SP_II_P01_4p - Introduction to Signal Processing 1...

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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!
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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.
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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
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SP_II_P01_4p - Introduction to Signal Processing 1...

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