Digital signal processing - Pseudo Inverse Method

[nikki92]

Digital signal processing -- Pseudo Inverse Method

Homework Statement

The Attempt at a Solution

(a)

A =the matrix with [ .4 0 0 0 0 0 0 0 0 0 0 0; .7 .4 0 0 0 0 0 0 0 0 0; -.1 .7 .4 0 0 0 0 0 0 0 0;... all the way down to 0 0 0 0 0 0 0 0 0 0 -.1] so it is 11 x9 .
w=[w0 w1 ... w8]' d0 = [1 0 0 0 0 0 0 0 0 0 0] ... to d10 = [ 0 0 0 0 0 0 0 0 0 0 1]

A'Aw=A'd and w=(A'A)^-1 A'd so dhat = A(A'A)^-1 A' d

do E = || d -dhat || ^2

So to get the optimum delay dn I calculate dhat_n = A(A'A)^-1 A' d for 0,1,2,3...10
and get E_n = || d_n - dhat_n ||^2 for each n and find which one gives the minimum?

I get a curve that is not concave when I do this. Shouldn't it be a concave up curve and not a decaying function? I am not sure what I did wrong and the entire problem relies on this part.

 

[KataKoniK]

(b)

The pseudo-inverse method in digital signal processing is a technique used to estimate the unknown signal from a set of measurements or observations. It is based on the concept of least squares, where the goal is to minimize the sum of squared errors between the estimated signal and the actual signal.

In the context of this problem, the matrix A represents a system of equations that relates the unknown signal (w) to the measured signal (d). By taking the pseudo-inverse of A, we can obtain an estimate for w that minimizes the sum of squared errors. This is because the pseudo-inverse method takes into account all the available information (the entire matrix A) to find the best estimate for w, rather than just using a single equation.

To solve for w, we use the formula w=(A'A)^-1 A'd, where A' is the transpose of A and d is the measured signal. This gives us the optimal weights for the unknown signal, which can then be used to estimate the signal at different time points.

In order to determine the best estimate for the unknown signal, we can calculate the error term E = || d - dhat || ^2 for different values of n, which represents the delay in the signal. The delay n that gives the minimum error represents the optimum delay for the signal.

However, it is important to note that the curve of E vs n may not always be concave. This is because the pseudo-inverse method takes into account all the available information, rather than just a single equation, which can result in a non-concave curve. In this case, the minimum value of E should still correspond to the optimum delay for the signal.

In conclusion, the pseudo-inverse method in digital signal processing is a powerful tool for estimating unknown signals and finding the optimum delay in a system. It takes into account all the available information and minimizes the sum of squared errors to provide the best estimate for the unknown signal.