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Fig.1 The scheme of the component code of the TC specified in the standard 3GPP-RAN, having the generator matrix G3epp=[11 13].

Figure 1 The scheme of the component code of the TC specified in the standard 3GPP-RAN, having the generator matrix G3epp=[11 13].

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The controller and observer canonical forms for the RSDBC encoder used in the standard Digital Video Broadcasting - Return Channel via Satellite (DVB RCS) [6] are presented in Fig. 2.
The controller and observer canonical forms for the RSDBC encoder used in the standard Digital Video Broadcasting - Return Channel via Satellite (DVB RCS) [6] are presented in Fig. 2.
Fig.2 The scheme of the component code of the TC specified in the standard DVB-RCS, having the generator matrix Gpyve=([13 15 11]. The generator matrix and the input-output relation of the system have the expressions: where g)(D)=1+ D+ D*,g,(D)=1+.D+D?+D* and g,(D)=1+D* +D®, x,(D) and x,(D) are the polynomials of the input sequences (their D transforms) and x (D) is the redundancy. Consensually with the previous paragraph we will use from now on the denotation (in decimal): Gpyg=[13 15 11], and the code word has the form y(D)=x(D)-Gpyg(D)=[x2(D) »(D) xo(D)].
Fig.2 The scheme of the component code of the TC specified in the standard DVB-RCS, having the generator matrix Gpyve=([13 15 11]. The generator matrix and the input-output relation of the system have the expressions: where g)(D)=1+ D+ D*,g,(D)=1+.D+D?+D* and g,(D)=1+D* +D®, x,(D) and x,(D) are the polynomials of the input sequences (their D transforms) and x (D) is the redundancy. Consensually with the previous paragraph we will use from now on the denotation (in decimal): Gpyg=[13 15 11], and the code word has the form y(D)=x(D)-Gpyg(D)=[x2(D) »(D) xo(D)].
in which fog=(fi-fo- | mod, fia=(1-2:/:)modN and frg=(N-fo)modN. It results that for the dual mirror equivalent TC of SC3gp,, denoted as IC3e,4, We have the following interleaver function: GENERATOR POLYNOMIALS UNDER MIRROR EQUIVALENCE Toppd(i) = N-TMopeN-i-1) -1= (foa + fiai + rari’) mod N, OSI<N, where u is a certain input sequence for the encoder defined by the matrix G.
in which fog=(fi-fo- | mod, fia=(1-2:/:)modN and frg=(N-fo)modN. It results that for the dual mirror equivalent TC of SC3gp,, denoted as IC3e,4, We have the following interleaver function: GENERATOR POLYNOMIALS UNDER MIRROR EQUIVALENCE Toppd(i) = N-TMopeN-i-1) -1= (foa + fiai + rari’) mod N, OSI<N, where u is a certain input sequence for the encoder defined by the matrix G.
Fig. 4 BER/FER versus SNR performances for the TCs: 3C3g,, (black), IC3eypa (blue), ICpyz (green) and SCpyzgq (red) ir the IDT case. _— a a ota For a single-binary TC to have a mirror equivalent it is necessary that its generator matrix to be of the form G=[9IU(go) go]. The dual TC has the generator matrix G=[go DU(go)]. The interleaver functions satisfy a general condition of the form (7). Particularizations of type (9) or (10) become applicable if QPP or ARP interleavers are used. For the duo-binary case, the generator matrix G may have the form [(go) gi gol, respectively [g: 2¢(go) go]. Table 1 shows the polynomials of a maximum degree of 5 (memory = m) whose mirror images polynomials are different but have the same degree. In addition, there are presented the g, polynomials for which 1 (g,) = gy as well. These cannot be used as go polynomials because it would result non efficient generator matrices (having two identical columns). There are also primitive polynomials marked in red. In the last two columns of table 1, we have indicated the number of recursive and systematic convolutional code pairs, which can
Fig. 4 BER/FER versus SNR performances for the TCs: 3C3g,, (black), IC3eypa (blue), ICpyz (green) and SCpyzgq (red) ir the IDT case. _— a a ota For a single-binary TC to have a mirror equivalent it is necessary that its generator matrix to be of the form G=[9IU(go) go]. The dual TC has the generator matrix G=[go DU(go)]. The interleaver functions satisfy a general condition of the form (7). Particularizations of type (9) or (10) become applicable if QPP or ARP interleavers are used. For the duo-binary case, the generator matrix G may have the form [(go) gi gol, respectively [g: 2¢(go) go]. Table 1 shows the polynomials of a maximum degree of 5 (memory = m) whose mirror images polynomials are different but have the same degree. In addition, there are presented the g, polynomials for which 1 (g,) = gy as well. These cannot be used as go polynomials because it would result non efficient generator matrices (having two identical columns). There are also primitive polynomials marked in red. In the last two columns of table 1, we have indicated the number of recursive and systematic convolutional code pairs, which can
Fig. 5 BER/FER versus SNR performances for the TCs: 3C3¢,, (black), IC3eynq (blue), SCpyg (green) and SCpyzgq (red) in the tail biting case.
Fig. 5 BER/FER versus SNR performances for the TCs: 3C3¢,, (black), IC3eynq (blue), SCpyg (green) and SCpyzgq (red) in the tail biting case.
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