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Analysis of LTI Systems and Sequence Transformations

This document presents various problems related to linear time-invariant (LTI) systems and discrete-time sequences. Problem 1 provides specific output values for a system and asks for the sequence y[n]. Problems 2.15 and 2.20 investigate properties of LTI systems, including causality, BIBO stability, and finding impulse responses. Problem 2.21 focuses on transformations of sequences, where sketches of different shifted and scaled versions of a given sequence x[n] are required.

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Analysis of LTI Systems and Sequence Transformations

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  1. Homework 1

  2. Problem 2.4 • Given: Find y[n] y[0]=0, y[1]=2, y[2]=3/2, y[3]=7/8, y[4]=15/32

  3. Problem 2.15 u[n] h[n] w[n] X[n] y[n] System LTI?, Causal?, BIBO?

  4. Problem 2.15 • For LTI, test: Let then But

  5. Problem 2.15 • Causal? From LTI test: , For n=0 Not Causal

  6. Problem 2.15 • BIBO? • Since h[n] is bounded and multiplying by u[n] will not cause it to become unbounded it is stable

  7. Problem 2.20 • Consider the following Causal LTI system: • Find impulse response, h[n] (b) Range of values for a? a>1

  8. Problem 2.21 • Given a sequence, x[n]: • (a) Sketch x[n-2]: • (b) Sketch x[4-n]: • (c) Sketch x[2n]: • (d) Sketch x[n]u[2-n]: • (e) Sketch x[n-1]

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