---
title: "Memory and memoryless systems; causality"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:memory-causality-dt"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/memory-causality-dt
site: BCT Engineering
---

# Memory and memoryless systems; causality

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2070 Bhadra, 3+4 marks
- Video: [2.2: Memory and Causality](https://www.youtube.com/watch?v=jkBTbO38jVo) — ECE Done Better
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/memory-causality-dt
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Define systems with memory and memory-less systems with examples. Explain the causality property of discrete time LTI systems.

## Short answer

A memoryless system's output depends only on current input, like y[n]=3x[n]; an LTI system is memoryless only if h[n]=K δ[n]. A causal LTI system must have h[n]=0 for n<0 (output independent of future inputs). All real-time systems are causal.

## Answer

#### Systems with memory and memoryless systems
A system is **memoryless (static)** if its output at any time $n$ depends only on the input at the **same** time $n$. A system **with memory (dynamic)** has an output that depends on past and/or future input values (or past outputs), so it must store information.

| Memoryless | With memory |
|---|---|
| $y[n]=3x[n]$ (amplifier) | $y[n]=x[n]-x[n-1]$ (first difference) |
| $y[n]=x^2[n]$ | $y[n]=\sum_{k=-\infty}^{n}x[k]$ (accumulator) |
| $y[n]=(2x[n]-x^2[n])^2$ | $y[n]=x[n+1]$ (advance) |
| Resistor: $v(t)=Ri(t)$ | Capacitor: $v(t)=\frac{1}{C}\int_{-\infty}^{t}i(\tau)d\tau$ |

For an **LTI** system, $y[n]=\sum_k h[k]x[n-k]$. This depends only on $x[n]$ if $h[k]=0$ for all $k\ne0$. So a DT LTI system is memoryless **only if**

$$
h[n]=K\delta[n] \quad\Rightarrow\quad y[n]=Kx[n]
$$

Any other $h[n]$ (e.g. $h[n]=\delta[n]-\delta[n-1]$) gives a system with memory.

#### Causality of discrete-time LTI systems
A system is **causal** if its output at time $n$ depends only on the present and past inputs ($x[n], x[n-1],\dots$), not on future inputs. Causal systems are "non-anticipative"; all real-time physical systems are causal.

For an LTI system,

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

The terms with $k<0$ use $x[n-k]$ with $n-k>n$, i.e. **future** inputs. For the output not to depend on them, those terms must vanish for every input:

$$
\boxed{h[n]=0 \quad \text{for } n<0}
$$

This is the necessary and sufficient condition. The convolution sum of a causal LTI system then becomes

$$
y[n]=\sum_{k=0}^{\infty}h[k]\,x[n-k]=\sum_{k=-\infty}^{n}x[k]\,h[n-k]
$$

Also, for a causal LTI system, **initial rest** holds: if $x[n]=0$ for $n<n_0$, then $y[n]=0$ for $n<n_0$.

#### Examples
- $h[n]=(0.5)^n u[n]$: zero for $n<0$, so **causal**.
- $h[n]=\delta[n+1]+\delta[n]$, i.e. $y[n]=x[n+1]+x[n]$: $h[-1]=1\ne0$, so **non-causal**.
- Accumulator $h[n]=u[n]$: causal, with memory.
- Ideal low pass filter $h[n]=\frac{\sin\omega_c n}{\pi n}$: non-zero for $n<0$, so non-causal (cannot work in real time).

Note: a memoryless system is always causal, but a causal system need not be memoryless.

## Check yourself

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

1. When is a system memoryless?
   A. Output at n depends only on the input at the same n
   B. Output depends only on past inputs
   C. Output is always zero
   D. Output depends on the previous output only
   Answer: A. Static systems use only the present input, e.g. y[n]=3x[n]. (Key point: Memoryless definition)

2. What condition on h[n] makes a DT LTI system memoryless?
   A. h[n]=K u[n]
   B. h[n]=K delta[n]
   C. h[n]=0 for n<0
   D. sum|h[n]| finite
   Answer: B. Only the k=0 term survives in the convolution sum, giving y[n]=Kx[n]. (Key point: Memoryless LTI)

3. What is the necessary and sufficient condition for a DT LTI system to be causal?
   A. h[n]=0 for n<0
   B. h[n]=0 for n>0
   C. h[n]=delta[n]
   D. h[n] is absolutely summable
   Answer: A. Terms with k<0 would use future inputs x[n-k], so they must vanish. (Key point: Causality condition)

4. Which classification is correct?
   A. y[n]=x[n+1]+x[n] is causal; the accumulator is memoryless
   B. Ideal low pass filter is causal and realizable
   C. y[n]=x[n+1]+x[n] is non-causal; the accumulator is causal with memory
   D. y[n]=x[n]-x[n-1] is memoryless
   Answer: C. The advance term uses a future input; the accumulator uses only past and present inputs but stores a sum. (Key point: Classifying systems)

## Related in Signals and Systems

- [Commutative property of CT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acommutative-property-ct) — Signals and Systems, chapter 5
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- [Causality condition for CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acausal-ct-lti-condition) — Signals and Systems, chapter 5
- [Properties of DT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-lti-properties) — Signals and Systems, chapter 6, asked 3×

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