SP_II_P01_4p

# SP_II_P01_4p - Introduction to Signal Processing 1...

This preview shows pages 1–4. Sign up to view the full content.

Introduction to Signal Processing 1 Copyright © 2005-2009 – Hayder Radha II. Time-Domain Analysis of Continuous- Time Systems In this course, we focus on systems that are both linear and time invariant (LTI). In general, there are two approaches for analyzing such systems: time-domain analysis and frequency-domain analysis . In this part of the course, we focus on time-domain analysis of LTI systems. In particular, we consider LTI Continuous-time Introduction to Signal Processing 2 Copyright © 2005-2009 – Hayder Radha (LTIC) system. Later in the course we will study LTI discrete-time system. The analysis of LTI systems is important because of many reasons; here is a few: 1. LTI systems are often very effective in the processing of signals (e.g. noise removal). 2. Many real world systems can be approximated as LTI systems (modeling). Introduction to Signal Processing 3 Copyright © 2005-2009 – Hayder Radha 3. Well developed tools exist for the analysis of LTI systems. We will start off by exploring these analysis tools in the time-domain. A. Convolution Integral and the Impulse Response We specifically begin by establishing the mathematical formula indicating the input-output relationship of an LTI system. Introduction to Signal Processing 4 Copyright © 2005-2009 – Hayder Radha Key Idea: Any input signal () x t can be represented as the sum of shifted and scaled impulse functions atb δ . This can be simply demonstrated from the sifting property : ()( ) x td x t τδ τ τ −∞ −= This can also be seen pictorially, which is more intuitive!

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Introduction to Signal Processing 5 Copyright © 2005-2009 – Hayder Radha To this end we approximate () x t with rectangular pulses: 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 t x(t) 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 t τ Introduction to Signal Processing 6 Copyright © 2005-2009 – Hayder Radha The rectangular pulses are defined as: ( ) 1 ( 1 ) pt n ut n ut n ττ ⎡⎤ −Δ = −Δ − − + Δ ⎣⎦ Δ Consequently, each pulse has a unit area. So, adding up and scaling all the rectangular pulses approximates () x t : ( ) ( ) n xt xn =−∞ ≈Δ Δ Δ Introduction to Signal Processing 7 Copyright © 2005-2009 – Hayder Radha As 0 Δ→ , the above approximation becomes exact: 0 () l im ( ) ( ) ( ) n x t d τδ τ τ =−∞ −∞ = Δ− Δ Δ =− Note that as 0 , the amplitude of and its width goes to zero. This is the sifting property in action. Introduction to Signal Processing 8 Copyright © 2005-2009 – Hayder Radha Now based on this key idea, we proceed as follows. 1. First, we need to evaluate the impulse response ht of the LTI system: th t δ →ℑ→ Simply stated, is the response (output) of the system as a result of applying a unit impulse at time zero.
Introduction to Signal Processing 9 Copyright © 2005-2009 – Hayder Radha 2. By the time-invariance property of the LTI system, we can conclude that applying a time-shifted impulse function will result in a time-shifted (by the same amount) impulse response: For an LTI system if: () th t δ →ℑ→ Then: nn t τ −→ Δ Δ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 06/08/2009 for the course ECE 366 taught by Professor Staff during the Spring '08 term at Michigan State University.

### Page1 / 14

SP_II_P01_4p - Introduction to Signal Processing 1...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online