21psd - EECS 501 Given: With: Note: WSS: DEF: Sx (ω) Rx...

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EECS 501 POWER SPECTRAL DENSITY Fall 2000 Given: x ( t ) is a real-valued 0-mean WSS random process (RP). With: Autocorrelation R x ( τ ) = E [ x ( t ) x ( t ± τ )] for any time t , Note: x ( t ) 0-mean R x ( τ ) = K x ( τ lag τ . WSS: E [ x ( t ) x ( s )] = R x ( t - s ); not function of t and s separately. DEF: Power spectral density (PSD) S x ( ω ) is defined as: S x ( ω ) = F{ R x ( τ ) } = R -∞ R x ( τ ) e - jωτ = 2 R 0 R x ( τ )cos( ωτ ) . R x ( τ ) = F - 1 { S x ( ω ) } = R -∞ S x ( ω ) e jωτ dω 2 π = R 0 S x ( ω )cos( ωτ ) π . Assume: x ( t ) is 2nd-order process : E [ x ( t ) 2 ] < (except: white). Properties of Power Spectral Density 1. x ( t ) real R x ( τ ) = R x ( - τ ) S x ( ω ) = S x ( - ω ) is real: F{ real , evenfunction } =real,even function cosine xform. 2. R x ( τ ) is positive semidefinite S x ( ω ) 0: σ 2 y ( t )= R f ( t ) x ( t ) dt = R -∞ R -∞ f ( t ) R x ( t - s ) f * ( s ) dtds 0. Proof: : Suppose ω o so that S x ( ω o ) < 0. Let f ( t ) = e o t . Then: σ 2 y = R R e o t R x ( t - s ) e - o s dtds = S x ( ω o ) · ∞ < 0. Proof: : For any real f ( t ), write f ( t ) = R F ( ω ) e jωt . Then: σ 2 y = R | F ( ω ) | 2 S x ( ω ) 0 using Parseval twice. Note: Much simpler in the frequency domain! (just nonnegative) 3. See table of properties on p.471 of Stark and Woods. 4. Averagepower = E [ x ( t ) 2 ] = σ 2 x ( t ) = 1 2 π R -∞ S x ( ω ) . Note: “Power” is the tendency of random | x ( t ) | to be large: Larger variance broader pdf RV tends to be larger. EX: Let v ( t )=random voltage across a resistor R = 1Ω. Average power= E [ v ( t ) 2 ]; large despite E [ v ( t )] = 0.
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EECS 501 PSD OF LINEAR TIME-INVARIANT Fall 2000 (LTI) SYSTEM OUTPUT WITH WSS PROCESS INPUT 1. LTI system; impulse response h ( t ): δ ( t ) | h ( t ) | h ( t ) 2. LTI system has transfer function H ( ω ) = F{ h ( t ) } : cos( ωt ) | h ( t ) | → | H ( ω ) | cos( ωt + ARG [ H ( ω )]) 3. WSS random processes: x ( t ) | h ( t ) | y ( t ) 4. IMPORTANT FORMULA: S y ( ω ) = | H ( ω ) | 2 S x ( ω ). From: Take F of R y ( τ ) = R R h ( u ) h ( v ) R x ( τ - u + v ) dudv . Note: Compare to random vectors: y = Ax K y = AK x A T . EX: x ( t ) | dy dt + ay ( t ) = x ( t ) | y ( t ) , a > 0 so stable. x ( t ) is a 0-mean uncorrelated WSS RP. What is S y ( ω )? 1. 0-mean WSS uncorrelated
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This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.

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21psd - EECS 501 Given: With: Note: WSS: DEF: Sx (ω) Rx...

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