---
title: "What is convolution; convolution sum"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:convolution-sum-what-is"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-sum-what-is
site: BCT Engineering
---

# What is convolution; convolution sum

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2078 Chaitra, 6 marks; 2073 Magh, 2+6 marks; 2072 Magh, 2+5 marks (asked 3 times)
- Video: [Tutorial: Convolution sum](https://www.youtube.com/watch?v=U3BwStzKvs0) — Rose-Hulman Online
- 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-sum-what-is
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

What is convolution? Obtain the expression for convolution sum.

## Short answer

Convolution gives the output of an LTI system as a weighted sum of input samples. The convolution sum is y[n] = ∑ x[k] h[n−k], where h[n] is the impulse response.

## Answer

#### Convolution

**Convolution** is the operation that gives the output of an LTI system from its input and its impulse response. For discrete time, the output is a weighted sum of the present and past (and possibly future) inputs, with the impulse response values as weights:

$$
y[n] = x[n] * h[n]
$$

Here $h[n]$ is the **impulse response**, the output when the input is the unit impulse $\delta[n]$.

#### Expression for the convolution sum

**Step 1: Represent the input with impulses.** Using the sifting property, any sequence is a sum of scaled, shifted impulses:

$$
x[n] = \sum_{k=-\infty}^{\infty} x[k]\,\delta[n-k]
$$

e.g. $\{\underset{\uparrow}{3}, 1\} = 3\delta[n] + \delta[n-1]$.

**Step 2: Response to one impulse.** By definition $\delta[n] \rightarrow h[n]$.

**Step 3: Time invariance.** $\delta[n-k] \rightarrow h[n-k]$.

**Step 4: Homogeneity.** $x[k]\,\delta[n-k] \rightarrow x[k]\,h[n-k]$ ($x[k]$ is a constant for each $k$).

**Step 5: Additivity.** The output for the sum of all these inputs is the sum of their outputs:

$$
y[n] = \sum_{k=-\infty}^{\infty} x[k]\,h[n-k]
$$

This is the **convolution sum**. With $m = n - k$ it can also be written $y[n] = \sum_{k} h[k]\,x[n-k]$ (commutative property).

For a causal system with a causal input: $y[n] = \sum_{k=0}^{n} x[k]\,h[n-k]$.

**Steps to evaluate:** fold $h[k] \rightarrow h[-k]$; shift to $h[n-k]$; multiply with $x[k]$; add all products; repeat for each $n$. If the lengths are $N_1$ and $N_2$, the output length is $N_1 + N_2 - 1$, starting at (start of $x$) + (start of $h$).

**Example:** $x[n] = \{\underset{\uparrow}{1}, 1\}$, $h[n] = \{\underset{\uparrow}{2}, 3\}$:
- $y[0] = 1\cdot2 = 2$
- $y[1] = 1\cdot3 + 1\cdot2 = 5$
- $y[2] = 1\cdot3 = 3$

So $y[n] = \{\underset{\uparrow}{2}, 5, 3\}$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the impulse response h[n]?
   A. The output when the input is the unit step
   B. The output when the input is delta[n]
   C. The input that makes output zero
   D. The derivative of the output
   Answer: B. h[n] is the response to delta[n]. (Key point: Impulse response)

2. Which equation is the convolution sum?
   A. y[n] = sum x[k] h[k]
   B. y[n] = sum x[n-k] h[n-k]
   C. y[n] = sum x[k] h[n-k]
   D. y[n] = sum x[k] + h[n-k]
   Answer: C. Each input sample x[k] weights a shifted copy h[n-k]. (Key point: Convolution sum)

3. Which property turns x[k] delta[n-k] into x[k] h[n-k]?
   A. Time invariance then homogeneity
   B. Additivity only
   C. Causality
   D. Stability
   Answer: A. delta[n-k] -> h[n-k] by time invariance; scaling by x[k] by homogeneity; additivity sums. (Key point: Steps of derivation)

4. For x = {1,1} and h = {2,3}, what is y[n]?
   A. {2, 3}
   B. {2, 5, 3}
   C. {3, 5, 2}
   D. {2, 6, 3}
   Answer: B. y[0] = 2, y[1] = 3 + 2 = 5, y[2] = 3. (Key point: Example)

## Related in Signals and Systems

- [Derive the convolution sum](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-sum-derivation) — Signals and Systems, chapter 6, asked 8×
- [Convolution sum; ideal LPF is non-causal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-sum-ideal-lpf-noncausal) — Signals and Systems, chapter 5
- [What is convolution; convolution integral](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-integral-what-is) — Signals and Systems, chapter 5
- [Convolution of δ-sums with h[n] = 2δ[n] + 2δ[n−1]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-delta-two-taps) — Signals and Systems, chapter 6
- [Convolve δ-sum with u[n] − u[n−1]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-delta-first-difference) — Signals and Systems, chapter 6

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- [X(z) from convolution integral: 1/(s²(s+1))](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aconvolution-integral-1-s2-s1) — Digital Control System, chapter 2
- [Convolution and Z-transform via convolution integral](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aconvolution-meaning-z-transform) — Digital Control System, chapter 2
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