Engineering Homework Help. EE 453 PSU Phase Constant and Toeplitz Matrix Exercises
Motion Deblurring (40 pts)
We will now return to some similar concepts that were first explored in Homework 1, but we now have
alternative methods we can use to solve these problems. In several sensing applications the source signal is
convolved with the impulse response of the medium that it travels through. Especially in imaging problems,
the impulse response is referred to as the point spread function (PSF). In the first homework, we used
regularization to deconvolve the resulting signals to determine the signals of interest. Here, we will use the
Wiener filter to accomplish similar goals.
1. In the motionBlur.mat file, there are three images (Image Original, Image A, and Image B) along
with a point spread function PSF. For this question, we will focus on Image A, which consists of the
original image undergoing motion blur which we will attempt to correct.
(a) Determine the Peak Signal-to-Noise Ratio (PSNR) of Image A compared to Image Original (You
may find the psnr function useful) and the Structural Similarity Index (You may find the ssim
function useful).
(b) Perform deconvolution on Image A using the Weiner filter and the specified PSF (You may find the
deconvwnr function useful).
(c) Display the deblurred image, and calculate the PSNR and SSIM for the deblurred image.
2. Now we will focus on Image B, which consists of the original image having undergone motion-blur, but
also the more realistic random noise as well. Luckily, if we have an estimate of the noise variance, we
can use that within the Wiener filter algorithm to better correct for it.
(a) Determine the PSNR and the SSIM of Image B compared to Image Original
(b) Perform deconvolution on Image B using the Wiener filter and the specified PSF, using an estimate
of the noise to signal ratio as 1e-5
σ2I
, where σ2
I is the variance of the image VAR{I(:)}
(c) Display the deblurred and denoised image, and calculate the PSNR and SSIM.
3. Often times while we can estimate the amount of noise in our systems, we cannot calculate them
extremely accurately. Let us determine the performance when we assume there is more noise than
there actually is (which may be the default safe assumption).
(a) Perform deconvolution on Image B using the Wiener filter and the specified PSF, using an estimate
of the noise to signal ratio as 1e-3
σ2I
.
(b) Display the deblurred and denoised image, and calculate the PSNR and SSIM.
4. Alternatively, let us evaluate the performance when we underestimate the noise.
(a) Perform deconvolution on Image B using the Wiener filter and the specified PSF, using an estimate
of the noise to signal ratio as 1e-7
σ2I
.
(b) Display the deblurred and denoised image, and calculate the PSNR and SSIM.Problems (60 pts)
1. For a uniform linear array with 5 elements, with kd = π and white noise with a variance of σ2
v = 0.1, we
want to determine the minimum variance distortionless response (MVDR) weights to point at θ = 45◦
.
(a) Calculate φ0
(b) Calculate the inverse correlation matrix R−1
vv
(c) Calculate the steering vector s(φ)
(d) Calculate the MVDR weights
2. Consider a Wiener filtering problem characterized by the following values for the correlation matrix R of
the input vector u[n] and the cross-correlation vector p between the input x[n] and the desired response
d[n]
R = 2 0.25
0.25 2
p = 0.3
0.1
(a) Find a suitable value for the step-size parameter µ that would ensure convergence of the method of
steepest descent
(b) Using the µ value calculated, determine the recursions for computing the weight elements w[n].
Assume w[0] = 0.
3. Suppose we have a signal s[n] with autocorrelation function Rss[k] = 0.95|k|. The signal is corrupted by
v[n], additive white noise with σ2
v = 2 (i.e. Rvv[k] = 2δ[k]). s[n] and v[n] are uncorrelated, zero-mean,
joint wide sense stationary random processes. Design a length N = 3 filter to approximate the signal
s[n] from the x[n] = s[n] + v[n] measurements.
(a) What is the Toeplitz correlation matrix Rxx[k]?
(b) What is the correlation vector p?
(c) What are the weights for the Wiener filter?
(d) What is the mean squared error (MSE) of the estimator?
4. Consider a case where you want to devleop a two-tap Wiener filter to convert the following input x[n]
into the output d[n]
x[n] = 2 cos2πn
N
d[n] = sin2πn
N
HINT: Using a temporary variable like θ =
2πN may prove helpful.
(a) Find the two-dimensional cross-correlation vector p = E{d[n]x[n − m]}
(b) Find the 2 × 2 auto-correlation matrix Rxx = E{x[n]x[n − m]
