--- title: "Convolve {1, 1, 2, −1} and {1, 2}" subject: "Signals and Systems" chapter: 6 key: "semester_4/signals_and_systems:convolution-1121-sketch" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-1121-sketch site: BCT Engineering --- # Convolve {1, 1, 2, −1} and {1, 2} Signals and Systems · Chapter 6: Discrete-Time Systems - Asked in: 2072 Asoj, 6 marks - Video: [#110 Solved problems -2 on Convolution Sum || EC Academy](https://www.youtube.com/watch?v=hl2T2adZWFI) — EC Academy - 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-1121-sketch - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6 ## Question The impulse response of a discrete time LTI system is h[n] = {1, 1, 2, −1} (↑ at the first 1, i.e. h[0] = 1). If input is x[n] = {1, 2} (↑ at 1, i.e. x[0] = 1), find the convolution sum and sketch the output y[n]. ## Short answer With x[n]=delta[n]+2 delta[n-1], the output is y[n]=h[n]+2h[n-1]. For h={1,1,2,-1} and x={1,2}, convolution gives y={1,3,4,3,-2} for n=0 to 4, with y[0]=1. ## Answer Given $h[n]=\{1,1,2,-1\}$ with $h[0]=1$ (first), so $n=0..3$, and $x[n]=\{1,2\}$ with $x[0]=1$, so $n=0,1$. Since $x[n]=\delta[n]+2\delta[n-1]$, $$ y[n]=x[n]*h[n]=h[n]+2h[n-1] $$ Output length $=2+4-1=5$, for $n=0$ to $4$. #### Tabular method | | $n=0$ | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | $h[n]$ | 1 | 1 | 2 | −1 | | | $2h[n-1]$ | | 2 | 2 | 4 | −2 | | $y[n]$ | **1** | **3** | **4** | **3** | **−2** | $$ \begin{aligned} y[0]&=1,\quad y[1]=1+2(1)=3,\quad y[2]=2+2(1)=4\\ y[3]&=-1+2(2)=3,\quad y[4]=2(-1)=-2 \end{aligned} $$ Sum check: $\sum y=9=(\sum x)(\sum h)=3\times3$. **Answer: $y[n]=\{1,\ 3,\ 4,\ 3,\ -2\}$ for $n=0,\dots,4$, with $y[0]=1$.** #### Sketch of y[n] ```text y[n] 4 o 3 o | o 1 o | | | 0 +---+---+---+---+--> n -2 o 0 1 2 3 4 ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. With x={1,2} and x[0]=1, y[n] equals what in terms of h? A. h[n]+2h[n-1] B. 2h[n]+h[n-1] C. h[n]+2h[n+1] D. h[n]-2h[n-1] Answer: A. x[n]=delta[n]+2delta[n-1]. (Key point: Decomposition) 2. h={1,1,2,-1}, x={1,2}, both starting at 0. What is the range of y? A. n=0 to 3 (4 samples) B. n=0 to 4 (5 samples) C. n=0 to 5 (6 samples) D. n=-1 to 3 (5 samples) Answer: B. Length is 2+4-1=5. (Key point: Length and range) 3. What is y[3]? A. -1 B. 4 C. 3 D. 1 Answer: C. y[3]=h[3]+2h[2]=-1+4=3. (Key point: A middle value) 4. What is y[n] for n=0..4? A. {1, 3, 4, 3, 2} B. {1, 2, 4, 3, -2} C. {1, 3, 4, 5, -2} D. {1, 3, 4, 3, -2} Answer: D. Adding rows h and 2h shifted: 1, 1+2, 2+2, -1+4, -2. Sum 9 = 3 x 3. (Key point: Final result) ## Related in Signals and Systems - [First-order system impulse and step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-system-response) — Signals and Systems, chapter 5 - [Derive the convolution sum](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-sum-derivation) — Signals and Systems, chapter 6, asked 8× - [Frequency response of a CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-frequency-response) — Signals and Systems, chapter 5, asked 3× - [Frequency response; ideal LPF impulse response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-cutoff-impulse) — Signals and Systems, chapter 5, asked 2× - [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× - [Impulse response; ideal LPF formula](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aimpulse-response-ideal-lpf) — Signals and Systems, chapter 5, asked 2× ## Related in other subjects - [Random process; LTI output is WSS if input is WSS](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arandom-process-lti-wssp) — Communication Systems, chapter 9 - [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 - [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× - [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