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LSE - Detection and Estimation(Spring 2009 LSE Least...

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Detection and Estimation (Spring, 2009) LSE NCTU EE P.1 Least Squares ~ The Least Squares Approach 9 Minimize the squared difference between the given data x[n] and the assumed signal or noiseless data. 9 No probabilistic assumptions have been made about the data x[n] 9 Usually applied in situations where a precise statistical characterization of the data is unknown or where an optimal estimator cannot be found or may be too complicated to apply in practice. MVU Criterion : Minimize ] ˆ [ ] ) ˆ [( 2 2 θ θ θ E E = need pdf of x[n] LSE (Least Squares Estimator) Criterion : Minimize = 1 0 2 ]) [ ] [ ( N n n s n x
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Detection & Estimation (Spring, 2009) LSE NCTU EE P.2 We choose our estimate of θ to make s[n] closest to the observed data x[n]. The performance of the LSE depends on the properties of the noise and the modeling errors. Example: DC Level Signal s[n] = A The estimator is not optimal! Remark: It is optimal only if x[n] = A + w[n], where w[n] is WGN. If the noise is not zero mean, x estimates A + E(w[n])!
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Detection & Estimation (Spring, 2009) LSE NCTU EE P.3 Remarks: 9 The underlying assumption is that the observed data are composed of a deterministic signal and zero mean noise. 9 If the signal model is incorrect, the modeling error would cause the LSE to be biased. Example: Sinusoidal Frequency Estimation f 0 is to be estimated. Highly nonlinear in f 0 A signal model that is linear in the unknown parameter is said to generate a linear least squares problem. In this case, J is a quadratic function. Example: Sinusoidal Amplitude Estimation If f 0 is known and A is to be estimated It is a linear LS problem. If both f 0 and A are to be estimated, J can be minimized in closed form with respect to A for a given f 0 . Reduce the problem to the minimization of J over f 0 . Separable least squares problem.
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Detection & Estimation (Spring, 2009) LSE NCTU EE P.4 ~ Linear Least Squares , h[n]: a known sequence If θ = A and h[n] = 1, Noiseless data J min = 0 If x[n] = A + w[n] and E(w 2 [n]) >> A In general,
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Detection & Estimation (Spring, 2009) LSE NCTU EE P.5 Extension: A vector parameter θ of dimension p × 1, The signal s = [s[0] s[1] … s[N-1]] T is linear in the unknown parameters. Remarks: 1. The equations are called the normal equations . 2. Same functional form as BLUE. If E( x ) = H θ and C x = σ 2 I BLUE. 3. Same functional form as efficient (MVU) estimator for linear model. If E( x ) = H θ , C x = σ 2 I, and x is Gaussian MVU
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Detection & Estimation (Spring, 2009) LSE NCTU EE P.6 Weighted LS For example, W is diagonal with [ W ] ii = w i > 0. Emphasize the contributions of those data samples that are more reliable. Example: x[n] = A + w[n], where w[n] is zero mean uncorrelated noise with variance σ n 2 . It is reasonable to choose w n = 1/ σ n 2 = = 1 0 2 n 2 ) ] [ ( ) ( N n A n x A J σ
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Detection & Estimation (Spring, 2009) LSE NCTU EE P.7 ~ Geometrical Interpretation The signal model is a linear combination of the “signal” vectors {h 1 , h 2 , …, h p } Example: Fourier Analysis f 0 is a known frequency. θ = [a b] T is to be estimated.
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