Definition - 6.1.1.4.1
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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
Examples & Analogies
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 & Applications
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 $s[0] = (1+0)/2 = 0.5$.
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.
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
Definition: The initial, changing part of the step response as the system adapts to the sudden input.
Term: Steady-State Output
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
Interactive tools to help you remember key concepts
Memory Tools
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"
Memory Tools
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
Glossary
- SteadyState Output
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.
- Complementary to `h[n]`
Offers intuitive view, while
h[n]is for mathematical completeness.
- 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.
- Definition
The final, constant value that the step response settles to after the transient phase.
- "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.