---
title: "Properties of DT LTI systems"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:dt-lti-properties"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dt-lti-properties
site: BCT Engineering
---

# Properties of DT LTI systems

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2078 Poush, 6 marks; 2079 Chaitra, 5 marks (asked 2 times)
- Video: [Linear Time-Invariant (LTI) Systems](https://www.youtube.com/watch?v=Y8iFJVmSQIk) — 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/dt-lti-properties
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

State and explain (describe) different properties of a discrete-time LTI system with suitable examples.

## Short answer

A DT LTI system is defined by h[n] and convolution. Properties: commutative, associative (cascade), distributive (parallel); memoryless iff h[n]=Kδ[n]; causal iff h[n]=0 for n<0; BIBO stable iff sum |h[n]| < ∞; invertible iff h*h_i=δ.

## Answer

A **discrete-time LTI system** is linear and time invariant. It is completely described by its impulse response $h[n]$, and its output is the convolution sum $y[n] = \sum_k x[k]\,h[n-k] = x[n]*h[n]$. Its main properties are listed below, with the test in terms of $h[n]$.

#### 1. Commutative property

$x[n]*h[n] = h[n]*x[n]$: input and impulse response can be interchanged.
Example: $\{1,1\}*\{1,2\} = \{1,2\}*\{1,1\} = \{1,3,2\}$.

#### 2. Associative property (cascade)

$\left(x*h_1\right)*h_2 = x*\left(h_1*h_2\right)$. Two systems in cascade equal one system with $h[n] = h_1[n]*h_2[n]$, and the order of the cascade does not matter.

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

#### 3. Distributive property (parallel)

$x*(h_1 + h_2) = x*h_1 + x*h_2$. Two systems in parallel equal one system with $h[n] = h_1[n] + h_2[n]$.

#### 4. Memory

The system is **memoryless** only if $h[n] = K\delta[n]$, i.e. $y[n] = Kx[n]$. Otherwise it has memory.
Example: $h[n] = \delta[n] + \delta[n-1]$ ($y[n] = x[n] + x[n-1]$) has memory.

#### 5. Causality

Causal if $h[n] = 0$ for $n < 0$, so the output uses no future inputs.
Example: $h[n] = (0.5)^n u[n]$ is causal; $h[n] = \delta[n+1]$ ($y[n] = x[n+1]$) is not.

#### 6. Stability (BIBO)

Stable if the impulse response is **absolutely summable**:

$$
\sum_{n=-\infty}^{\infty} |h[n]| < \infty
$$

Example: $h[n] = (0.5)^n u[n]$ gives $\sum = 2$, stable; $h[n] = u[n]$ (accumulator) gives an infinite sum, unstable. Every FIR system is stable.

#### 7. Invertibility

Invertible if an inverse system $h_i[n]$ exists with $h[n]*h_i[n] = \delta[n]$.
Example: the accumulator $h[n] = u[n]$ has inverse $h_i[n] = \delta[n] - \delta[n-1]$ (first difference).

#### 8. Identity and unit step response

$x[n]*\delta[n] = x[n]$ and $x[n]*\delta[n-k] = x[n-k]$. The step response is the running sum of $h[n]$: $s[n] = \sum_{k=-\infty}^{n} h[k]$.

| Property | Condition on $h[n]$ |
|---|---|
| Memoryless | $h[n] = K\delta[n]$ |
| Causal | $h[n] = 0$, $n < 0$ |
| Stable | $\sum \lvert h[n]\rvert < \infty$ |
| Invertible | $h * h_i = \delta[n]$ |
| Cascade | $h_1 * h_2$ |
| Parallel | $h_1 + h_2$ |

## Check yourself

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

1. Two LTI systems h1 and h2 in cascade are equivalent to one system with:
   A. h = h1 + h2
   B. h = h1 * h2 (convolution), and the order does not matter
   C. h = h1 x h2 (point-wise product)
   D. h = h1 only
   Answer: B. By the associative (and commutative) property, a cascade has the convolution of the impulse responses. (Key point: Cascade)

2. A DT LTI system is memoryless only if:
   A. h[n] = 0 for n < 0
   B. h[n] = K delta[n]
   C. h[n] = u[n]
   D. sum |h[n]| is finite
   Answer: B. Then y[n] = K x[n], depending only on the present input. (Key point: Memoryless condition)

3. Which impulse response is unstable?
   A. h[n] = (0.5)^n u[n]
   B. h[n] = delta[n] + delta[n-1]
   C. h[n] = u[n] (accumulator)
   D. h[n] = delta[n+1]
   Answer: C. BIBO stability needs sum|h[n]| < infinity; u[n] sums to infinity. The others sum to 2, 2 and 1. (Key point: Stability example)

4. What is the inverse system of the accumulator h[n] = u[n]?
   A. delta[n] + delta[n-1]
   B. delta[n] - delta[n-1] (first difference)
   C. u[-n]
   D. delta[n+1]
   Answer: B. u[n] * (delta[n] - delta[n-1]) = u[n] - u[n-1] = delta[n], as required for an inverse. (Key point: Invertibility)

## Close variants set in other exams

- [Define LTI; properties of DT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adefine-lti-dt-properties) — Signals and Systems, chapter 6, asked 3×

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