Convolutional vs block coder; (7,4) cyclic code for 1010

Why convolutional coders are better suited than block coders

• Continuous operation: they encode a serial bit stream directly; block coders must wait for $k$ bits, adding delay and buffering.
• Memory: each output depends on the present and $K-1$ previous bits, so redundancy is spread over a long span, giving good performance against random noise.
• Soft-decision decoding: the Viterbi algorithm easily uses soft (analog) values, giving about 2 dB extra gain, which is hard for algebraic block decoders.
• Simple hardware: shift registers and XOR gates; widely used in satellite, GSM and Wi-Fi links.

(7,4) cyclic Hamming code, $g(x)=1+x+x^3$, message 1010

Convention (Haykin): $(m_0m_1m_2m_3)=1010$, ascending powers:

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

Non-systematic:

$$\begin{aligned} c(x)&=m(x)g(x)=(1+x^2)(1+x+x^3)\\ &=1+x+x^3+x^2+x^3+x^5\\ &=1+x+x^2+x^5 \end{aligned}$$

$(c_0c_1\dots c_6)$ = 1110010

Systematic:

1. $x^{n-k}m(x)=x^3(1+x^2)=x^3+x^5$
2. Divide by $g(x)$:

              x^2
          ------------------
x^3+x+1 ) x^5 +       x^3
          x^5 +       x^3 + x^2
          ---------------------
                            x^2   <- remainder

3. $b(x)=x^2$, parity bits $(b_0b_1b_2)=001$.
4. $c(x)=b(x)+x^3m(x)=x^2+x^3+x^5$.

$(b_0b_1b_2\,m_0m_1m_2m_3)$ = 001 1010

Check: $c(x)=x^2g(x)=x^2+x^3+x^5$, divisible by $g(x)$, so valid.