VIPSolutionsgive answer in 2 step with explanation at the end of each step and final answer at the end:18.13 (2) (fe) A zero mean signal with PSD Ps(f) = 2 2cos(2nf) is embedded in white noise with variance of, = 1. Plot the frequency response of the optimal Wiener smoother. Also, compute the minimum MSE. Hint: For the MSE use a “sum” approximation to the integral (see Problem 1.14).
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give answer in 2 step with explanation at the end of each step and final answer at the end:
18.13 (2) (fe) A zero mean signal with PSD Ps(f) = 2 2cos(2nf) is embedded
in white noise with variance of, = 1. Plot the frequency response of the
optimal Wiener smoother. Also, compute the minimum MSE. Hint: For the
MSE use a “sum” approximation to the integral (see Problem 1.14).
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Asked by: Pagal
Created at: 2025-07-30 18:41:09
Asked by: Pagal
Created at: 2025-07-30 18:41:09
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Let's solve the problem **step by step** as requested.
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## **Step 1: Frequency Response of the Optimal Wiener Smoother**
The optimal Wiener filter (smoother) in the frequency domain for a signal with power spectral density \( P_S(f) \) embedded in white noise with variance \( \sigma_W^2 \) is given by:
\[
H(f) = \frac{P_S(f)}{P_S(f) + P_W(f)}
\]
where \( P_W(f) = \sigma_W^2 \) is the power spectral density of the white noise.
Given:
- \( P_S(f) = 2 - 2\cos(2\pi f) \)
- \( \sigma_W^2 = 1 \Rightarrow P_W(f) = 1 \)
So,
\[
H(f) = \frac{2 - 2\cos(2\pi f)}{2 - 2\cos(2\pi f) + 1} = \frac{2 - 2\cos(2\pi f)}{3 - 2\cos(2\pi f)}
\]
**Explanation:**
We substituted the given PSDs into the Wiener filter formula to get the frequency response.
---
## **Step 2: Compute the Minimum MSE**
The minimum mean square error (MSE) for the Wiener filter is given by:
\[
\text{MSE}_{\text{min}} = \int_{-1/2}^{1/2} \frac{P_S(f) P_W(f)}{P_S(f) + P_W(f)} df
\]
Plugging in the given values:
\[
\text{MSE}_{\text{min}} = \int_{-1/2}^{1/2} \frac{(2 - 2\cos(2\pi f)) \cdot 1}{2 - 2\cos(2\pi f) + 1} df
= \int_{-1/2}^{1/2} \frac{2 - 2\cos(2\pi f)}{3 - 2\cos(2\pi f)} df
\]
**Sum Approximation:**
If we approximate the integral as a sum over \( N \) uniformly spaced points in \([-1/2, 1/2]\):
\[
\text{MSE}_{\text{min}} \approx \frac{1}{N} \sum_{k=0}^{N-1} \frac{2 - 2\cos(2\pi f_k)}{3 - 2\cos(2\pi f_k)}
\]
where \( f_k = -\frac{1}{2} + \frac{k}{N} \), for \( k = 0, 1, ..., N-1 \).
**Explanation:**
We used the Wiener filter MSE formula and substituted the given PSD and noise variance. The sum approximation allows us to numerically estimate the MSE.
---
## **Final Answers:**
1. **Frequency response of the optimal Wiener smoother:**
\[
\boxed{
H(f) = \frac{2 - 2\cos(2\pi f)}{3 - 2\cos(2\pi f)}
}
\]
2. **Minimum MSE (as a sum):**
\[
\boxed{
\text{MSE}_{\text{min}} \approx \frac{1}{N} \sum_{k=0}^{N-1} \frac{2 - 2\cos(2\pi f_k)}{3 - 2\cos(2\pi f_k)}
}
\]
where \( f_k = -\frac{1}{2} + \frac{k}{N} \).
If you'd like, I can help plot \( H(f) \) or compute a numerical value for the MSE for a given \( N \).
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