--- title: "Convolve {3, 5, 6, 9} and {−1, 3, 5}" subject: "Signals and Systems" chapter: 6 key: "semester_4/signals_and_systems:convolution-3569" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-3569 site: BCT Engineering --- # Convolve {3, 5, 6, 9} and {−1, 3, 5} Signals and Systems · Chapter 6: Discrete-Time Systems - Asked in: 2081 Chaitra, 6 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-3569 - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6 ## Question Find convolution between the signals x[n] = {3, 5, 6, 9} and h[n] = {−1, 3, 5}. Choose your appropriate origin. Show the output signal. ## Short answer Tabular convolution (both origins at n=0) gives y[n]={-3,4,24,34,57,45} and y[0]=-3. ## Answer **Chosen origin:** first sample of each sequence at $n = 0$: $x[n] = \{\underset{\uparrow}{3}, 5, 6, 9\}$ ($0 \le n \le 3$) and $h[n] = \{\underset{\uparrow}{-1}, 3, 5\}$ ($0 \le n \le 2$). $$ y[n] = \sum_k x[k]\,h[n-k] $$ Output starts at $n = 0$, length $= 4 + 3 - 1 = 6$, so $0 \le n \le 5$. #### Tabular method | Row | $n=0$ | $n=1$ | $n=2$ | $n=3$ | $n=4$ | $n=5$ | |---|---|---|---|---|---|---| | $x[0]\,h[n]$ | -3 | 9 | 15 | | | | | $x[1]\,h[n-1]$ | | -5 | 15 | 25 | | | | $x[2]\,h[n-2]$ | | | -6 | 18 | 30 | | | $x[3]\,h[n-3]$ | | | | -9 | 27 | 45 | | $y[n]$ | **-3** | **4** | **24** | **34** | **57** | **45** | #### Direct working - $y[0] = 3(-1) = -3$ - $y[1] = 3(3) + 5(-1) = 4$ - $y[2] = 3(5) + 5(3) + 6(-1) = 24$ - $y[3] = 5(5) + 6(3) + 9(-1) = 34$ - $y[4] = 6(5) + 9(3) = 57$ - $y[5] = 9(5) = 45$ **Sum check:** $(\sum x)(\sum h) = 23 \times 7 = 161$ and $-3 + 4 + 24 + 34 + 57 + 45 = 161$. **Answer (output signal):** $$ y[n] = \{\underset{\uparrow}{-3},\ 4,\ 24,\ 34,\ 57,\ 45\} $$ Stem plot drawn sideways (each # = 3; left of | is negative): ```text n y[n] 0 -3 #| 1 4 |# 2 24 |######## 3 34 |########### 4 57 |################### 5 45 |############### ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. x = {3,5,6,9} (4 samples) and h = {-1,3,5} (3 samples), both starting at 0. The output is: A. n = 0 to 5 B. n = 0 to 6 C. n = 0 to 3 D. n = 1 to 6 Answer: A. Length = 4 + 3 - 1 = 6, so n = 0 to 5. (Key point: Output range) 2. Which gives y[2]? A. 3(5) + 5(3) + 6(-1) = 24 B. 3(-1) + 5(3) + 6(5) = 42 C. 3(5) + 5(3) = 30 D. 6(5) + 9(3) = 57 Answer: A. y[2] = x[0]h[2] + x[1]h[1] + x[2]h[0] = 15 + 15 - 6 = 24. (Key point: Value y[2]) 3. The sum of y must be: A. 30 B. 161 C. 23 + 7 = 30 D. 16 Answer: B. (3+5+6+9)(-1+3+5) = 23 x 7 = 161. (Key point: Sum check) 4. What is y[0]? A. 3 B. -3 C. 9 D. 0 Answer: B. y[0] = x[0]h[0] = 3 x (-1) = -3. (Key point: First sample) ## Related in Signals and Systems - [Convolve {1, 2, 2, 1} and {1, 2, 2, −1}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-1221) — Signals and Systems, chapter 6, asked 2× - [Convolve {2, 1, 5, 7} and {1, 3, 5, 9, 6}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-2157) — Signals and Systems, chapter 6, asked 2× - [Convolve {2, 0, −1, 2} and {1, 2, −1, 3}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-20-12) — Signals and Systems, chapter 6, asked 2× - [Convolve {1, 2, 3, 4} and {1, 2, −3}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-1234-12-3) — Signals and Systems, chapter 6, asked 2× - [Convolve {3, 1, 5} and {1, 4, −2, 3}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-315) — Signals and Systems, chapter 6 - [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× ## 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 - [System types; output as convolution](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-causal-convolution-proof) — Communication Systems, chapter 1 - [TTL input and output signal levels](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Attl-signal-levels) — Digital Logic, chapter 10, asked 2× - [Signal flow graph with G3 subtracted at output](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Asfg-g1g2g3-h-2069) — Control Systems, chapter 3