1. **Problem Statement:** We have a rate-\(\frac{1}{2}\) convolutional code with constraint length 4. The first generator polynomial \(h_1\) is given by 10000 and the second generator polynomial \(h_2\) is 01011. The input sequence is \(\{1 \quad 0 \quad 1 \quad 1 \quad 0 \quad 1 \quad 0 \quad 1 \quad 1 \quad 0\}\). 2. **Understanding the Code:** - Rate \(\frac{1}{2}\) means for each input bit, two output bits are generated. - Constraint length 4 means the encoder has 4 memory elements (including the current input bit). - The generator polynomials define how the input bits and memory bits are combined to produce output bits. 3. **Generator Polynomials:** - \(h_1 = 10000\) means the first output bit depends only on the current input bit (the first '1') and no memory bits. - \(h_2 = 01011\) means the second output bit depends on the second, fourth, and fifth bits in the shift register. 4. **Shift Register Setup:** We consider the input bit and 3 previous bits (total 4 bits) for convolution. Since \(h_1\) and \(h_2\) are 5 bits long, we assume the first bit corresponds to the current input, and the next 4 bits correspond to memory bits. 5. **Input Sequence with Zero Padding:** To handle initial memory, pad input with 4 zeros at the start: \[ \{0,0,0,0,1,0,1,1,0,1,0,1,1,0\} \] 6. **Calculate Output Bits:** For each input bit starting from index 4 (the first actual input), calculate two output bits: - \(y_1[n] = \sum_{i=0}^4 h_1[i] \cdot x[n-i] \mod 2\) - \(y_2[n] = \sum_{i=0}^4 h_2[i] \cdot x[n-i] \mod 2\) Since \(h_1 = 1 0 0 0 0\), only \(x[n]\) matters for \(y_1[n]\). For \(h_2 = 0 1 0 1 1\), \(y_2[n] = x[n-1] + x[n-3] + x[n-4] \mod 2\). 7. **Step-by-step Calculation:** Indexing \(n\) from 4 to 13 (10 input bits): | n | x[n] | y_1[n] = x[n] | y_2[n] = x[n-1]+x[n-3]+x[n-4] mod 2 | |---|-------|--------------|------------------------------------| | 4 | 1 | 1 | 0+0+0=0 | | 5 | 0 | 0 | 1+0+0=1 | | 6 | 1 | 1 | 0+1+0=1 | | 7 | 1 | 1 | 1+0+1=0 | | 8 | 0 | 0 | 1+1+1=1 | | 9 | 1 | 1 | 0+1+0=1 | |10 | 0 | 0 | 1+0+1=0 | |11 | 1 | 1 | 0+1+0=1 | |12 | 1 | 1 | 1+0+1=0 | |13 | 0 | 0 | 1+1+0=0 | 8. **Output Sequence:** Concatenate \(y_1[n]\) and \(y_2[n]\) for each \(n\): \[ \{(1,0), (0,1), (1,1), (1,0), (0,1), (1,1), (0,0), (1,1), (1,0), (0,0)\} \] Or as a single sequence: \[ 1,0,0,1,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,0 \] **Final answer:** The output sequence is \(\{1,0,0,1,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,0\}\).