Question

Two sequences \(x_1[n]\) and \(x_2[n]\) are described as follows: \[ x_1[0] = x_2[0] = 1, x_1[1] = x_2[2] = 2, x_1[2] = x_2[1] = 1 \] \[ x_1[n] = x_2[n] = 0 \text{for all } n<0 \text{ and } n>2 \] If \(x[n]\) is obtained by convolving \(x_1[n]\) with \(x_2[n]\), which of the following equations is/are TRUE?

• A. \(x[2] = x[3]\)
• B. \(x[1] = 2\)
• C. \(x[4] = 3\)
• D. \(x[2] = 5\)

Hint:

- Always remember: The convolution length for two finite sequences of length \(L_1\) and \(L_2\) is \(L_1 + L_2 - 1\). - Carefully align terms when performing the summation to avoid missing products. - Symmetry in sequences often leads to equal values in convolution outputs.

Answer 1

The Correct Option is A, D

Step 1: Recall convolution definition. \[ x[n] = (x_1 * x_2)[n] = \sum_{k=-\infty}^{\infty} x_1[k] \, x_2[n-k] \] Since both \(x_1[n]\) and \(x_2[n]\) are nonzero only for \(0 \leq n \leq 2\), the support of \(x[n]\) is \(0 \leq n \leq 4\). Step 2: Compute values of \(x[n]\). \underline{For \(n=0\):} \[ x[0] = x_1[0]x_2[0] = 1 \cdot 1 = 1 \] \underline{For \(n=1\):} \[ x[1] = x_1[0]x_2[1] + x_1[1]x_2[0] = (1)(1) + (2)(1) = 1+2=3 \] \underline{For \(n=2\):} \[ x[2] = x_1[0]x_2[2] + x_1[1]x_2[1] + x_1[2]x_2[0] = (1)(2) + (2)(1) + (1)(1) = 2+2+1 = 5 \] \underline{For \(n=3\):} \[ x[3] = x_1[1]x_2[2] + x_1[2]x_2[1] = (2)(2) + (1)(1) = 4+1=5 \] \underline{For \(n=4\):} \[ x[4] = x_1[2]x_2[2] = (1)(2)=2 \] Thus, the sequence is: \[ x[0]=1,\; x[1]=3,\; x[2]=5,\; x[3]=5,\; x[4]=2 \] Step 3: Verify given statements.
- (A) \(x[2]=x[3] \Rightarrow 5=5\), True
- (B) \(x[1]=2\). But actually \(x[1]=3\). False
- (C) \(x[4]=3\). But actually \(x[4]=2\). False
- (D) \(x[2]=5\). Correct. True
So the correct ones are (A) and (D). Final Answer: (A) and (D)