Virtual Labs

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

1. In Wiener filtering, under what condition does the optimal filter become a simple gain factor (i.e., scalar multiplication)?

a: When the noise is white and the signal is deterministic. Explanation

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b: When both signal and noise are uncorrelated white processes. Explanation

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c: When the autocorrelation matrix becomes diagonal. Explanation

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d: When the signal and noise have equal power spectra. Explanation

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2. What is the form of the Wiener filter H(f) in the frequency domain for stationary input and noise?

a: H(f) = S_x(f) / S_y(f) Explanation

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b: H(f) = S_d(f) / (S_d(f) + S_n(f)) Explanation

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c: H(f) = S_{dx}(f) / S_x(f) Explanation

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d: H(f) = |X(f)|^2 / |Y(f)|^2 Explanation

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3. The Wiener-Hopf equation arises from which of the following optimization principles?

a: Least squares estimation of noise. Explanation

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b: Maximizing output SNR. Explanation

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c: Orthogonality principle in MMSE estimation. Explanation

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d: Minimizing filter energy subject to distortion constraints. Explanation

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4. In the frequency domain Wiener filter design, what condition leads to the filter behaving as an ideal pass-through (i.e., H(f) ≈ 1)?

a: When the noise power spectral density is much greater than the signal power spectral density. Explanation

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b: When the signal power spectral density dominates the noise power spectral density at all frequencies. Explanation

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c: When the signal and noise power spectra are equal at all frequencies. Explanation

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d: When both signal and noise are uncorrelated and zero-mean. Explanation

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5. In practice, what issue arises when estimating the Wiener filter coefficients using sample data?

a: High computational complexity in estimating Fourier transforms. Explanation

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b: Noise whitening can distort the desired signal. Explanation

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c: Sample autocorrelation matrix may be ill-conditioned or non-invertible. Explanation

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d: The Wiener filter only works on infinite-length signals. Explanation

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6. Which of the following is **not** an assumption made in the derivation of the Wiener filter?

a: Signal and noise are uncorrelated. Explanation

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b: Signal and noise are jointly Gaussian. Explanation

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c: The system is linear and time-invariant. Explanation

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d: Mean square error is the cost function. Explanation

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7. Which of the following best describes the effect of the Wiener filter on the input spectrum?

a: It normalizes the signal spectrum. Explanation

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b: It suppresses frequencies where the SNR is low. Explanation

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c: It equalizes the spectrum of the signal. Explanation

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d: It boosts the noise spectrum to enhance clarity. Explanation

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8. In adaptive implementations, which algorithm is considered a stochastic approximation of the Wiener solution?

a: LMS (Least Mean Squares) algorithm. Explanation

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b: Kalman filter. Explanation

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c: Matched filter algorithm. Explanation

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d: Fourier series estimator. Explanation

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9. For colored noise, the optimal Wiener filter requires knowledge of:

a: Only the signal power spectrum. Explanation

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b: Only the noise autocorrelation. Explanation

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c: Both signal and noise power spectra. Explanation

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d: The amplitude spectrum of the desired signal. Explanation

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10. What is a major limitation of the classical Wiener filter in real-time applications?

a: It introduces significant phase distortion. Explanation

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b: It assumes non-causal filtering which is non-implementable in real-time. Explanation

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c: It increases the noise floor significantly. Explanation

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d: It only works for band-limited signals. Explanation

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