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?
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?
3. The Kalman filter can be viewed as a generalisation of the Wiener filter that handles which situation?
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?
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
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)?
7. According to the orthogonality principle, the Wiener estimation error is orthogonal to
8. In causal Wiener prediction, the optimum one‑step predictor minimises which quantity?
9. When the estimated autocorrelation matrix is ill‑conditioned, a practical way to stabilise the Wiener solution is to solve
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?