--- title: "Convolve {1, 2, 2, 3} and {1, 4, 3}" subject: "Signals and Systems" chapter: 6 key: "semester_4/signals_and_systems:convolution-1223-origin" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-1223-origin site: BCT Engineering --- # Convolve {1, 2, 2, 3} and {1, 4, 3} Signals and Systems · Chapter 6: Discrete-Time Systems - Asked in: 2071 Magh, 7 marks - Video: [Method to Find Discrete Convolution](https://www.youtube.com/watch?v=Nj8no2GMm8s) — TutorialsPoint - 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-1223-origin - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6 ## Question Perform the convolution and draw output y[n]: x[n] = {1, 2, 2, 3} (↑ at the first 2, i.e. x[0] = 2) and h[n] = {1, 4, 3} (↑ at 1, i.e. h[0] = 1). ## Short answer Here x[-1]=1, x[0]=2, x[1]=2, x[2]=3 and h={1,4,3} starting at n=0. Convolution by the tabular method gives y={1,6,13,17,18,9} for n=-1 to 4, so y[0]=6. ## Answer Given $x[n]=\{1,2,2,3\}$ with $x[0]$ = the first 2, so $x[-1]=1,\ x[0]=2,\ x[1]=2,\ x[2]=3$. $h[n]=\{1,4,3\}$ with $h[0]=1$, so $h[0]=1,\ h[1]=4,\ h[2]=3$. Output starts at $n=-1+0=-1$, ends at $n=2+2=4$; length $4+3-1=6$. $y[n]=\sum_k x[k]h[n-k]$. #### Tabular method | $k$, $x[k]$ | $n=-1$ | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---|---| | $-1$, 1 | 1 | 4 | 3 | | | | | $0$, 2 | | 2 | 8 | 6 | | | | $1$, 2 | | | 2 | 8 | 6 | | | $2$, 3 | | | | 3 | 12 | 9 | | $y[n]$ | **1** | **6** | **13** | **17** | **18** | **9** | #### Step values $$ \begin{aligned} y[-1]&=x[-1]h[0]=1\\ y[0]&=x[-1]h[1]+x[0]h[0]=4+2=6\\ y[1]&=x[-1]h[2]+x[0]h[1]+x[1]h[0]=3+8+2=13\\ y[2]&=x[0]h[2]+x[1]h[1]+x[2]h[0]=6+8+3=17\\ y[3]&=x[1]h[2]+x[2]h[1]=6+12=18\\ y[4]&=x[2]h[2]=9 \end{aligned} $$ Sum check: $\sum y=64=(\sum x)(\sum h)=8\times8$. **Answer: $y[n]=\{1,\ 6,\ 13,\ 17,\ 18,\ 9\}$ for $n=-1,\dots,4$, so $y[0]=6$.** #### Output plot ```text y[n] 18 o 17 o | 13 o | | 9 | | | o 6 o | | | | 1 o | | | | | 0 +---+---+---+---+---+--> n -1 0 1 2 3 4 ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. x={1,2,2,3} with x[0] being the first 2. Which indices does x occupy? A. n=-1 to 2 B. n=0 to 3 C. n=-2 to 1 D. n=1 to 4 Answer: A. One sample (the 1) lies before the origin, so x[-1]=1, x[0]=2, x[1]=2, x[2]=3. (Key point: Locating the origin) 2. With x on n=-1..2 and h on n=0..2, what is the range of y? A. n=0 to 5 B. n=-1 to 4 (6 samples) C. n=-1 to 5 D. n=-1 to 3 Answer: B. Start -1+0=-1, end 2+2=4, length 4+3-1=6. (Key point: Output range) 3. What is y[0]? A. 2 B. 4 C. 6 D. 8 Answer: C. y[0]=x[-1]h[1]+x[0]h[0]=4+2=6. (Key point: Value of y[0]) 4. What is y[n] for n=-1..4? A. {1, 6, 13, 17, 18, 8} B. {1, 6, 12, 17, 18, 9} C. {2, 6, 13, 17, 18, 9} D. {1, 6, 13, 17, 18, 9} Answer: D. Sum 1+6+13+17+18+9=64 = 8 x 8. (Key point: Final result) ## Related in Signals and Systems - [Convolve {1,2,1,−1} and {1,2,3,1}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-origin-given) — Signals and Systems, chapter 6, asked 2× - [Convolve {1, 1, 2, −1} and {1, 2}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-1121-sketch) — Signals and Systems, chapter 6 - [Convolve {1, 2, 1} and {1, 2, 3, 1}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-121-1231) — Signals and Systems, chapter 6 - [Convolve {1, 2, 0, −1} and {2, 0, 2}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-1-2-0-minus1) — Signals and Systems, chapter 6, asked 2× - [Convolve {3, 5, 6, 9} and {−1, 3, 5}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-3569) — Signals and Systems, chapter 6 - [Circular convolution of two sequences](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acircular-convolution) — Signals and Systems, chapter 3 ## Related in other subjects - [System types; output as convolution](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-causal-convolution-proof) — Communication Systems, chapter 1 - [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 - [G-S first iteration, 3-bus with PV bus](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Ags-first-iteration-2076-asoj) — Power System Analysis II, chapter 2 - [Output of printf statements](https://www.bctengineer.com/question.md?key=semester_1%2Fcomputer_programming%3Aprintf-output-a-b) — Computer Programming, chapter 4