Ve451Su2010C5 - Ve451 Lecture Notes Dianguang Ma Summer...

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Unformatted text preview: Ve451 Lecture Notes Dianguang Ma Summer 2010 Chapter 5 Frequency-Domain Analysis of LTI Systems Input-Output Correlation Functions and Spectra 2 ( ) ( )* ( ) ( ) ( ) ( ) ( ) ( )* ( ) ( ) ( ) ( ) xy xx xy xx yy hh xx yy xx r l h l r l S H S r l r l r l S H S ϖ ϖ ϖ ϖ ϖ ϖ = ↔ = = ↔ = System Identification If the input has a flat spectrum, i.e., ( ) constant, then ( ) ( ) ( ) ( ) 1 1 hence ( ) ( ) or ( ) ( ) The relation implies that ( ) can be determined by exciting the sy xx x xy xx x yz yx x x S S S H S H S H S h l r l S S h n ϖ π ϖ π ϖ ϖ ϖ ϖ ϖ ϖ = =- ≤ ≤ = = = = stem by a spectrally flat signal, and crosscorrelating the input with the output of the system. Correlation Functions and Power Spectra for Random Input Signals Consider a discrete-time LTI system with unit sample response ( ) and frequency response ( ). Let ( ) be a sample function of a stationary random process and let ( ) denote the response of the sys h n H f x n y n [ ] [ ] tem. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0) k y k k x x x k k y n h k x n k m E y n E h k x n k h k E x n k h k m m h k m H ∞ =-∞ ∞ ∞ =-∞ =-∞ ∞ ∞ =-∞ =-∞ =- =- =- = = = ∑ ∑ ∑ ∑ ∑ @ [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) yy k i k i k i xx k i l E y n y n l E h k x n k h i x n i l h k h i E x n k x n i l h k h i E x n k x n k k i l h k h i k i l γ γ ∞ ∞ =-∞ =-∞ ∞ ∞ =-∞ =-∞ ∞ ∞ =-∞ =-∞ ∞ ∞ =-∞ =-∞ + =-- + =-- + =-- +- + =- + ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ @ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) j l yy yy l j l xx l k i j l xx k i l j k i xx k i j k j i xx k i l e h k h i k i l e h k h j k i l e h k h i e h k e h i e ϖ ϖ ϖ ϖ ϖ ϖ ϖ γ γ γ ϖ ϖ ∞- =-∞ ∞ ∞ ∞- =-∞ =-∞ =-∞ ∞ ∞ ∞- =-∞ =-∞ =-∞ ∞ ∞- =-∞ =-∞ ∞ ∞- =-∞ =-∞ Γ = =- + =- + = Γ = Γ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ 2 ( ) ( ) xx H ϖ ϖ = Γ ∑ [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )* ( ) ( ) ( ) ( ) yx k k k xx k k xx yx xx l E y n x n l E h k x n k x n l E h k x n k x n l h k E x n k x n l h k E x n k x n k k l h k l k h l l H γ γ γ ϖ ϖ ϖ ∞ =-∞ ∞ ∞ =-∞ =-∞ ∞ ∞ =-∞ =-∞ - =-- ÷ =-- =-- =-- +- =- = Γ = Γ ∑ ∑ ∑ ∑ ∑ @ Digital Resonators • A digital resonator is a special two-pole bandpass filter with the pair of complex- conjugate poles located near the unit circle....
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Ve451Su2010C5 - Ve451 Lecture Notes Dianguang Ma Summer...

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