National Aviation University Aviation radio-electronic complex department Bulgarian Academy of Sciences Institute of Information and Communication Technologies Advanced computing for Innovation Y.D. Chyrka Harmonic signal parameters estimation Supervisor: I.G. Prokopenko
Background Postgraduate student Working as a scientist Working as an engineer Study at the IMT Student in the NAU 2005 2007 2009 2011 2013 2
Outline Publications : Harmonic signals parameters estimation 4 International conferences (SMSDP-2010, SPS-2011, Automatics-2011, IRS-2012) (include 1 in Scopus) Stochastic NonStochastic 1 International article (JET) (include 1 in Scopus) approach approach 5 Ukrainian conferences 3 Ukrainian articles 2 International conferences (SPS-2013, IRS-2013) (include 2 in Scopus) Synchronization 1 International article (TKEA) systems 1 Ukrainian conference 3 International conferences (SPS-2009, SPS-2013, IRS-2013) (include 2 in Scopus) Signal Detection 2 Ukrainian conferences 1 Ukrainian article 1 Ukrainian conference Clusterization by 1 Ukrainian patent ordered statistics 2 Ukrainian conferences EEG processing 1 Ukrainian patent Total: 6 Scopus publications Local projects: 9 International conferences 2 International articles Radar simulation complex for laboratory works 10 Ukrainian conferences Simulation modeling program for a production line 4 Ukrainian articles 2 Ukrainian patents 3
Scopus publications 4
Goal and a signal model GOAL: to improve efficiency of parameters estimation on a limited observation interval . < 2 f / T ob d The additive model of counts sample: = + η + ξ = u s 1 , , i N i i i i 1. The digitized harmonic signal: [ ] ( ) γ = ω = ρ γ − + ϕ d / cos 1 f s i i , – the normalized frequency τ 1 η 2. The white Gaussian noise: i ξ , = ÷ 0 . 95 1 . 0 r 3. The correlated interference: i c There is no any a priori information about signal, noise and interference parameters. Harmonic signal parameters estimation 5
Frequency estimation = α − − α = γ = s s s 2 cos( ) 3 The recurrent form of a sinewave: , i , N , − 1 2 i i i α − α − = 2 2 2 0 B 1) Without interference The quadratic equation: [ ] − − 2 2 N N Step1 – ( ) ( ) ( ) ∑ ∑ = + − + 2 2 , ..., 2 / 2 B x x x x x x x x x − + − + − 0 N 1 i 1 i 1 i i 1 i i i 1 = = 1 1 i i ( ) α + − = ± + γ ∗ = α ∗ ( , ) 2 2 B B arccos / 2 Step2 – Step3 – normed frequency estimation: 1 , 2 ( ) α + = γ < γ < π 2 cos 0 / 2 , 1 α + α + = 2 0 A B C 2) With interference The quadratic equation: [ ] − 1 N ( ) ( )( ) ∑ Step 1 : = − − − − + − 2 A x x x x x x x x − − − − − − − i 2 i 1 i 2 i 1 i i 1 i 2 i 3 = i 3 [ ] [ ] − − N 1 1 ( ) ( ) N ∑ ∑ ( )( ) ( ) = − 2 − − + − 2 = − − + − − − + − 2 2 2 B x x x x x x C x x x x x x x x x x − − − − − − − − − − − − − 2 1 1 2 3 i i i i i i i 2 i 1 i i 1 i 2 i 3 i i 1 i 2 i 3 = = i 3 i 3 ( ) 2 − α = − + 4 / 2 B B AC A Step 2 : ( ) ∗ ∗ γ = α α 1 = < < γ γ π arccos / 2 2 cos 0 Step 3 : , , Harmonic signal parameters estimation 6
Frequency estimation = 10 dB P S /P Simulation parameters: N = 32 , η − − N 1 N 1 ( ) ∑ ∑ ∗ α = + 2 x x x / x − − − AR-est.: AR i i 2 i 1 i 1 = = i 2 i 1 The normed shift of estimations The normed st. dev of estimations. “k” – Our method; “ а ” – AR. Harmonic signal parameters estimation 7
