Page 1 MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Page 2 MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lectures 8 - 9: Signal Detection in Noise and the Matched Filter Eytan Modiano Aero-Astro Dept. Eytan Modiano Slide 1 Page 3 MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lectures 8 - 9: Signal Detection in Noise and the Matched Filter Eytan Modiano Aero-Astro Dept. Eytan Modiano Slide 1 Noise in communication systems S(t) Channel r(t) r(t) = S(t) + n(t) n(t) • Noise is additional “unwanted” signal that interferes with the transmitted signal – Generated by electronic devices • The noise is a random process – Each “sample” of n(t) is a random variable • Typically, the noise process is modeled as “Additive White Gaussian Noise” (AWGN) – White: Flat frequency spectrum – Gaussian: noise distribution Eytan Modiano Slide 2 Page 4 MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lectures 8 - 9: Signal Detection in Noise and the Matched Filter Eytan Modiano Aero-Astro Dept. Eytan Modiano Slide 1 Noise in communication systems S(t) Channel r(t) r(t) = S(t) + n(t) n(t) • Noise is additional “unwanted” signal that interferes with the transmitted signal – Generated by electronic devices • The noise is a random process – Each “sample” of n(t) is a random variable • Typically, the noise process is modeled as “Additive White Gaussian Noise” (AWGN) – White: Flat frequency spectrum – Gaussian: noise distribution Eytan Modiano Slide 2 Random Processes • The auto-correlation of a random process x(t) is de?ned as – R xx (t 1 ,t 2 ) = E[x(t 1 )x(t 2 )] • A random process is Wide-sense-stationary (WSS) if its mean and auto-correlation are not a function of time. That is – m x (t) = E[x(t)] = m – R xx (t 1 ,t 2 ) = R x (t), where t = t 1 -t 2 • If x(t) is WSS then: – R x (t) = R x (-t) – | R x (t)| <= |R x (0)| (max is achieved at t = 0) • The power content of a WSS process is: P x = E[ lim t!" 1 T x 2 (t)dt = #T / 2 T / 2 $ lim t!" 1 T R x (0)dt = #T / 2 T / 2 $ R x (0) Eytan Modiano Slide 3 Page 5 MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lectures 8 - 9: Signal Detection in Noise and the Matched Filter Eytan Modiano Aero-Astro Dept. Eytan Modiano Slide 1 Noise in communication systems S(t) Channel r(t) r(t) = S(t) + n(t) n(t) • Noise is additional “unwanted” signal that interferes with the transmitted signal – Generated by electronic devices • The noise is a random process – Each “sample” of n(t) is a random variable • Typically, the noise process is modeled as “Additive White Gaussian Noise” (AWGN) – White: Flat frequency spectrum – Gaussian: noise distribution Eytan Modiano Slide 2 Random Processes • The auto-correlation of a random process x(t) is de?ned as – R xx (t 1 ,t 2 ) = E[x(t 1 )x(t 2 )] • A random process is Wide-sense-stationary (WSS) if its mean and auto-correlation are not a function of time. That is – m x (t) = E[x(t)] = m – R xx (t 1 ,t 2 ) = R x (t), where t = t 1 -t 2 • If x(t) is WSS then: – R x (t) = R x (-t) – | R x (t)| <= |R x (0)| (max is achieved at t = 0) • The power content of a WSS process is: P x = E[ lim t!" 1 T x 2 (t)dt = #T / 2 T / 2 $ lim t!" 1 T R x (0)dt = #T / 2 T / 2 $ R x (0) Eytan Modiano Slide 3 Power Spectrum of a random process • If x(t) is WSS then the power spectral density function is given by: S x (f) = F[R x (t)] • The total power in the process is also given by: P x = S x ( f)df = !" " # R x (t)e ! j2$ft dt !" " # % & ' ' ( ) * * !" " # df = R x (t)e ! j2$ft df !" " # % & ' ' ( ) * * !" " # dt = R x (t) e ! j2$ft df !" " # % & ' ' ( ) * * !" " # dt = R x (t)+(t) !" " # dt = R x (0) Eytan Modiano Slide 4Read More

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