Convolutional vs block codes; (7,4) cyclic code for 1011

Why a convolutional coder is better than a block coder

• It works on a continuous stream, so no need to wait for a full block; encoding delay and buffering are small.
• It has memory: each output depends on several past bits, so redundancy is spread over many bits and burst-like noise is handled better with interleaving.
• Soft-decision Viterbi decoding is easy, giving 2 to 3 dB more coding gain than hard-decision block decoding at the same rate.
• Simple shift-register encoder and good performance at low rates (used in GSM, satellite, Wi-Fi).

(7,4) cyclic Hamming code, $g(x)=1+x+x^3$, message 1011

Convention (Haykin): message $(m_0m_1m_2m_3)=1011$ in ascending powers, so

$$m(x)=1+x^2+x^3$$

Modulo-2 arithmetic.

Non-systematic: $c(x)=m(x)g(x)$

$$\begin{aligned} c(x)&=(1+x^2+x^3)(1+x+x^3)\\ &=1+x+x^3\;+\;x^2+x^3+x^5\;+\;x^3+x^4+x^6\\ &=1+x+x^2+x^3+x^4+x^5+x^6 \end{aligned}$$

Non-systematic code vector $(c_0\dots c_6)$ = 1111111.

Systematic: $c(x)=b(x)+x^{n-k}m(x)$, with $b(x)$ = remainder of $x^3m(x)/g(x)$.

$$x^3m(x)=x^3+x^5+x^6$$

              x^3 + x^2 + x + 1
          ---------------------------
x^3+x+1 ) x^6 + x^5 +       x^3
          x^6 +       x^4 + x^3
          -------------------------
                x^5 + x^4
                x^5 +       x^3 + x^2
                -----------------------
                      x^4 + x^3 + x^2
                      x^4 +       x^2 + x
                      -------------------
                            x^3 +       x
                            x^3 +       x + 1
                            -----------------
                                          1

$b(x)=1$, so parity bits $(b_0b_1b_2)=100$.

$$c(x)=1+x^3+x^5+x^6$$

Systematic code vector $(b_0b_1b_2\,m_0m_1m_2m_3)$ = 100 1011.

Check: $c(x)=(1+x+x^2+x^3)g(x)$, which expands to $1+x^3+x^5+x^6$, so it is a valid codeword.