---
title: "CT vs DT signals; u[n] = Σδ[n−k]"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:ct-dt-signals-step-impulse-sum"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ct-dt-signals-step-impulse-sum
site: BCT Engineering
---

# CT vs DT signals; u[n] = Σδ[n−k]

Signals and Systems · Chapter 1: Signals

- Asked in: 2073 Bhadra, 2+4 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/ct-dt-signals-step-impulse-sum
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Differentiate between continuous time and discrete time signals with examples. Show that u[n] = Σ_{k=0}^{∞} δ[n−k]

## Answer

#### CT vs DT signals

| Continuous-time signal | Discrete-time signal |
|---|---|
| Defined for every $t$ | Defined only at integer $n$ |
| Written $x(t)$ | Written $x[n]$ |
| Analog in nature | Obtained by sampling or naturally discrete |
| Processed by analog circuits | Processed by digital hardware |
| Uses integrals, derivatives | Uses sums, differences |
| e.g. $\sin 100\pi t$, speech voltage | e.g. $(0.5)^n u[n]$, daily temperature |

#### Show that $u[n] = \sum_{k=0}^{\infty}\delta[n-k]$

Expand the right-hand side:

$$
\sum_{k=0}^{\infty}\delta[n-k] = \delta[n] + \delta[n-1] + \delta[n-2] + \cdots
$$

$\delta[n-k] = 1$ only when $n = k$, otherwise 0.

- **For $n < 0$:** $n - k < 0$ for every $k \ge 0$, so every term is 0. Sum $= 0$.
- **For $n \ge 0$:** exactly one term, the one with $k = n$, equals 1; all others are 0. Sum $= 1$.

$$
\sum_{k=0}^{\infty}\delta[n-k] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases} = u[n]
$$

Hence proved. (Equivalently, putting $m = n-k$: $u[n] = \sum_{m=-\infty}^{n}\delta[m]$, the running sum of the impulse.) Graphically, the step is a train of unit impulses at $n = 0, 1, 2, \ldots$:

```text
   o  o  o  o  o ...   = delta[n] + delta[n-1] + ...
   |  |  |  |  |
 --+--+--+--+--+--> n
   0  1  2  3  4
```

## Check yourself

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

1. How does a DT signal differ from a CT signal?
   A. It is defined for every real t
   B. It is defined only at integer values of n
   C. It must be periodic
   D. It can only be real-valued
   Answer: B. x[n] is a sequence, while x(t) is defined at every instant. (Key point: CT vs DT definition)

2. When is δ[n-k] equal to 1?
   A. For all n ≥ k
   B. For all n
   C. Only when n = k
   D. Only when n = 0
   Answer: C. δ[n-k] is a unit impulse located at n = k. (Key point: Impulse property)

3. What is Σ δ[n-k] over k ≥ 0 for n < 0?
   A. 1
   B. n
   C. Infinite
   D. 0
   Answer: D. For n < 0, n - k < 0 for all k ≥ 0, so every impulse term is zero. (Key point: Sum for n < 0)

4. Why does the sum equal 1 for n ≥ 0?
   A. Exactly one term, k = n, is 1 and the rest are 0
   B. All terms equal 1
   C. Terms add up to n
   D. Terms cancel to 1 in pairs
   Answer: A. Only the impulse at k = n is non-zero, so the sum is u[n]. (Key point: Sum for n ≥ 0)

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