Crucial Relationship to Impulse Response - 6.1.1.4.2 | Module 6: Time Domain Analysis of Discrete-Time Systems | Signals and Systems
K12 Students

Academics

AI-Powered learning for Grades 8–12, aligned with major Indian and international curricula.

Academics

Grades

Grade 8 Grade 9 Grade 10 Grade 11 Grade 12

Curriculum

CBSE ICSE IB
Professionals

Professional Courses

Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.

Professional Courses
Games

Interactive Games

Fun, engaging games to boost memory, math fluency, typing speed, and English skillsβ€”perfect for learners of all ages.

games
Typing
Memory
Math
English Adventures
Knowledge

6.1.1.4.2 - Crucial Relationship to Impulse Response

Practice

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

For any Discrete-Time Linear Time-Invariant (DT-LTI) system, the **step response `s[n]` and the impulse response `h[n]` are fundamentally and directly related**. * The **step response `s[n]` is the running sum (accumulation) of the impulse response `h[n]`**: $s[n] = \\sum\_{k=-\\infty}^{n} h[k]$. * Conversely, the **impulse response `h[n]` is the first difference of the step response `s[n]`**: $h[n] = s[n] - s[n-1]$. This bidirectional relationship allows one to be derived from the other and highlights their intrinsic connection within LTI system analysis.

Audio Book

Dive deep into the subject with an immersive audiobook experience.

Step Response and Impulse Response: Crucial Relationship

Unlock Audio Book

Signup and Enroll to the course for listening the Audio Book

6.1.1.4.2 Crucial Relationship to Impulse Response: Given the intrinsic relationship between $u[n]$ and $\delta[n]$, there exists a direct and important relationship between $s[n]$ and $h[n]$ for any LTI system:
* Step Response from Impulse Response: The step response is obtained by computing the running sum (accumulation) of the impulse response:
$$s[n]=\sum_{k=-\infty}^{n} h[k]$$
* Impulse Response from Step Response: Conversely, the impulse response can be obtained by taking the first difference of the step response:
$$h[n]=s[n]-s[n-1]$$

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

  • s[n] from h[n]: Accumulation (running sum) of h[n].

  • $s[n] = \sum\_{k=-\infty}^{n} h[k]$

  • h[n] from s[n]: First difference of s[n].

  • $h[n] = s[n] - s[n-1]$

  • Why: Direct consequence of $u[n] = \sum \delta[k]$ and $\delta[n] = u[n] - u[n-1]$, combined with LTI properties.

  • Practical Value: Enables measurement of s[n] (often easier) and derivation of h[n] for analytical purposes; offers intuitive visualization of transient behavior.

  • Examples

  • Example 1: Finding Step Response from Impulse Response

  • Given h[n]: Let $h[n]$ be a system that averages the current and previous sample: $h[n] = 0.5\delta[n] + 0.5\delta[n-1]$.

  • In sequence form: $h[n] = {\underline{0.5}, 0.5, 0, 0, ...}$ (underline at $n=0$).

  • Find s[n] using $s[n] = \sum\_{k=-\infty}^{n} h[k]$:

  • For $n \< 0$: $s[n] = 0$ (since $h[k]=0$ for $k\<0$).

  • For $n = 0$: $s[0] = h[0] = 0.5$.

  • For $n = 1$: $s[1] = h[0] + h[1] = 0.5 + 0.5 = 1$.

  • For $n = 2$: $s[2] = h[0] + h[1] + h[2] = 0.5 + 0.5 + 0 = 1$.

  • For $n \> 1$: $s[n] = 1$ (since $h[k]=0$ for $k\>1$, the sum beyond $k=1$ does not change).

  • Result s[n]: $s[n] = {\underline{0.5}, 1, 1, 1, ...}$ for $n \ge 0$.

  • This shows the output rising to 0.5 at $n=0$ and then settling to 1.0.

  • Example 2: Finding Impulse Response from Step Response

  • Given s[n]: Let $s[n]$ be the step response of a simple delay system: $s[n] = u[n-1]$.

