1. **Problem Statement:** Find the convolution of the discrete sequences $X(n) = \{1, 2, -1, 3\}$ and $h(n) = \{2, -1, 1, 1\}$. 2. **Formula for Convolution:** The convolution $y(n)$ of two sequences $X(n)$ and $h(n)$ is given by $$y(n) = \sum_{k=-\infty}^{\infty} X(k) h(n-k)$$ Since $X(n)$ and $h(n)$ are finite sequences, the sum limits reduce accordingly. 3. **Step-by-step Calculation:** - Length of $X(n)$ is 4, length of $h(n)$ is 4, so $y(n)$ length is $4 + 4 - 1 = 7$. - Compute each $y(n)$ for $n=0$ to $6$: $$y(0) = X(0)h(0) = 1 \times 2 = 2$$ $$y(1) = X(0)h(1) + X(1)h(0) = 1 \times (-1) + 2 \times 2 = -1 + 4 = 3$$ $$y(2) = X(0)h(2) + X(1)h(1) + X(2)h(0) = 1 \times 1 + 2 \times (-1) + (-1) \times 2 = 1 - 2 - 2 = -3$$ $$y(3) = X(0)h(3) + X(1)h(2) + X(2)h(1) + X(3)h(0) = 1 \times 1 + 2 \times 1 + (-1) \times (-1) + 3 \times 2 = 1 + 2 + 1 + 6 = 10$$ $$y(4) = X(1)h(3) + X(2)h(2) + X(3)h(1) = 2 \times 1 + (-1) \times 1 + 3 \times (-1) = 2 - 1 - 3 = -2$$ $$y(5) = X(2)h(3) + X(3)h(2) = (-1) \times 1 + 3 \times 1 = -1 + 3 = 2$$ $$y(6) = X(3)h(3) = 3 \times 1 = 3$$ 4. **Final Answer:** $$y(n) = \{2, 3, -3, 10, -2, 2, 3\}$$ This is the convolution result of the two sequences.