SP_IV_P03_4p - Introduction to Signal Processing 1...

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

View Full Document Right Arrow Icon
Introduction to Signal Processing 1 Copyright © 2005-2009 – Hayder Radha D. Signal Transmission Through Linear Time-Invariant Systems If () x t and yt are the input and output of an LTIC system with impulse response ht then recall that the output of the system ( zero-state response ) can be evaluated as the convolution of the input ( ) x t and the impulse response ( ) : Introduction to Signal Processing 2 Copyright © 2005-2009 – Hayder Radha () () () * xt ht = . where ()( ) x ht d τ ττ −∞ =− * y tx t h t = xt Introduction to Signal Processing 3 Copyright © 2005-2009 – Hayder Radha Let ( ) x t , , and have corresponding Fourier transform functions ( ) X ω , ( ) Y , and ( ) H respectively: ( ) x tX Y H Introduction to Signal Processing 4 Copyright © 2005-2009 – Hayder Radha Now, by using the time-convolution property, we have the following: () () YH X ωω = . YX H = H X
Background image of page 1

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

View Full DocumentRight Arrow Icon
Introduction to Signal Processing 5 Copyright © 2005-2009 – Hayder Radha In other words, the output of a continuous-time LTI system in the frequency domain () Y ω can be obtained by multiplying the Fourier transform of the input X by the Fourier transform of the impulse response ( ) H . Note that we can evaluate the time-domain output yt by taking the inverse Fourier transform of ( ) Y . () ( ) 1 Y ⎡⎤ ⎣⎦ =F Introduction to Signal Processing 6 Copyright © 2005-2009 – Hayder Radha Evaluating the output of the system first in the frequency domain (i.e. by using ( ) ( ) ( ) YH X ωω = first) and then evaluating 1 Y provides an alternative approach to performing time-domain convolution () () () * xt ht = . Introduction to Signal Processing 7 Copyright © 2005-2009 – Hayder Radha Example Find the zero state response of an LTI system with frequency response, 1 2 H j = + to an input t x te u t = . Introduction to Signal Processing 8 Copyright © 2005-2009 – Hayder Radha Solution: We are already given the Fourier transform of the impulse response in the frequency domain. We need to evaluate the Fourier transform of the input. Recall that: at x u t = 1 X aj = + Therefore, when 1 a = , we have t x u t = 1 1 X j = +
Background image of page 2
Introduction to Signal Processing 9 Copyright © 2005-2009 – Hayder Radha Hence, and therefore, () () () 1 (2 )(1 ) YH X jj ω ωω = = ++ This can be expanded into (Using partial fractions ), 11 (1 ) ) Y =− And, 2 () ( ) () tt yt e e ut −− . Introduction to Signal Processing 1 0 Copyright © 2005-2009 – Hayder Radha Heuristic Understanding of Linear System Response/ Time-Frequency Duality in the Transform Operations The linear system response of a system to arbitrary input can be found using time or frequency-domain methods. Time-Domain: Convolution integral Sum of (scaled/ shifted) impulse responses. Frequency-Domain: Fourier Integral Sum of sinusoids / everlasting exponentials. Introduction to Signal Processing 1 1 Copyright © 2005-2009 – Hayder Radha th t δ Impulse response of a system is ht .
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
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 / 15

SP_IV_P03_4p - Introduction to Signal Processing 1...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online