  • In sequence form: $s[n] = {..., 0, \underline{0}, 1, 1, 1, ...}$ (underline at $n=0$).

  • Find h[n] using $h[n] = s[n] - s[n-1]$:

  • For $n \< 0$: $h[n] = s[n] - s[n-1] = 0 - 0 = 0$.

  • For $n = 0$: $h[0] = s[0] - s[-1] = 0 - 0 = 0$.

  • For $n = 1$: $h[1] = s[1] - s[0] = 1 - 0 = 1$.

  • For $n \> 1$: $h[n] = s[n] - s[n-1] = 1 - 1 = 0$.

  • Result h[n]: $h[n] = {..., 0, 0, \underline{0}, 1, 0, 0, ...}$ (underline at $n=0$).

  • This confirms that the impulse response is indeed $h[n] = \delta[n-1]$, which is the impulse response for a unit delay system.

  • Flashcards

  • Term: s[n] = Ξ£ h[k]

  • Definition: The step response is the running sum (accumulation) of the impulse response.

  • Term: h[n] = s[n] - s[n-1]

  • Definition: The impulse response is the first difference of the step response.

  • Term: Accumulation

  • Definition: The operation converting h[n] to s[n].

  • Term: First Difference

  • Definition: The operation converting s[n] to h[n].

  • Term: Practical Significance

  • Definition: Allows derivation of h[n] from easily measurable s[n], and provides intuitive transient visualization.

  • Memory Aids

  • "Step = Sum": The word "Step" has an "S", and "Sum" has an "S". The Step Response is the Sum (accumulation) of the Impulse Response.

  • "Impulse = Instantaneous Difference": An impulse is an instantaneous spike. How do you get an instantaneous change from a gradually changing step? By taking the difference between two consecutive steps. The impulse is the "instantaneous change" in the step.

  • Analogy: Odometer vs. Speedometer:

  • Odometer (total distance): Analogous to Step Response ($s[n]$) - it accumulates mileage (input) over time.

  • Speedometer (instantaneous speed): Analogous to Impulse Response ($h[n]$) - it shows the rate of change of distance.

  • How do they relate? Your total distance (odometer) is the sum of all your instantaneous speeds over time. Your instantaneous speed (speedometer) is the difference in distance over a very short time interval.

  • This analogy perfectly captures s[n] = Ξ£ h[k] and h[n] = s[n] - s[n-1].

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example 1: Finding Step Response from Impulse Response

  • Given h[n]: Let $h[n]$ be a system that averages the current and previous sample: $h[n] = 0.5\delta[n] + 0.5\delta[n-1]$.

  • In sequence form: $h[n] = {\underline{0.5}, 0.5, 0, 0, ...}$ (underline at $n=0$).

  • Find s[n] using $s[n] = \sum\_{k=-\infty}^{n} h[k]$:

  • For $n \< 0$: $s[n] = 0$ (since $h[k]=0$ for $k\<0$).

  • For $n = 0$: $s[0] = h[0] = 0.5$.

  • For $n = 1$: $s[1] = h[0] + h[1] = 0.5 + 0.5 = 1$.

  • For $n = 2$: $s[2] = h[0] + h[1] + h[2] = 0.5 + 0.5 + 0 = 1$.

  • For $n \> 1$: $s[n] = 1$ (since $h[k]=0$ for $k\>1$, the sum beyond $k=1$ does not change).

  • Result s[n]: $s[n] = {\underline{0.5}, 1, 1, 1, ...}$ for $n \ge 0$.

  • This shows the output rising to 0.5 at $n=0$ and then settling to 1.0.

  • Example 2: Finding Impulse Response from Step Response

  • Given s[n]: Let $s[n]$ be the step response of a simple delay system: $s[n] = u[n-1]$.

  • In sequence form: $s[n] = {..., 0, \underline{0}, 1, 1, 1, ...}$ (underline at $n=0$).

  • Find h[n] using $h[n] = s[n] - s[n-1]$:

  • For $n \< 0$: $h[n] = s[n] - s[n-1] = 0 - 0 = 0$.

