Question

The optimal Wiener filter can be designed if noise is a stationary random process that is statistically:

• A. Independent of the stationary signal
• B. Dependent on the non-stationary signal
• C. Independent of the non-stationary signal
• D. Dependent on the stationary signal

Hint:

The classic optimal Wiener filter requires all processes to be wide-sense stationary (WSS). It separates them under the assumption that the noise is completely independent of the target signal.

Answer 1

The Correct Option is A

Concept: The Wiener filter is a classic linear filter optimized to minimize the mean square error (MSE) between a desired target signal and a noisy received signal. To mathematically solve the foundational Wiener-Hopf equations and determine fixed, optimal filter coefficients, the system must operate under strict statistical constraints.

Step 1: Core operational assumptions of the standard Wiener filter.
The derivation of the standard Wiener filter relies on two primary statistical assumptions:
• Both the useful underlying signal, $s[n]$, and the interfering corrupting noise, $v[n]$, must be wide-sense stationary (WSS) random processes. This means their statistical properties—such as mean, variance, and autocorrelation functions—remain completely constant and invariant over time.
• The noise process must be completely uncorrelated with, and statistically independent of, the target signal.

Step 2: Examining the impact of independence on the filter derivation.
Let the received noisy signal be $x[n] = s[n] + v[n]$. The design of the filter depends on the cross-correlation function between the input and the desired target signal, $R_{xs}[k]$. Because the noise is assumed to be statistically independent and has a zero mean value, the expectation simplifies cleanly: \[ R_{xs}[k] = E[x[n]s[n-k]] = E[(s[n] + v[n])s[n-k]] = E[s[n]s[n-k]] + E[v[n]s[n-k]] \] Due to complete statistical independence, the joint expectation splits into separate products: \[ E[v[n]s[n-k]] = E[v[n]] \cdot E[s[n-k]] = 0 \cdot E[s[n-k]] = 0 \] This simplifies the cross-correlation equation to: \[ R_{xs}[k] = R_{ss}[k] \] This simplification allows the optimal transfer function to be derived directly from the power spectral densities: \[ H_{\text{Wiener}}(\omega) = \frac{P_{ss}(\omega)}{P_{ss}(\omega) + P_{vv}(\omega)} \] This mathematical derivation shows that the process requires the noise to be stationary and independent of the stationary signal, which matches Option (A).