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Introduction & Overview
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Step Response Definition
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6.1.1.4.1 Definition: The step response, denoted as $s[n]$, is defined as the output sequence of a DT-LTI system when the discrete-time unit step function $u[n]$ is applied as its input. Thus, if the input is $x[n]=u[n]$, then the corresponding output of the system is $y[n]=s[n]$.
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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Input: Always the unit step function ($u[n]$).
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Output: Defined as $s[n]$.
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Purpose: Shows system's reaction to sudden, sustained input.
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Visualization: Good for observing transient behavior and steady-state.
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Complementary to
h[n]: Offers intuitive view, whileh[n]is for mathematical completeness. -
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Examples
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Example 1: Step Response of a Unit Delay System
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System:
y[n] = x[n-1] -
Input: $x[n] = u[n]$
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Output ($s[n]$): $y[n] = u[n-1]$
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Definition Applied: The step response $s[n]$ for this system is
u[n-1]. This means the output is zero for $n \< 1$, and then becomes 1 for $n \ge 1$. It graphically shows the delayed "turn-on" effect. -
Example 2: Step Response of a Two-Point Averaging System
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System:
y[n] = (x[n] + x[n-1]) / 2 -
Input: $x[n] = u[n]$
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Output ($s[n]$):
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For $n \< 0$, $u[n]=0$ and $u[n-1]=0$, so $s[n]=0$.
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For $n = 0$, $u[0]=1$ and $u[-1]=0$, so $s[0] = (1+0)/2 = 0.5$.
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For $n \ge 1$, $u[n]=1$ and $u[n-1]=1$, so $s[n] = (1+1)/2 = 1$.
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Definition Applied: The step response $s[n]$ is ${0, \underline{0.5}, 1, 1, 1, ...}$ where the underline is at $n=0$. This graphically shows that the system gradually transitions to the steady-state value of 1, taking one sample to reach it fully.
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Flashcards
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Term: Step Response ($s[n]$)
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Definition: The output of a DT-LTI system when the discrete-time unit step function ($u[n]$) is applied as its input.
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Term: Unit Step Function ($u[n]$)
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Definition: The specific input signal used to define the step response; 0 for negative time, 1 for non-negative time.
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Term: Transient Behavior
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Definition: The initial, changing part of the step response as the system adapts to the sudden input.
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Term: Steady-State Output
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Definition: The final, constant value that the step response settles to after the transient phase.
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Memory Aids
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"Step Input = Step Output": Just like the impulse response is triggered by an impulse, the step response is triggered by a step input. It's a direct mapping.
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"Switch On": Think of the step input $u[n]$ as literally "switching on" a constant input. The step response $s[n]$ then shows how the system "powers up" and stabilizes in response to that switch being thrown.
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"S" for Step and Sustained: The "S" in $s[n]$ reminds you it's the response to a Step input, which is a Sustained input, showing the Settling behavior.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example 1: Step Response of a Unit Delay System
-
System:
y[n] = x[n-1] -
Input: $x[n] = u[n]$
-
Output ($s[n]$): $y[n] = u[n-1]$
-
Definition Applied: The step response $s[n]$ for this system is
u[n-1]. This means the output is zero for $n \< 1$, and then becomes 1 for $n \ge 1$. It graphically shows the delayed "turn-on" effect. -
Example 2: Step Response of a Two-Point Averaging System
-
System:
y[n] = (x[n] + x[n-1]) / 2 -
Input: $x[n] = u[n]$
-
Output ($s[n]$):
-
For $n \< 0$, $u[n]=0$ and $u[n-1]=0$, so $s[n]=0$.
-
For $n = 0$, $u[0]=1$ and $u[-1]=0$, so $s[0] = (1+0)/2 = 0.5$.
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For $n \ge 1$, $u[n]=1$ and $u[n-1]=1$, so $s[n] = (1+1)/2 = 1$.
-
Definition Applied: The step response $s[n]$ is ${0, \underline{0.5}, 1, 1, 1, ...}$ where the underline is at $n=0$. This graphically shows that the system gradually transitions to the steady-state value of 1, taking one sample to reach it fully.
-
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Flashcards
-
Term: Step Response ($s[n]$)
-
Definition: The output of a DT-LTI system when the discrete-time unit step function ($u[n]$) is applied as its input.
-
Term: Unit Step Function ($u[n]$)
-
Definition: The specific input signal used to define the step response; 0 for negative time, 1 for non-negative time.
-
Term: Transient Behavior
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Definition: The initial, changing part of the step response as the system adapts to the sudden input.
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Term: Steady-State Output
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Definition: The final, constant value that the step response settles to after the transient phase.
-
-
Memory Aids
-
"Step Input = Step Output": Just like the impulse response is triggered by an impulse, the step response is triggered by a step input. It's a direct mapping.
-
"Switch On": Think of the step input $u[n]$ as literally "switching on" a constant input. The step response $s[n]$ then shows how the system "powers up" and stabilizes in response to that switch being thrown.
-
"S" for Step and Sustained: The "S" in $s[n]$ reminds you it's the response to a Step input, which is a Sustained input, showing the Settling behavior.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π§ Other Memory Gems
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Just like the impulse response is triggered by an impulse, the step response is triggered by a step input. It's a direct mapping.
- "Switch On"
π§ Other Memory Gems
-
The "S" in $s[n]$ reminds you it's the response to a Step input, which is a Sustained input, showing the Settling behavior.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: SteadyState Output
Definition:
The constant or repeating value that a system's output eventually reaches after the transient behavior has subsided, usually in response to a constant or periodic input.
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Term: Complementary to `h[n]`
Definition:
Offers intuitive view, while
h[n]is for mathematical completeness. -
Term: Definition Applied
Definition:
The step response $s[n]$ is ${0, \underline{0.5}, 1, 1, 1, ...}$ where the underline is at $n=0$. This graphically shows that the system gradually transitions to the steady-state value of 1, taking one sample to reach it fully.
-
Term: Definition
Definition:
The final, constant value that the step response settles to after the transient phase.
-
Term: "S" for Step and Sustained
Definition:
The "S" in $s[n]$ reminds you it's the response to a Step input, which is a Sustained input, showing the Settling behavior.
6.1.1.4.1 Definition Step Response (s[n])
This section provides the precise definition of the discrete-time step response, denoted as $s[n]$. Understanding this specific output is important for visualizing how a DT-LTI system behaves when subjected to a sudden and sustained input.