Chapter Questions
Design an encoder of the binary convolutional $(2,1)$-code with the generator polynomial $g_0(x)=1+x+x^3+x^4+x^6+x^7+x^9$. Encode 110 .
Find a five-row generator matrix for the code in 14A.
Design an encoder of a binary convolutional $(3,2)$-code of memory $m=2$.
The encoder in Figure 4 in 14.2 uses five shift registers. Design an encoder of the same code using four shift register states only.
Describe the generator polynomials of the convolutional code in Figure 11.
Find a five-row generator matrix for the code in 14E. Encode 11001.
Design an encoder for the (2,3)-code with the generator polynomials $g_0(x)=1+x+x^3+x^5+x^6, g_1(x)=x+x^2+x^3+x^4$, and $g_2(x)=$ $1+x^2+x^3+x^5+x^6$.
Use the Viterbi algorithnm to decode $w=01000001000 \ldots$ provided that the code of Example 2 of 14.4 is used (and $b=7$ ).
Draw the trellis of the $(2,1)$-code with $g_0(x)=1+x^2+x^3+x^5$. Use the Viterbi algorithm to decode $\mathbf{w}=1000001000001000 \ldots$.
Generalize the concept of a trellis diagram to $(n, k)$-codes. Draw the trellis of the $(2,2)$-code in Figure 12.