--- title: "Convolve |n| and 2n sequences" subject: "Signals and Systems" chapter: 6 key: "semester_4/signals_and_systems:convolution-abs-n-ramp" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-abs-n-ramp site: BCT Engineering --- # Convolve |n| and 2n sequences Signals and Systems · Chapter 6: Discrete-Time Systems - Asked in: 2078 Chaitra, 8 marks - Video: [Discrete Time Convolution](https://www.youtube.com/watch?v=A6qR7UhLvng) — 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-abs-n-ramp - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6 ## Question Find and plot the convolution between x[n] = |n| for −2 ≤ n ≤ 2, 0 otherwise and y[n] = 2n for 2 ≤ n ≤ 5, 0 otherwise. ## Short answer Convolving x[n] = |n| (-2 ≤ n ≤ 2) and h[n] = 2n (2 ≤ n ≤ 5) gives c[n] = {8, 16, 22, 32, 24, 20, 26, 20} for n = 0 to 7. ## Answer Write both sequences with their sample positions. (The second signal is called $y[n]$ in the question; to avoid confusion it is renamed $h[n]$ and the result is $c[n]=x[n]*h[n]$.) $$ x[n]=|n|:\quad x[-2]=2,\ x[-1]=1,\ x[0]=0,\ x[1]=1,\ x[2]=2 $$ $$ h[n]=2n:\quad h[2]=4,\ h[3]=6,\ h[4]=8,\ h[5]=10 $$ The result starts at $n=-2+2=0$, ends at $n=2+5=7$, and has $5+4-1=8$ samples. #### Tabular method Rows are $x[k]\,h[n-k]$ for each $k$: | $k$, $x[k]$ | $n=0$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---|---| | $-2$, 2 | 8 | 12 | 16 | 20 | | | | | | $-1$, 1 | | 4 | 6 | 8 | 10 | | | | | $0$, 0 | | | 0 | 0 | 0 | 0 | | | | $1$, 1 | | | | 4 | 6 | 8 | 10 | | | $2$, 2 | | | | | 8 | 12 | 16 | 20 | | $c[n]$ | **8** | **16** | **22** | **32** | **24** | **20** | **26** | **20** | Example check: $c[3]=x[-2]h[5]+x[-1]h[4]+x[0]h[3]+x[1]h[2]=20+8+0+4=32$. Sum check: $\sum c = 168 = (\sum x)(\sum h)=6\times 28$. **Answer: $c[n]=\{8,16,22,32,24,20,26,20\}$ for $n=0,1,\dots,7$**, zero elsewhere. #### Plot ```text c[n] 32 o 26 | o 24 | o | 22 o | | | 20 | | | o | o 16 o | | | | | | 8 o | | | | | | | 0 +---+---+---+---+---+---+---+--> n 0 1 2 3 4 5 6 7 ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. x occupies n = -2 to 2 and h occupies n = 2 to 5. The output spans: A. n = 0 to 7 B. n = -2 to 5 C. n = 0 to 6 D. n = 2 to 7 Answer: A. Start = -2 + 2 = 0, end = 2 + 5 = 7, giving 8 samples. (Key point: Output range) 2. What is the row for k = 0 in the tabular method? A. All zeros, because x[0] = |0| = 0 B. Equal to h C. Negative of h D. Twice h Answer: A. x[0] = 0 so that row contributes nothing. (Key point: Zero sample) 3. c[3] = A. x[-2]h[5] + x[-1]h[4] + x[0]h[3] + x[1]h[2] = 32 B. x[-2]h[5] + x[-1]h[4] = 28 C. x[1]h[2] + x[2]h[1] = 4 D. 40 Answer: A. 20 + 8 + 0 + 4 = 32. (Key point: Value c[3]) 4. sum c must equal: A. 34 B. 168 C. 28 D. 6 Answer: B. (sum x)(sum h) = 6 x 28 = 168. (Key point: Sum check) ## Related in Signals and Systems - [Circular convolution of two sequences](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acircular-convolution) — Signals and Systems, chapter 3 - [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 {−0.5, 1, −1, 0.5} and {1, 2, −1, 0, 2, −1}](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-plot-six-sample) — Signals and Systems, chapter 6 - [Convolution of two rectangular pulses](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-two-rect-pulses) — Signals and Systems, chapter 5, asked 3× - [Convolution of e^(0.5t) window with a pulse](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-growing-exponential-window) — Signals and Systems, chapter 5, asked 3× ## Related in other subjects - [Linear convolution via circular convolution](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alinear-conv-via-circular) — Digital Signal Analysis and Processing, chapter 7 - [Convolution theorem and inverse Laplace](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aconvolution-theorem-and-inverse-laplace-four) — Mathematics III, chapter 3