---
title: "Commutative, associative, distributive properties"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:lti-convolution-properties"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/lti-convolution-properties
site: BCT Engineering
---

# Commutative, associative, distributive properties

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2072 Asoj, 2+6 marks
- Video: [| LTI System Properties - Commutative - Associative-Distributive |](https://www.youtube.com/watch?v=mOhwe2VPSAo) — Dr.Santosh Tondare Engineering Tutorials
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/lti-convolution-properties
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

What is LTI system? Explain the commutative, associative and distributive properties of discrete time LTI systems.

## Short answer

An LTI system is linear and time-invariant, described by y=x*h. Convolution is commutative (x*h=h*x), associative ((x*h1)*h2=x*(h1*h2), so cascades combine as h1*h2), and distributive (x*(h1+h2)=x*h1+x*h2, so parallel systems add as h1+h2).

## Answer

#### LTI system
A **linear time-invariant (LTI)** system obeys superposition (linearity) and its behaviour does not change with time (a shifted input gives an equally shifted output). It is fully described by its impulse response $h[n]$, and

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

#### 1. Commutative property
$$
x[n]*h[n]=h[n]*x[n]
$$

Proof: in $\sum_k x[k]h[n-k]$ put $m=n-k$: it becomes $\sum_m h[m]x[n-m]=h[n]*x[n]$.
Meaning: the roles of input and impulse response can be swapped without changing the output.

#### 2. Associative property
$$
\{x[n]*h_1[n]\}*h_2[n]=x[n]*\{h_1[n]*h_2[n]\}
$$

Meaning: two LTI systems in **cascade** are equivalent to a single system with $h[n]=h_1[n]*h_2[n]$. With the commutative property, the order of the cascade can also be changed.

```text
 x -->[h1]-->[h2]--> y
   ==  x -->[h1*h2]--> y
   ==  x -->[h2]-->[h1]--> y
```

#### 3. Distributive property
$$
x[n]*\{h_1[n]+h_2[n]\}=x[n]*h_1[n]+x[n]*h_2[n]
$$

Proof: $\sum_k x[k](h_1[n-k]+h_2[n-k])=\sum_k x[k]h_1[n-k]+\sum_k x[k]h_2[n-k]$.
Meaning: two LTI systems in **parallel** (same input, outputs added) are equivalent to one system with $h=h_1+h_2$.

```text
       +-->[h1]--+
 x ----+         (+)--> y   ==  x -->[h1+h2]--> y
       +-->[h2]--+
```

#### Example
$h_1=\delta[n]+\delta[n-1]$, $h_2=\delta[n]-\delta[n-1]$:
- cascade: $h_1*h_2=\delta[n]-\delta[n-2]$;
- parallel: $h_1+h_2=2\delta[n]$.

These properties let a block diagram of many LTI systems be reduced to a single equivalent $h[n]$.

## Check yourself

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

1. What does x*h = h*x mean?
   A. Input and impulse response can be swapped without changing output
   B. The system must be causal
   C. Output is independent of the input
   D. Two systems in parallel can be swapped only
   Answer: A. Convolution is symmetric in its two arguments. (Key point: Commutative meaning)

2. Two LTI systems in cascade are equivalent to what single system?
   A. h = h1+h2
   B. h = h1*h2
   C. h = h1 - h2
   D. h = h1 x h2 sample by sample
   Answer: B. The associative property gives an equivalent response equal to the convolution of the two. (Key point: Cascade)

3. Two LTI systems in parallel (same input, outputs added) are equivalent to what?
   A. h = h1*h2
   B. h = h1/h2
   C. h = h1+h2
   D. h = max(h1,h2)
   Answer: C. The distributive property: x*(h1+h2)=x*h1+x*h2. (Key point: Parallel)

4. For h1=delta[n]+delta[n-1] and h2=delta[n]-delta[n-1], what are the cascade and parallel responses?
   A. Cascade 2delta[n]; parallel delta[n]-delta[n-2]
   B. Cascade delta[n]+delta[n-2]; parallel 2delta[n]
   C. Cascade delta[n]-delta[n-2]; parallel delta[n]
   D. Cascade delta[n]-delta[n-2]; parallel 2delta[n]
   Answer: D. (1+z^-1)(1-z^-1)=1-z^-2 for cascade, and the sum is 2delta[n]. (Key point: Example)

## Related in Signals and Systems

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