also called matrix direct product, is an mp X nq matrix with elements defined by cab = where a = p(i - 1) + and = 1. The matrix direct product provides the matrix of the linear transformation induced by the vector space tensor product of the original vector spaces. Assuming that operators S:V1 W1 and T:V2 W2 are given by S(x) = Ax and T(y) = By, then T:V1 V2 W1 x W2 is determined by SOT(xy) (Ax) X 520 Chapter 4 Size-2 polar encoder Size-8 polar encoder Size-4 polar encoder Figure 4.45 Polar encoder of different sizes [32]. Successive Channel Index selection Polar encoder cancellation Index selection decoder Encoder Decoder Figure 4.46 High-level polar code encoding and SC list decoding [40]. bits are obtained as XORs of corresponding bits from the outputs of the two N = 2 kernels, while the second N = 2 of the kernel encoded bits are equal to the output of the second N = 2 kernel [35]. Polar codes were introduced as non-systematic codes. Any linear code can be transformed from non-systematic to a systematic code. The systematic polar encoding can be performed using the standard non-systematic polar encoding apparatus in a three-phase operation as follows (see Fig. 4.46): 1. The vector u = (ua, is encoded in the standard non-systematic fashion producing the vector u. 2. The frozen bit positions A in the vector u are set to zero UAC=0 (the frozen bits are always set to zero here). 3. The modified vector u is encoded in the standard non-systematic fashion producing the codeword X which is a systematic polar codeword, in the sense that the information bits UA appear in the final codeword X in the information bits position . New Radio Access Physical Layer Aspects (Part 2) 521 The main advantage of the systematic polar codes is that its BER performance is better than non-systematic polar codes. However, both systematic and non-systematic polar codes 