Definition - 6.1.1.1.1
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Audio Book
Dive deep into the subject with an immersive audiobook experience.
The Discrete-Time Impulse Function: Definition
Chapter 1 of 1
π Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
6.1.1.1.1 Definition: The discrete-time unit impulse function, most commonly denoted as $\delta[n]$, is a remarkably simple yet extraordinarily powerful sequence. Its definition is precise: $\delta[n]=1$ when the integer time index $n$ is exactly 0. $\delta[n]=0$ for all other integer values of $n$ (i.e., for $n \neq 0$).
Detailed Explanation
Examples & Analogies
Key Concepts
-
Value at $n=0$: Always 1.
-
Value at $n \neq 0$: Always 0.
-
Fundamental Building Block: The most basic signal in discrete time.
-
Analogue: Discrete-time version of the continuous-time Dirac delta function.
-
Graphical Representation: A single spike at the origin.
-
-
Examples
-
Simple Sequence:
-
$\delta[n] = {..., 0, 0, \underline{1}, 0, 0, ...}$
-
The
$\\underline{1}$indicates the value at $n=0$. -
Specific Values:
-
$\delta[0] = 1$
-
$\delta[1] = 0$
-
$\delta[-1] = 0$
-
$\delta[5] = 0$
-
$\delta[-100] = 0$
-
Plot:
-
Amplitude
-
^
-
|
-
1 .
-
| .
-
| .
-
+--------------------> n (Time Index)
-
... -2 -1 0 1 2 ...
-
| .
-
| .
-
0 . . . . . . . . . .
-
This shows a single vertical line of height 1 at $n=0$, with all other amplitudes being zero.
-
-
Flashcards
-
Term: Discrete-Time Unit Impulse Function
-
Definition: A signal, $\delta[n]$, that is 1 at $n=0$ and 0 for all other integer values of $n$.
-
Term: $\delta[0]$
-
Definition: The value of the unit impulse function at $n=0$, which is always 1.
-
Term: $\delta[n]$ for $n \neq 0$
-
Definition: The value of the unit impulse function at any time index other than 0, which is always 0.
-
Term: Unit Sample Sequence
-
Definition: Another name for the discrete-time unit impulse function.
-
-
Memory Aids
-
"The Origin Spike": Imagine a flat line (zero amplitude) everywhere, except for a single, sharp spike (amplitude 1) directly at the origin (time $n=0$). This simple visual captures the entire definition.
-
"Only at Zero": The word "impulse" implies a very short, localized event. In discrete time, the "shortest" means "only at one sample." And by convention, that one sample is at $n=0$. So, "only at zero, equals one."
Examples & Applications
Simple Sequence:
$\delta[n] = {..., 0, 0, \underline{1}, 0, 0, ...}$
The $\\underline{1}$ indicates the value at $n=0$.
Specific Values:
$\delta[0] = 1$
$\delta[1] = 0$
$\delta[-1] = 0$
$\delta[5] = 0$
$\delta[-100] = 0$
Plot:
Amplitude
^
|
1 .
| .
| .
+--------------------> n (Time Index)
... -2 -1 0 1 2 ...
| .
| .
0 . . . . . . . . . .
This shows a single vertical line of height 1 at $n=0$, with all other amplitudes being zero.
Flashcards
Term: Discrete-Time Unit Impulse Function
Definition: A signal, $\delta[n]$, that is 1 at $n=0$ and 0 for all other integer values of $n$.
Term: $\delta[0]$
Definition: The value of the unit impulse function at $n=0$, which is always 1.
Term: $\delta[n]$ for $n \neq 0$
Definition: The value of the unit impulse function at any time index other than 0, which is always 0.
Term: Unit Sample Sequence
Definition: Another name for the discrete-time unit impulse function.
Memory Aids
"The Origin Spike": Imagine a flat line (zero amplitude) everywhere, except for a single, sharp spike (amplitude 1) directly at the origin (time $n=0$). This simple visual captures the entire definition.
"Only at Zero": The word "impulse" implies a very short, localized event. In discrete time, the "shortest" means "only at one sample." And by convention, that one sample is at $n=0$. So, "only at zero, equals one."
Memory Aids
Interactive tools to help you remember key concepts
Memory Tools
Imagine a flat line (zero amplitude) everywhere, except for a single, sharp spike (amplitude 1) directly at the origin (time $n=0$). This simple visual captures the entire definition.
- "Only at Zero"
Flash Cards
Glossary
- Origin
The point where the time index $n=0$.
- Graphical Representation
A single spike at the origin.
- Plot
- Definition
Another name for the discrete-time unit impulse function.
- "Only at Zero"
The word "impulse" implies a very short, localized event. In discrete time, the "shortest" means "only at one sample." And by convention, that one sample is at $n=0$. So, "only at zero, equals one."