--- title: "Convolution of e^(0.5t) window with a pulse" subject: "Signals and Systems" chapter: 5 key: "semester_4/signals_and_systems:convolution-growing-exponential-window" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-growing-exponential-window site: BCT Engineering --- # Convolution of e^(0.5t) window with a pulse Signals and Systems · Chapter 5: Continuous-Time Systems - Asked in: 2071 Magh, 7 marks - 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-growing-exponential-window - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5 ## Question Find convolution between two signals x(t) = e^(0.5t) for 0 < t < 5, 0 otherwise and h(t) = 1 for 1 < t < 3, 0 otherwise. ## Short answer Sliding the rectangular window h across x(t)=e^(0.5t) for 08. ## Answer Use the convolution integral, sliding the pulse $h$ across the fixed input $x$: $$ y(t) = \int_{-\infty}^{\infty} x(\tau)\,h(t-\tau)\,d\tau $$ **Given:** - $x(\tau) = e^{0.5\tau}$ for $0 < \tau < 5$. - $h(t-\tau) = 1$ when $1 < t-\tau < 3$, i.e. $t-3 < \tau < t-1$. So the shifted window is $(t-3,\ t-1)$, of width 2, and it must overlap $(0,\ 5)$. ```text x(tau): |0=========5| window: (t-3 ----- t-1) slides right as t increases ``` **Useful integral:** $\int_{a}^{b} e^{0.5\tau}d\tau = 2\left(e^{0.5b} - e^{0.5a}\right)$. **Case 1: $t < 1$** ($t-1 < 0$): no overlap, $y(t) = 0$. **Case 2: $1 < t < 3$** (partial overlap, window enters): limits $0$ to $t-1$. $$ y(t) = \int_0^{t-1} e^{0.5\tau}d\tau = 2\left(e^{0.5(t-1)} - 1\right) $$ **Case 3: $3 < t < 6$** (window fully inside): limits $t-3$ to $t-1$. $$ \begin{aligned} y(t) &= 2\left(e^{0.5(t-1)} - e^{0.5(t-3)}\right) \\ &= 2\left(1 - e^{-1}\right)e^{0.5(t-1)} \approx 1.2642\,e^{0.5(t-1)} \end{aligned} $$ **Case 4: $6 < t < 8$** (window leaving): limits $t-3$ to $5$. $$ y(t) = 2\left(e^{2.5} - e^{0.5(t-3)}\right) $$ **Case 5: $t > 8$** ($t-3 > 5$): no overlap, $y(t) = 0$. **Answer:** $$ y(t) = \begin{cases} 0, & t < 1 \\ 2\left(e^{0.5(t-1)} - 1\right), & 1 \le t < 3 \\ 2\left(e^{0.5(t-1)} - e^{0.5(t-3)}\right), & 3 \le t < 6 \\ 2\left(e^{2.5} - e^{0.5(t-3)}\right), & 6 \le t < 8 \\ 0, & t \ge 8 \end{cases} $$ **Key values (continuity check):** | $t$ | $y(t)$ | |---|---| | 1 | 0 | | 3 | $2(e - 1) = 3.437$ | | 6 | $2(e^{2.5} - e^{1.5}) = 15.402$ | | 8 | 0 | The pieces meet at $t = 3$ and $t = 6$. The output rises from 0 at $t=1$, peaks at about 15.40 at $t = 6$, and falls to 0 at $t = 8$. The total duration is $5 + 2 = 7$ (from $0+1$ to $5+3$), as expected for convolution. ```text y(t) 15.4| * | .-' \ | .--' \ 3.4| .--* \ | .-' \ 0 +--*-----+-------------*--> t 1 3 6 8 ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For h(t - tau) = 1 when 1 < t - tau < 3, over what range of tau is it non-zero? A. t + 1 < tau < t + 3 B. 1 < tau < 3 C. t - 3 < tau < t - 1 D. t - 1 < tau < t + 3 Answer: C. 1 < t - tau < 3 gives t - 3 < tau < t - 1, a window of width 2 sliding right. (Key point: Window limits) 2. For 1 < t < 3 the window partly overlaps x(tau) on (0, 5). What is y(t)? A. 2(e^(0.5(t-1)) - 1) B. 2(e^(0.5t) - 1) C. e^(0.5(t-1)) - 1 D. 2(e^(0.5(t-3)) - 1) Answer: A. The limits are 0 to t-1, and integral of e^(0.5 tau) = 2(e^(0.5(t-1)) - 1). (Key point: Case 2 result) 3. Over which range of t does the window lie fully inside (0, 5), and what are the limits? A. 1 < t < 3, limits 0 to t-1 B. 3 < t < 6, limits t-3 to t-1 C. 6 < t < 8, limits t-3 to 5 D. 3 < t < 8, limits 0 to 5 Answer: B. Need t-3 >= 0 and t-1 <= 5, i.e. 3 <= t <= 6, with limits t-3 to t-1. (Key point: Fully inside range) 4. Where is y(t) non-zero, and what is its peak value (at t = 6)? A. From t = 1 to 8, peak about 15.40 B. From t = 0 to 5, peak about 3.44 C. From t = 1 to 8, peak about 3.44 D. From t = 0 to 8, peak about 15.40 Answer: A. Duration is 5 + 2 = 7, from 0+1 to 5+3, and y(6) = 2(e^2.5 - e^1.5) = 15.40. (Key point: Output duration) ## Close variants set in other exams - [Convolution of two rectangular pulses](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-two-rect-pulses) — Signals and Systems, chapter 5, asked 3× ## Related in Signals and Systems - [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× - [FT of periodic signals; rect pulse and constant](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-periodic-rect-constant) — Signals and Systems, chapter 3 - [Circular convolution of two sequences](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acircular-convolution) — Signals and Systems, chapter 3 - [Energy or power: e^(−5t)u(t+2)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-shifted-exponential) — Signals and Systems, chapter 1, asked 5× - [Energy and power of 2e^(−5t)u(t−1)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-2e-5t) — Signals and Systems, chapter 1 - [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× ## 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 - [Cohen-Sutherland: two lines vs window A-D](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acohen-sutherland-two-lines-window-a) — Computer Graphics, 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 - [Pulse TF for two sampler configurations](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apulse-tf-sampler-configurations) — Digital Control System, chapter 3, asked 2×