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Introduction & Overview
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The Discrete-Time Impulse Function: Definition
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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
No detailed explanation available.
Examples & Analogies
No real-life example available.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Value at $n=0$: Always 1.
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Value at $n \neq 0$: Always 0.
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Fundamental Building Block: The most basic signal in discrete time.
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Analogue: Discrete-time version of the continuous-time Dirac delta function.
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Graphical Representation: A single spike at the origin.
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Examples
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Simple Sequence:
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$\delta[n] = {..., 0, 0, \underline{1}, 0, 0, ...}$
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The
$\\underline{1}$indicates the value at $n=0$. -
Specific Values:
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$\delta[0] = 1$
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$\delta[1] = 0$
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$\delta[-1] = 0$
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$\delta[5] = 0$
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$\delta[-100] = 0$
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Plot:
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Amplitude
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^
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|
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1 .
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| .
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| .
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+--------------------> n (Time Index)
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... -2 -1 0 1 2 ...
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| .
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| .
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0 . . . . . . . . . .
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This shows a single vertical line of height 1 at $n=0$, with all other amplitudes being zero.
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Flashcards
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Term: Discrete-Time Unit Impulse Function
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Definition: A signal, $\delta[n]$, that is 1 at $n=0$ and 0 for all other integer values of $n$.
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Term: $\delta[0]$
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Definition: The value of the unit impulse function at $n=0$, which is always 1.
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Term: $\delta[n]$ for $n \neq 0$
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Definition: The value of the unit impulse function at any time index other than 0, which is always 0.
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Term: Unit Sample Sequence
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Definition: Another name for the discrete-time unit impulse function.
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Memory Aids
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"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.
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"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 & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Simple Sequence:
-
$\delta[n] = {..., 0, 0, \underline{1}, 0, 0, ...}$
-
The
$\\underline{1}$indicates the value at $n=0$. -
Specific Values:
-
$\delta[0] = 1$
-
$\delta[1] = 0$
-
$\delta[-1] = 0$
-
$\delta[5] = 0$
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$\delta[-100] = 0$
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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
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π§ Other Memory Gems
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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
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Origin
Definition:
The point where the time index $n=0$.
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Term: Graphical Representation
Definition:
A single spike at the origin.
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Term: Plot
Definition:
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Term: Definition
Definition:
Another name for the discrete-time unit impulse function.
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Term: "Only at Zero"
Definition:
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."
6.1.1.1.1 Definition The Discrete-Time Impulse Function (Unit Sample Sequence)
This section focuses on the precise definition of the discrete-time unit impulse function, often called the unit sample sequence. Understanding this basic signal is paramount as it forms the foundational element for analyzing discrete-time systems.