Crucial Relationship to Impulse Response

Introduction & Overview

Read summaries of the section's main ideas at different levels of detail.

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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.

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• 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.

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• Flashcards

• Term: s[n] = Σ 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.

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• 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] = Σ h[k] and h[n] = s[n] - s[n-1].

Examples & Applications

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Glossary