Let's solve **3b** step by step using the **convolution integral**. --- ## **Step 1: Write Given Functions** **Given:** - \( x(t) = -\mu(t) + \mu(t-1) \) - \( h(t) = 2\mu(t) - 2\mu(t-1) \) Where \( \mu(t) \) is the **unit step function**. --- ## **Step 2: Interpret the Functions** Let's rewrite these in a more intuitive form: ### **For \( x(t) \):** \[ x(t) = \begin{cases} & t < \\ -1 & \leq t < 1 \\ & t \geq 1 \end{cases} \] This is a pulse from \( t= \) to \( t=1 \), with value \(-1\). ### **For \( h(t) \):** \[ h(t) = \begin{cases} & t < \\ 2 & \leq t < 1 \\ & t \geq 1 \end{cases} \] This is a pulse from \( t= \) to \( t=1 \), with value \(2\). --- ## **Step 3: Write the Convolution Integral** \[ y(t) = (x * h)(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau \] --- ## **Step 4: Find the Limits of Integration** Both \( x(\tau) \) and \( h(t-\tau) \) are only nonzero for \( \leq \tau < 1 \) and \( \leq t-\tau < 1 \): - For \( x(\tau) \neq \), \( \leq \tau < 1 \) - For \( h(t-\tau) \neq \), \( \leq t-\tau < 1 \implies t-1 < \tau \leq t \) So, the effective limits are: \[ \tau \in [\max(, t-1), \min(1, t)] \] and \( x(\tau) = -1 \), \( h(t-\tau) = 2 \) in this interval. --- ## **Step 5: Write the Integral** \[ y(t) = \int_{\max(, t-1)}^{\min(1, t)} (-1) \cdot 2 \; d\tau = -2 \int_{\max(, t-1)}^{\min(1, t)} d\tau \] \[ y(t) = -2 \left[ \min(1, t) - \max(, t-1) \right] \] --- ## **Step 6: Piecewise Solution** Let's break it into cases for \( t \): ### **Case 1: \( t < \)** - The interval is empty. - \( y(t) = \) ### **Case 2: \( \leq t < 1 \)** - \( \max(, t-1) = \) - \( \min(1, t) = t \) - \( y(t) = -2 [ t - ] = -2t \) ### **Case 3: \( 1 \leq t < 2 \)** - \( \max(, t-1) = t-1 \) - \( \min(1, t) = 1 \) - \( y(t) = -2 [ 1 - (t-1) ] = -2 [2 - t] = -4 + 2t \) ### **Case 4: \( t \geq 2 \)** - The interval is empty. - \( y(t) = \) --- ## **Step 7: Final Answer** \[ \boxed{ y(t) = \begin{cases} & t < \\ -2t & \leq t < 1 \\ -4 + 2t & 1 \leq t < 2 \\ & t \geq 2 \end{cases} } \] --- ### **Summary Table** | \( t \) range | \( y(t) \) | |---------------------|--------------------| | \( t < \) | | | \( \leq t < 1 \) | \(-2t\) | | \( 1 \leq t < 2 \) | \(-4 + 2t\) | | \( t \geq 2 \) | | --- **This is the convolution result!**