  • For $n = 0$: $h[0] = s[0] - s[-1] = 0 - 0 = 0$.

  • For $n = 1$: $h[1] = s[1] - s[0] = 1 - 0 = 1$.

  • For $n \> 1$: $h[n] = s[n] - s[n-1] = 1 - 1 = 0$.

  • Result h[n]: $h[n] = {..., 0, 0, \underline{0}, 1, 0, 0, ...}$ (underline at $n=0$).

  • This confirms that the impulse response is indeed $h[n] = \delta[n-1]$, which is the impulse response for a unit delay system.

  • Flashcards

  • Term: s[n] = Ξ£ h[k]

  • Definition: The step response is the running sum (accumulation) of the impulse response.

  • Term: h[n] = s[n] - s[n-1]

  • Definition: The impulse response is the first difference of the step response.

  • Term: Accumulation

  • Definition: The operation converting h[n] to s[n].

  • Term: First Difference

  • Definition: The operation converting s[n] to h[n].

  • Term: Practical Significance

  • Definition: Allows derivation of h[n] from easily measurable s[n], and provides intuitive transient visualization.

  • Memory Aids

  • "Step = Sum": The word "Step" has an "S", and "Sum" has an "S". The Step Response is the Sum (accumulation) of the Impulse Response.

  • "Impulse = Instantaneous Difference": An impulse is an instantaneous spike. How do you get an instantaneous change from a gradually changing step? By taking the difference between two consecutive steps. The impulse is the "instantaneous change" in the step.

  • Analogy: Odometer vs. Speedometer:

  • Odometer (total distance): Analogous to Step Response ($s[n]$) - it accumulates mileage (input) over time.

  • Speedometer (instantaneous speed): Analogous to Impulse Response ($h[n]$) - it shows the rate of change of distance.

  • How do they relate? Your total distance (odometer) is the sum of all your instantaneous speeds over time. Your instantaneous speed (speedometer) is the difference in distance over a very short time interval.

  • This analogy perfectly captures s[n] = Ξ£ h[k] and h[n] = s[n] - s[n-1].

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🧠 Other Memory Gems

  • The word "Step" has an "S", and "Sum" has an "S". The Step Response is the Sum (accumulation) of the Impulse Response.
    - **"Impulse = Instantaneous Difference"

🎨 Fun Analogies

  • Odometer vs. Speedometer:
    * Odometer** (total distance)

🧠 Other Memory Gems

  • Analogous to Impulse Response ($h[n]$) - it shows the rate of change of distance.
    * How do they relate? Your total distance (odometer) is the     sum     of all your instantaneous speeds over time. Your instantaneous speed (speedometer) is the difference in distance over a very short time interval.
    * This analogy perfectly captures s[n] = Ξ£ h[k] and h[n] = s[n] - s[n-1].

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Linearity and TimeInvariance (LTI)

    Definition:

    The fundamental properties of systems that enable this direct relationship between h[n] and s[n].

  • Term: Practical Value

    Definition:

    Enables measurement of s[n] (often easier) and derivation of h[n] for analytical purposes; offers intuitive visualization of transient behavior.

  • Term: Result `h[n]`

    Definition:

    $h[n] = {..., 0, 0, \underline{0}, 1, 0, 0, ...}$ (underline at $n=0$).

  • Term: Definition

    Definition:

    Allows derivation of h[n] from easily measurable s[n], and provides intuitive transient visualization.

  • Term: Analogy: Odometer vs. Speedometer

    Definition:

    • Odometer (total distance): Analogous to Step Response ($s[n]$) - it accumulates mileage (input) over time.

6.1.1.4.2 Crucial Relationship to Impulse Response

This section explores the vital and direct relationship between the discrete-time step response $s[n]$ and the discrete-time impulse response $h[n]$ for any LTI system. This relationship is not merely a mathematical curiosity; it is a fundamental connection that deepens our understanding of LTI system behavior and provides practical methods for derivation.