Significance for LTI Systems (The Ultimate System Characterization)

Introduction & Overview

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Key Concepts

• h[n] completely and uniquely characterizes any DT-LTI system.

• No other independent information is needed.

• This is a direct result of linearity and time-invariance.

• It enables computation of any output y[n] for any input x[n] via the convolution sum.

• h[n] is the "fingerprint" or "DNA" of the LTI system.

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• Examples

• Example 1: Simple Unit Delay System

• System Description: y[n] = x[n-1] (The output is simply the input delayed by one sample).

• Finding h[n]: If x[n] = δ[n], then y[n] = δ[n-1]. So, h[n] = δ[n-1].

• Significance: Knowing h[n] = δ[n-1] immediately tells you this system is a pure delay. If you're given any input, say x[n] = {..., 1, 2, 3, ...}, you instantly know the output y[n] will be x[n-1] = {..., 0, 1, 2, 3, ...}. No further system analysis is needed. The impulse response completely describes its behavior.

• Example 2: Two-Point Averaging System

• System Description: y[n] = (x[n] + x[n-1]) / 2 (The output is the average of the current and previous input samples).

• Finding h[n]: If x[n] = δ[n], then y[n] = (δ[n] + δ[n-1]) / 2. So, h[n] = 0.5δ[n] + 0.5δ[n-1].

• Significance: h[n] = {0.5, 0.5} at n=0, 1 (and zero elsewhere). This h[n] tells you:

• The system has a "memory" of one past sample.

• It weights the current and previous inputs equally (0.5 each).

• You can now predict the output for any input, for example, if x[n] = u[n] (a step input), you can compute y[n] = u[n] * (0.5δ[n] + 0.5δ[n-1]). The h[n] provided the full "recipe" for this averaging filter.

• Example 3: Unknown LTI System

• Scenario: You are given an unknown DT-LTI system. You apply $\\delta[n]$ as input and observe the output y[n] = h[n] = {1, 0.5, 0.25} for n=0, 1, 2 and 0 elsewhere.

• Significance: You now know everything about this system's behavior. For instance:

• You know it's a Finite Impulse Response (FIR) filter because h[n] has finite duration.

• You know it's causal because h[n] = 0 for n < 0.

• You know it's BIBO stable because the sum of |h[n]| is 1 + 0.5 + 0.25 = 1.75 < ∞.

• You can compute its response to any input signal x[n] using convolution y[n] = x[n] * h[n]. For example, y[n] will be 1*x[n] + 0.5*x[n-1] + 0.25*x[n-2]. The impulse response is the complete mathematical model of the system.

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• Flashcards

• Term: Impulse Response h[n]

• Definition: The unique output of an LTI system to a unit impulse input.

• Term: Ultimate System Characterization

• Definition: The property that h[n] completely describes all input-output behavior of an LTI system.

• Term: Linearity's Role (in h[n] significance)

• Definition: Allows decomposition of input into scaled impulses and summing of scaled impulse responses.

• Term: Time-Invariance's Role (in h[n] significance)

• Definition: Ensures shifted input impulses produce correspondingly shifted impulse responses.

• Term: Convolution Sum

• Definition: The mathematical operation ($y[n] = x[n] \* h[n]$) that computes output from input and impulse response.

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• Memory Aids

• "The System's DNA": Just like DNA contains all the genetic information for an organism, the impulse response h[n] contains all the information about how an LTI system will behave. If you have the DNA, you know everything.

• "The Master Key": The impulse response is the master key to unlocking any LTI system's behavior. Once you have it, you can open any door (find the output for any input).

• "LTI = Convolution": This is a fundamental equation. The very fact that LTI systems are described by convolution (which uses h[n]) inherently means h[n] is the core. If you remember that LTI is defined by convolution, and convolution requires h[n], you'll always recall its significance.

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