--- title: "Unit step signal and relation to impulse" subject: "Digital Signal Analysis and Processing" chapter: 1 key: "semester_7/digital_signal_analysis_and_processing:unit-step-relation-impulse" source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/unit-step-relation-impulse site: BCT Engineering --- # Unit step signal and relation to impulse Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems - Asked in: 2073 Chaitra, 1+2 marks - Video: [Relation between Unit Impulse and Unit Step Signals](https://www.youtube.com/watch?v=SWt2PYiGgKQ) — Neso Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/unit-step-relation-impulse - Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1 ## Question Define and plot a discrete time unit step signal. Explain its relation with unit impulse signal. ## Short answer Unit step u[n]=1 for n≥0, 0 otherwise. Unit impulse δ[n]=1 at n=0, 0 elsewhere. Relation: δ[n]=u[n]−u[n−1] (first difference); u[n]=Σ_{k=-∞}^{n} δ[k] (running sum). ## Answer #### Unit step signal The discrete-time unit step is u[n] = 1 for n ≥ 0, and 0 for n < 0. ```text u[n] 1 | o o o o o ... | | | | | | 0 +--o--o--o--+--+--+--+--+--- n -3 -2 -1 0 1 2 3 4 ``` (The samples are 0 for n < 0 and 1 for n ≥ 0; the step starts at n = 0.) #### Unit impulse δ[n] = 1 for n = 0, and 0 otherwise. #### Relation between u[n] and δ[n] 1. **Impulse as the first difference of the step:** δ[n] = u[n] − u[n − 1] For n = 0: 1 − 0 = 1; for n ≥ 1: 1 − 1 = 0; for n < 0: 0 − 0 = 0. 2. **Step as the running sum of the impulse:** u[n] = Σ(k = −∞ to n) δ[k] = Σ(m = 0 to ∞) δ[n − m] The sum is 0 until n reaches 0 and 1 from then on. These are the discrete counterparts of the continuous relations δ(t) = du(t)/dt and u(t) = ∫δ(τ)dτ: the difference replaces the derivative and the sum replaces the integral. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The discrete unit step u[n] equals A. 1 for n > 0, 0 otherwise B. 1 only at n = 0 C. 1 for n >= 0, 0 for n < 0 D. n for n >= 0 Answer: C. u[n] starts at n = 0 and stays at 1 afterwards. (Key point: Unit step definition) 2. Which expresses delta[n] using the step? A. u[n] + u[n-1] B. u[n] - u[n-1] C. u[n] u[n-1] D. u[n+1] + u[n] Answer: B. The first difference u[n] - u[n-1] is 1 only at n = 0. (Key point: Impulse from step) 3. The step as a running sum of the impulse is A. u[n] = sum of delta[k] from k = -infinity to n B. u[n] = delta[n] - delta[n-1] C. u[n] = n delta[n] D. u[n] = sum of delta[k] from k = 0 to n-1 Answer: A. The accumulated sum is 0 until n reaches 0 and 1 from then on. (Key point: Step from impulse) 4. The discrete relations correspond to the continuous relation A. delta(t) = integral of u B. u(t) = d delta/dt C. delta(t) = u(t)^2 D. delta(t) = du/dt, with difference for derivative and sum for integral Answer: D. Difference replaces derivative and summation replaces integration. 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