Let's use the singularity functions \( u(t) \) (unit step) and \( r(t) \) (unit ramp), scaled and shifted as needed. ### a. \( x_a(t) \) This is a triangular pulse that starts at \( t = -1 \), peaks at \( t = 0 \), and ends at \( t = 1 \). - From \( t = -1 \) to \( t = 0 \), it ramps up: \( r(t+1) \) - From \( t = 0 \) to \( t = 1 \), it ramps down: \( -r(t) \) - After \( t = 1 \), subtract the ramp: \( r(t-1) \) Combining: \[ \boxed{x_a(t) = r(t+1) - 2r(t) + r(t-1)} \] --- ### b. \( x_b(t) \) This is a rectangular pulse from \( t = -1 \) to \( t = 1 \). - Turns on at \( t = -1 \): \( u(t+1) \) - Turns off at \( t = 1 \): \( u(t-1) \) Combining: \[ \boxed{x_b(t) = u(t+1) - u(t-1)} \] --- ### c. \( x_c(t) \) This is a ramp that starts at \( t = 0 \) and continues indefinitely. - \( r(t) \) is a ramp starting at \( t = 0 \). So: \[ \boxed{x_c(t) = r(t)} \] --- ### d. \( x_d(t) \) This is an "inverted V" centered at \( t = 0 \), with slopes 1 to the left and -1 to the right. - Left of \( t = 0 \), it is \( r(t+1) \). - Right of \( t = 0 \), it decreases: \( -r(t) \). Combine: \[ \boxed{x_d(t) = r(t+1) - 2r(t)} \] --- ### Summary Table | Part | Signal Expression | |------|-------------------| | a | \( r(t+1) - 2r(t) + r(t-1) \) | | b | \( u(t+1) - u(t-1) \) | | c | \( r(t) \) | | d | \( r(t+1) - 2r(t) \) | If you need the derivation for any part, feel free to ask!