--- title: "Convolve {−0.5, 1, −1, 0.5} and {1, 2, −1, 0, 2, −1}" subject: "Signals and Systems" chapter: 6 key: "semester_4/signals_and_systems:convolution-plot-six-sample" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-plot-six-sample site: BCT Engineering --- # Convolve {−0.5, 1, −1, 0.5} and {1, 2, −1, 0, 2, −1} Signals and Systems · Chapter 6: Discrete-Time Systems - Asked in: 2079 Jestha, 8 marks - Video: [Discrete Time Convolution (Tabular Method)](https://www.youtube.com/watch?v=C_kwjJLepLg) — Neso 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-plot-six-sample - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6 ## Question Find and plot the convolution sum of the signals x[n] = {−0.5, 1, −1, 0.5} and y[n] = {1, 2, −1, 0, 2, −1}. ## Short answer Using convolution sum (tabular method), the result is z[n] = {−0.5, 0, 1.5, −2.5, 1, 2, −3, 2, −0.5} for n from 0 to 8. ## Answer No origin is marked, so take the first sample of each at $n = 0$: $x[n] = \{\underset{\uparrow}{-0.5}, 1, -1, 0.5\}$ ($0 \le n \le 3$) and $y[n] = \{\underset{\uparrow}{1}, 2, -1, 0, 2, -1\}$ ($0 \le n \le 5$). Let the result be $z[n] = x[n]*y[n] = \sum_k x[k]\,y[n-k]$. Output starts at $n = 0$; length $4 + 6 - 1 = 9$, so $0 \le n \le 8$. #### Tabular method Each row is one sample of $x$ times $y$ shifted to that position (last row is the column sum $z[n]$): | Row | $n=0$ | $n=1$ | $n=2$ | $n=3$ | $n=4$ | $n=5$ | $n=6$ | $n=7$ | $n=8$ | |---|---|---|---|---|---|---|---|---|---| | $x[0]\,y[n]$ | -0.5 | -1 | 0.5 | 0 | -1 | 0.5 | | | | | $x[1]\,y[n-1]$ | | 1 | 2 | -1 | 0 | 2 | -1 | | | | $x[2]\,y[n-2]$ | | | -1 | -2 | 1 | 0 | -2 | 1 | | | $x[3]\,y[n-3]$ | | | | 0.5 | 1 | -0.5 | 0 | 1 | -0.5 | | $z[n]$ | **-0.5** | **0** | **1.5** | **-2.5** | **1** | **2** | **-3** | **2** | **-0.5** | #### Direct working - $z[0] = -0.5(1) = -0.5$ - $z[1] = -0.5(2) + 1(1) = 0$ - $z[2] = -0.5(-1) + 1(2) + (-1)(1) = 1.5$ - $z[3] = -0.5(0) + 1(-1) + (-1)(2) + 0.5(1) = -2.5$ - $z[4] = -0.5(2) + 1(0) + (-1)(-1) + 0.5(2) = 1$ - $z[5] = -0.5(-1) + 1(2) + (-1)(0) + 0.5(-1) = 2$ - $z[6] = 1(-1) + (-1)(2) + 0.5(0) = -3$ - $z[7] = (-1)(-1) + 0.5(2) = 2$ - $z[8] = 0.5(-1) = -0.5$ **Sum check:** $(\sum x)(\sum y) = 0 \times 3 = 0$, and $-0.5 + 0 + 1.5 - 2.5 + 1 + 2 - 3 + 2 - 0.5 = 0$. **Answer:** $$ z[n] = \{\underset{\uparrow}{-0.5},\ 0,\ 1.5,\ -2.5,\ 1,\ 2,\ -3,\ 2,\ -0.5\} $$ #### Plot Stem plot drawn sideways (each # = 0.5; left of | is negative): ```text n z[n] 0 -0.5 #| 1 0.0 | 2 1.5 |### 3 -2.5 #####| 4 1.0 |## 5 2.0 |#### 6 -3.0 ######| 7 2.0 |#### 8 -0.5 #| ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. x has 4 samples and y has 6 samples. The convolution z has: A. 10 samples B. 9 samples, n = 0 to 8 C. 24 samples D. 6 samples Answer: B. Length = 4 + 6 - 1 = 9. (Key point: Output length) 2. What is z[3]? A. -2.5 B. 2.5 C. -1.5 D. 0.5 Answer: A. z[3] = -0.5(0) + 1(-1) + (-1)(2) + 0.5(1) = -2.5. (Key point: Value z[3]) 3. sum x = -0.5 + 1 - 1 + 0.5 = 0. What must sum z be? A. 3 B. 0 C. -3 D. 1.5 Answer: B. sum z = (sum x)(sum y) = 0 x 3 = 0. (Key point: Sum check) 4. What is z[8]? A. -0.5 B. 0.5 C. -1 D. 2 Answer: A. z[8] = x[3]y[5] = 0.5 x (-1) = -0.5. 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