Amplitude and initial phase estimation 1) Without interference − − − N 1 N 1 N 1 ∑ ∑ ∑ ρ = + − γ + γ γ = γ * 2 2 2 sin ( ) sin( ) cos( ) sin( ) A A amplitude A i A i i x i x y i x y ( ) = = = ⇒ ϕ = − i 0 i 0 i 0 * arctan A A initial phase − − − y x N 1 N 1 N 1 ∑ ∑ ∑ γ γ + γ = γ = ρ ϕ = ρ ϕ 2 cos( ) sin( ) cos ( ) cos( ) cos , sin A i i A i x i A A x y i x y = = = i 0 i 0 i 0 2) With interference − − − − N 1 N 1 N 1 N 1 ∑ ∑ ∑ ∑ γ + γ γ + γ = γ 2 sin ( ) sin( ) cos( ) sin( ) sin( ) ; A i A i i A i x i x y z i = = = = ρ = + − i 0 i 0 i 0 i 0 * 2 2 A A amplitude x y − − − − ( ) N 1 N 1 N 1 N 1 ∑ ∑ ∑ ∑ γ γ + γ + γ = γ ⇒ ϕ = − 2 * cos( ) sin( ) cos ( ) cos( ) cos( ) ; arctan A i i A i A i x i A A initial phase x y z i y x = = = = = ρ ϕ = ρ ϕ = ξ i 0 i 0 i 0 i 0 cos , sin , A A A − − − x y z 1 1 1 N N N ∑ ∑ ∑ γ + γ + ⋅ = sin( ) cos( ) . A i A i A N x x y z i = = = 0 0 0 i i i Harmonic signal parameters estimation 8
Likelihood function analysis − 1 N ∑ Λ ϕ ξ = − + ϕ − ξ ρ, γ, ρ γ 2 ( , | , ..., ) ( sin( ) ) x x x i 0 - 1 N i = 0 i { } ∗ ∗ ∗ ∗ ϕ = Λ ϕ ρ γ ξ ρ, γ, ξ , , , arg min ( , | , ..., ) x x 0 N - 1 ϕ ρ, γ, ξ , rad rad ∆ γ ≈ 2 π / N Frequency interval between local minima: Harmonic signal parameters estimation 9
Optimization algorithm Harmonic signal parameters estimation 10
Optimization algorithm efficiency % Frequency, [rad] Harmonic signal parameters estimation 11
Estimations censoring We propose and investigate a method of reducing the measuring system sensitivity to the appearance of gross errors by an a posteriori censoring of results of frequency measurements. ∆ ≤ ∆ * 0 , , f f reliable estimation s ∆ = cr с * ( ) f ∆ > ∆ , С * , , f f unreliable estimation s fl cr We consider that a more reliable result is the P significantly one at which the signal power S σ . In this case the 2 exceeds the noise power following decision rule for erroneous estimate is useful. ( ) * / σ ∗ < 2 P d S Harmonic signal parameters estimation 12
Guaranteed estimation of frequency Goal : to get guaranteed estimations of parameters as corresponding limited sets ∈ ω ∈ ω ω ϕ ∈ ϕ ϕ [ , ], [ , ], [ , ] A A A . Maximum amplitude of an interferences is limited and a priori known . Solution of N-2 compatible inequalities system gives us bounds of frequency estimations + + ≤ ε = − y 2 y c y 4 n 2 ,..., N 1 − − n n 1 n 2 , Harmonic signal parameters estimation 13
Guaranteed estimation of amplitude and phase ( Ω Π n ( Ω Π − 1 Ω ) ) ( ) P Let us to define quadrangle made by crossing of strips and n n ~ ~ ~ Π Ω = Ω + Ω − ≤ ε ( ) { : sin cos } A A n A n y 1 2 n n ~ ~ ~ = ϕ = ϕ = T cos sin [ , ] A A A A A A A 1 2 1 2 ( Ω ) Also we will define as minimal rectangle for each . P P , + 1 n n n − 1 N ∈ ∈ A P P + n , n 1 = 0 n ≤ ≤ min max P A P i i = = 1 ,..., 4 1 ,..., 4 i i ϕ ≤ ϕ ≤ ϕ min max i i = = 1 ,..., 4 1 ,..., 4 i i Harmonic signal parameters estimation 14
Simulation results ~ = ~ = = = ω + ϕ + ξ ω ϕ = ξ n = 32 5 , sin( ) , 0 , 5 , 0 , 5 , 0 , 1 , N A y A n n n Harmonic signal parameters estimation 15
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