--- title: "Output for a decaying stem-plot h[n]" subject: "Signals and Systems" chapter: 6 key: "semester_4/signals_and_systems:convolution-stem-plot-response" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-stem-plot-response site: BCT Engineering --- # Output for a decaying stem-plot h[n] Signals and Systems · Chapter 6: Discrete-Time Systems - Asked in: 2070 Bhadra, 6+2 marks - 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-stem-plot-response - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6 ## Question A discrete time LTI system has an impulse response as shown below: [Figure: stem plot of h[n] = 2, 1, 0.5, 0.25, 0.125 at n = 0, 1, 2, 3, 4 and 0 elsewhere (shown from n = −2 to 6)]. If the input to the given system is x[n] = {−0.25, 0.5, 1, −0.5, 0, 0.25} (↑ at 1, i.e. x[0] = 1), calculate and plot the output of the system. ## Short answer Convolve to get y[n] = { -0.5, 0.75, 2.375, 0.1875, 0.09375, 0.5625, 0.25, 0.0625, 0.0625, 0.03125 } for n=-2..7. Sum check: ∑y=3.875=(∑x)(∑h). ## Answer From the figure, $h[n]=\{2,\ 1,\ 0.5,\ 0.25,\ 0.125\}$ for $n=0..4$, i.e. $h[n]=2(0.5)^n$ for $0\le n\le4$. The input is $x[n]=\{-0.25,\ 0.5,\ 1,\ -0.5,\ 0,\ 0.25\}$ with $x[0]=1$, so $x[-2]=-0.25,\ x[-1]=0.5,\ x[0]=1,\ x[1]=-0.5,\ x[2]=0,\ x[3]=0.25$. The output $y[n]=\sum_k x[k]h[n-k]$ starts at $n=-2+0=-2$ and ends at $n=3+4=7$ (length $6+5-1=10$). #### Tabular method Each row is $x[k]\cdot h[n-k]$: | $k$, $x[k]$ | $n=-2$ | −1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---|---|---|---| | −2, −0.25 | −0.5 | −0.25 | −0.125 | −0.0625 | −0.03125 | | | | | | | −1, 0.5 | | 1 | 0.5 | 0.25 | 0.125 | 0.0625 | | | | | | 0, 1 | | | 2 | 1 | 0.5 | 0.25 | 0.125 | | | | | 1, −0.5 | | | | −1 | −0.5 | −0.25 | −0.125 | −0.0625 | | | | 2, 0 | | | | | 0 | 0 | 0 | 0 | 0 | | | 3, 0.25 | | | | | | 0.5 | 0.25 | 0.125 | 0.0625 | 0.03125 | | $y[n]$ | **−0.5** | **0.75** | **2.375** | **0.1875** | **0.09375** | **0.5625** | **0.25** | **0.0625** | **0.0625** | **0.03125** | #### Some values worked out $$ \begin{aligned} y[0]&=x[-2]h[2]+x[-1]h[1]+x[0]h[0]=-0.125+0.5+2=2.375\\ y[1]&=x[-2]h[3]+x[-1]h[2]+x[0]h[1]+x[1]h[0]=-0.0625+0.25+1-1=0.1875\\ y[3]&=x[-1]h[4]+x[0]h[3]+x[1]h[2]+x[2]h[1]+x[3]h[0]=0.0625+0.25-0.25+0+0.5=0.5625 \end{aligned} $$ Sum check: $\sum y=3.875=(\sum x)(\sum h)=1\times3.875$. **Answer:** | $n$ | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |---|---|---|---|---|---|---|---|---|---|---| | $y[n]$ | −0.5 | 0.75 | 2.375 | 0.1875 | 0.09375 | 0.5625 | 0.25 | 0.0625 | 0.0625 | 0.03125 | and $y[n]=0$ elsewhere. #### Plot of y[n] (Not to scale for the small values.) ```text y[n] 2.375 o 0.75 o | 0.5625 | | o 0.25 | | | o 0.1875 | | o | | 0.09375 | | | o | | 0.0625 | | | | | | o o 0.03125 | | | | | | | | o 0 +----+----+----+----+----+----+----+----+----+--> n -0.5 o -2 -1 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. What impulse response is read from the figure? A. h[n]=2(0.5)^n for 0<=n<=4, i.e. {2,1,0.5,0.25,0.125} B. h[n]=0.5^n for n>=0 C. h[n]={2,1,0.5,0.25,0.125} for n=-2..2 D. h[n]=2(0.5)^n for all n Answer: A. Values are 2 at n=0 and halve each step to n=4. (Key point: Reading h[n]) 2. x={-0.25,0.5,1,-0.5,0,0.25} with x[0]=1. Which indices does x occupy? A. n=0 to 5 B. n=-3 to 2 C. n=-2 to 3 D. n=-1 to 4 Answer: C. Two samples precede the 1, so the first is at n=-2. (Key point: Input indices) 3. What is the range and length of y? A. n=-2 to 6, 9 samples B. n=0 to 7, 8 samples C. n=-2 to 8, 11 samples D. n=-2 to 7, 10 samples Answer: D. Start -2+0, end 3+4=7, length 6+5-1=10. (Key point: Output range) 4. What is y[0]? A. 2 B. 2.375 C. 2.5 D. 0.75 Answer: B. y[0]=x[-2]h[2]+x[-1]h[1]+x[0]h[0]=-0.125+0.5+2=2.375. 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