Estimate the signal from its noisy observation using a linear filter designed by minimizing the mean square error (Wiener Filter)

1. Which statement best compares a matched filter and a Wiener filter for detecting random signals in additive noise?
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2. In discrete‑time causal Wiener filtering, the optimum transfer function can be written H(z) = B(z) / A_star(z) where A(z) A_star(z) equals the observation spectrum S_x. What does the polynomial B(z) represent?
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3. The Kalman filter can be viewed as a generalisation of the Wiener filter that handles which situation?
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4. For image deblurring, the two‑dimensional Wiener deconvolution filter is H(u,v) = H_psf_conj(u,v) / ( |H_psf(u,v)|^2 + K ). What is the physical meaning of the constant K?
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5. Given an observed signal x[n] = g[n] * d[n] + v[n] with channel frequency response G(f), the non‑causal Wiener filter that estimates d[n] has frequency response
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6. Solving the Toeplitz system R w = p for an N‑tap Wiener predictor by brute‑force inversion costs O(N^3) operations. Which recursion lowers this to O(N^2)?
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7. According to the orthogonality principle, the Wiener estimation error is orthogonal to
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8. In causal Wiener prediction, the optimum one‑step predictor minimises which quantity?
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9. When the estimated autocorrelation matrix is ill‑conditioned, a practical way to stabilise the Wiener solution is to solve
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10. Which theorem guarantees that a causal Wiener filter exists whenever the observation power spectrum can be written as |A(e^jω)|^2 with log‑integrable magnitude?
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