--- title: "LTI output for h(t)=e^(-2t), rectangular input" subject: "Data Communication" chapter: 2 key: "semester_5/data_communication:lti-output-rect-input" source: https://www.bctengineer.com/q/semester_5/data_communication/lti-output-rect-input site: BCT Engineering --- # LTI output for h(t)=e^(-2t), rectangular input Data Communication · Chapter 2: Signals and Systems - Asked in: 2074 Asoj, 8 marks - Video: [Signals and Systems 46 the output of a Continuous Time LTI System Example](https://www.youtube.com/watch?v=Dxk3nNUHWAE) — Himanshu Kaushik | DigiiMento GATE, NET, CSE Prep - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/data_communication/lti-output-rect-input - Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2 ## Question Find the output of the LTI system having impulse response h(t) = e^(-2t); t > 0 to the input x(t) = 0 for t<0, 1 for 0≤t≤1, 0 for 11. ## Answer Given h(t) = e^(-2t) for t > 0 and x(t) = 1 for 0 ≤ t ≤ 1, else 0. y(t) = x(t) * h(t) = ∫ x(τ) h(t - τ) dτ, with h(t - τ) = e^(-2(t - τ)) for τ < t. ```text x(t): 1 |‾‾‾‾‾| h(t): 1 |\ | | | \__ --+-----+---> t +-------> t 0 1 0 ``` #### Region 1: t < 0 No overlap, so y(t) = 0. #### Region 2: 0 ≤ t ≤ 1 The overlap is from τ = 0 to τ = t: y(t) = ∫₀^t e^(-2(t - τ)) dτ = e^(-2t) [e^(2τ)/2]₀^t = e^(-2t)(e^(2t) - 1)/2 **y(t) = (1 - e^(-2t)) / 2** #### Region 3: t > 1 The overlap is from τ = 0 to τ = 1: y(t) = e^(-2t) [e^(2τ)/2]₀^1 = e^(-2t)(e² - 1)/2 **y(t) = (e² - 1) e^(-2t) / 2** At t = 1 both give y(1) = (1 - e^(-2))/2 = 0.4323, so the output is continuous. **Answer:** ```text 0 t < 0 y(t) = (1 - e^(-2t))/2 0 <= t <= 1 (e^2 - 1) e^(-2t)/2 t > 1 ``` The output rises to 0.432 at t = 1 and then decays exponentially. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the output of an LTI system found from x(t) and h(t)? A. y(t) = x(t) h(t) B. y(t) = integral of x(tau) h(tau) dtau C. y(t) = dx/dt times h(t) D. y(t) = integral of x(tau) h(t-tau) dtau Answer: D. The output is the convolution of input and impulse response. (Key point: Convolution formula) 2. For 0 <= t <= 1, what are the limits of the convolution integral? A. tau = 0 to tau = t B. tau = 0 to tau = 1 C. tau = t to tau = 1 D. tau = -infinity to 0 Answer: A. x(tau) is nonzero from 0 and h(t-tau) needs tau < t, so overlap is 0 to t. (Key point: Overlap region) 3. What is y(t) for 0 <= t <= 1? A. (e^2 - 1)e^(-2t)/2 B. (1 - e^(-2t))/2 C. 1 - e^(-2t) D. e^(-2t)/2 Answer: B. Integrating e^(-2(t-tau)) from 0 to t gives (1 - e^(-2t))/2. (Key point: Result for 0<=t<=1) 4. For t > 1 the output is (e^2 - 1)e^(-2t)/2. How does it behave? A. It stays constant at 0.432 B. It keeps rising C. It decays exponentially after reaching 0.432 at t = 1 D. It becomes zero immediately at t = 1 Answer: C. The overlap is fixed at 0 to 1, so only e^(-2t) varies and the output decays. (Key point: Result for t>1) ## Close variants set in other exams - [Output of LTI with h(t) = e^(-at)u(t) and e^(at)u(-t)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aconvolution-exp-u-and-u-neg) — Data Communication, chapter 2, asked 3× ## Related in Data Communication - [Define LTI system and impulse response](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-impulse-response-define) — Data Communication, chapter 2, asked 2× - [LTI output for h(t)=e^(-at)u(t), x(t)=u(t)-u(t-2)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-output-u-minus-u2) — Data Communication, chapter 2 - [Check LTI: y(t) = 2x(t) + 3](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-check-2x-plus-3) — Data Communication, chapter 2 ## Related in other subjects - [Output of e^(−2t)u(t) to a rectangular input](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-exponential-rect-input) — Signals and Systems, chapter 5, asked 4× - [LTI output for 2δ(n) − δ(n−1) input](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alti-output-impulse-input) — Digital Signal Analysis and Processing, chapter 1 - [Parseval's theorem; output PSD of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aparseval-power-lti-psd) — Communication Systems, chapter 1 - [LTI output of a WSS input is WSS](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-wssp-output) — Communication Systems, chapter 9