--- title: "Output for h[n] = {1, 2, 2, 5}, x[n] = {1, 3}" subject: "Signals and Systems" chapter: 6 key: "semester_4/signals_and_systems:convolution-1225" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-1225 site: BCT Engineering --- # Output for h[n] = {1, 2, 2, 5}, x[n] = {1, 3} Signals and Systems · Chapter 6: Discrete-Time Systems - Asked in: 2072 Magh, 8 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-1225 - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6 ## Question Consider a system with impulse response h[n] = {1, 2, 2, 5}. Determine the output y[n] for input x[n] = {1, 3} ## Short answer Since x[n]=delta[n]+3 delta[n-1], the output is y[n]=h[n]+3h[n-1]. For h={1,2,2,5} and x={1,3}, the convolution gives y={1,5,8,11,15} for n=0 to 4. ## Answer Take $h[0]$ and $x[0]$ as the first samples (no origin is marked): $h[n]=\{1,2,2,5\}$, $n=0..3$ and $x[n]=\{1,3\}$, $n=0,1$. The output $y[n]=x[n]*h[n]=\sum_k x[k]h[n-k]$ has $2+4-1=5$ samples, $n=0..4$. Because $x[n]=\delta[n]+3\delta[n-1]$, the output is $$ y[n]=h[n]+3h[n-1] $$ #### Tabular method | | $n=0$ | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | $h[n]$ | 1 | 2 | 2 | 5 | | | $3h[n-1]$ | | 3 | 6 | 6 | 15 | | $y[n]$ | **1** | **5** | **8** | **11** | **15** | #### Step values $$ \begin{aligned} y[0]&=1\\ y[1]&=2+3(1)=5\\ y[2]&=2+3(2)=8\\ y[3]&=5+3(2)=11\\ y[4]&=3(5)=15 \end{aligned} $$ Sum check: $\sum y=40=(\sum x)(\sum h)=4\times10$. **Answer: $y[n]=\{1,\ 5,\ 8,\ 11,\ 15\}$ for $n=0,\dots,4$** (↑ at 1). ```text y[n] 15 o 11 o | 8 o | | 5 o | | | 1 o | | | | 0 +---+---+---+---+--> n 0 1 2 3 4 ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. h has 4 samples and x has 2 samples. How many samples does y=x*h have? A. 5 B. 6 C. 8 D. 4 Answer: A. Length is 2+4-1=5. (Key point: Output length) 2. Since x[n]=delta[n]+3delta[n-1], what is y[n]? A. h[n]+3h[n+1] B. h[n]+3h[n-1] C. 3h[n]+h[n-1] D. h[n]*3h[n-1] Answer: B. Each impulse in x contributes a scaled and shifted copy of h. (Key point: Decomposition of x) 3. With h={1,2,2,5} and x={1,3}, what is y[1]? A. 2 B. 3 C. 5 D. 8 Answer: C. y[1]=h[1]+3h[0]=2+3=5. (Key point: Value of y[1]) 4. What is y[n] for n=0..4? A. {1, 5, 8, 11, 5} B. {1, 3, 8, 11, 15} C. {1, 5, 8, 10, 15} D. {1, 5, 8, 11, 15} Answer: D. y[0]=1, y[1]=5, y[2]=2+6=8, y[3]=5+6=11, y[4]=15. Sum 40 = 4 x 10. (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 - [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× - [System output using Fourier transforms](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aoutput-via-fourier-transform) — 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× - [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× - [Impulse response of y[n] − (3/4)y[n−1] + (1/8)y[n−2] = 2x[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asecond-order-difference-equation) — Signals and Systems, chapter 6, asked 2× ## Related in other subjects - [Matched filter impulse response is reverse delayed input](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-reverse-delayed) — Communication Systems, chapter 9, asked 2× - [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× - [Output for e^(jπn) input, h=(1/2)^n u[n]](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Alti-output-complex-exp-half) — Digital Signal Analysis and Processing, chapter 3 - [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