Principles of Modern Radar - Volume 2 1891121537

6.7 Phase Error Effects 249dB re: Ideal Peak0−10−20−30−402Mean PSR (σ κ = 0.3)Single RealizationIdeal PSRFIGURE 6-20 Theexpected PSRcontaminated by aphase error whosevariance is0.3 radians (solidline), a singlerealization (dottedline), and the idealPSR (dashed line).−50−15 −10 −5 0 5 10 15Normalized Cross−RangeThe average effect on the PSR is found by taking the expected value of the magnitudesquaredof the corrupted PSR, E{|F c (x)| 2 }. Taking the Fourier transform gives:E { |F c (x)| 2} = E { F c (x) · ¯F c (x) }= E { F { f c (θ) ∗ ¯f c (−θ) }}= E { F { [ f (θ)g(θ)] ∗ [ ¯f (θ)ḡ(θ) ]}}= E {F {[rect(θ/θ int )g(θ)] ∗ [rect(θ/θ int )ḡ(θ)]}}= E {Fθ int · {triangle(θ/θ int ) · [g(θ) ∗ ḡ(θ)]}}= Fθ int · {triangle(θ/θ int ) · E {[g(θ) ∗ ḡ(θ)]}}= Fθ int · {triangle(θ/θint ) · R gg (θ) }= δx −2 · sinc 2 (x/δx) (6.36)where R gg is the autocorrelation of g, S gg is its power spectral density [33], and the scalingbetween x and θ int is found in Equation 7.26 and the discussion in Section 7.5.2. We thusfind that the expected value of the corrupted magnitude-squared PSR is described by theideal PSR convolved with the power spectral density of the autocorrelation of the phaseerror function g(θ).To find the autocorrelation R gg we employ the fact that g(θ) can be expressed as thesum of its mean and the centered version of itself, g(θ) = μ g + (g(θ) − μ g ) = μ g + v(θ).R gg = E {g(θ)ḡ(θ + ζ)}= E { (v(θ) + μ g )(¯v(θ + ζ)+ ¯μ g ) }= E { }v(θ)¯v(θ + ζ)+ v(θ) μ¯g + ¯v(θ + ζ)μ g + μ g ¯μ g= σg 2 δ(θ) + ∣∣ ∣μ g 2The Fourier transform of this is simply:(6.37)S gg (x) = σ 2 g + ∣ ∣μ g∣ ∣2 δ(x) (6.38)Inserting (6.38) into the last line of (6.36) we find that the mean effect of a random phaseerror is to scale the squared ideal PSR |F(x)| 2 by μ g while adding background noise equalto the variance of g(x) times the total energy of the ideal PSR F(x),E{|F c (x)| 2 }=μ 2 g |F(x)|2 + σg 2 /δx (6.39)