--- title: "Convolution of two rectangular pulses" subject: "Signals and Systems" chapter: 5 key: "semester_4/signals_and_systems:convolution-two-rect-pulses" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-two-rect-pulses site: BCT Engineering --- # Convolution of two rectangular pulses Signals and Systems · Chapter 5: Continuous-Time Systems - Asked in: 2080 Asoj, 6 marks; 2079 Asoj, 8 marks (asked 2 times) - Video: [Convolution Example-Two Rectangular Pulses (Edited)](https://www.youtube.com/watch?v=4-FS5GN_vFE) — Darryl Morrell - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-two-rect-pulses - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5 ## Question Find the convolution between the signals x(t) = 1 for −1 < t < 1, 0 otherwise and h(t) = 1 for −1 < t < 1, 0 otherwise. ## Short answer Convolve two unit rectangular pulses each of width 2 and amplitude 1. The output is a triangular pulse y(t) = 2 - |t| for |t| < 2, zero otherwise. Height 2, base from -2 to 2. ## Answer $y(t) = x(t) * h(t) = \displaystyle\int_{-\infty}^{\infty} x(\tau)\,h(t-\tau)\,d\tau$, with $x(\tau) = 1$ for $-1 < \tau < 1$, and $h(t - \tau) = 1$ for $-1 < t - \tau < 1$, i.e. $t - 1 < \tau < t + 1$. The result is non-zero only where the two pulses overlap. Each has width 2, so the output extends from $t = -2$ to $t = 2$. ```text x(tau) h(t-tau) _________ _________ | | | | ---+----+----+-- --+----+----+--> tau -1 0 1 t-1 t t+1 ``` #### Case 1: $t < -2$ $t + 1 < -1$: no overlap, $y(t) = 0$. #### Case 2: $-2 \le t < 0$ (partial overlap from left) Overlap from $\tau = -1$ to $\tau = t + 1$: $$ y(t) = \int_{-1}^{t+1} 1\,d\tau = t + 2 $$ #### Case 3: $0 \le t < 2$ (partial overlap from right) Overlap from $\tau = t - 1$ to $\tau = 1$: $$ y(t) = \int_{t-1}^{1} 1\,d\tau = 2 - t $$ #### Case 4: $t \ge 2$ $t - 1 > 1$: no overlap, $y(t) = 0$. #### Result $$ y(t) = \begin{cases} t + 2, & -2 \le t < 0 \\ 2 - t, & 0 \le t < 2 \\ 0, & |t| \ge 2 \end{cases} \;=\; (2 - |t|)\ \text{for } |t| < 2 $$ ```text y(t) 2 | /\ | / \ 1 | / \ | / \ ---+--/---+----\---> t -2 0 2 ``` **Answer: a triangular pulse of height 2 at $t = 0$, base from $-2$ to $2$.** Convolving two equal rectangles always gives a triangle whose width is the sum of the widths (2 + 2 = 4) and whose peak equals the area of overlap (2). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Two rectangles of width 2 are convolved. The output is non-zero for: A. -2 < t < 2 B. -1 < t < 1 C. 0 < t < 2 D. -4 < t < 4 Answer: A. Output width is the sum of widths (4), centered at 0. (Key point: Extent of output) 2. For -2 <= t < 0, y(t) is: A. 2 - t B. t + 2 C. 2 D. t - 2 Answer: B. Overlap from τ = -1 to t+1 has length t + 2. (Key point: Case t between -2 and 0) 3. The final y(t) is: A. A rectangle of height 2 B. A triangle of height 1 C. A triangle of height 2 and base from -2 to 2 D. A triangle of height 2 and base from -1 to 1 Answer: C. y(t) = 2 - |t| for |t| < 2. (Key point: Shape) 4. The peak at t = 0 equals: A. The full overlap area, 2 B. 1 C. 4 D. 0 Answer: A. Both pulses fully overlap over length 2 with height 1 each. (Key point: Peak) ## Close variants set in other exams - [Convolution of e^(0.5t) window with a pulse](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-growing-exponential-window) — Signals and Systems, chapter 5, asked 3× ## Related in Signals and Systems - [FS of unit-amplitude rectangular pulses](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Arectangular-pulse-train-fs) — Signals and Systems, chapter 2 - [Circular convolution of two sequences](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acircular-convolution) — Signals and Systems, chapter 3 - [Inverse FT of a rectangular spectrum](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Ainverse-ft-rectangular-spectrum) — Signals and Systems, chapter 3 - [FT of periodic signals; pulse example](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-periodic-pulse-example) — Signals and Systems, chapter 3, asked 2× - [CTFT convolution property](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-convolution-property) — Signals and Systems, chapter 3, asked 12× - [DTFT convolution property](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-convolution-property) — Signals and Systems, chapter 3, asked 12× ## Related in other subjects - [Periodic and non-periodic signals; LTI output is convolution](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aperiodic-nonperiodic-convolution) — Communication Systems, chapter 1 - [Moment distribution: rectangular frame, two horizontal loads](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Amoment-dist-rectangular-frame-25kn-50kn) — Theory of Structures II, chapter 3 - [Graphical auto-correlation, cross-correlation and convolution](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Agraphical-correlation-convolution) — Communication Systems, chapter 1 - [Convolution theorem and inverse Laplace](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aconvolution-theorem-and-inverse-laplace-four) — Mathematics III, chapter 3