16
I agree that the windowing filter design method is not one of the most important design methods anymore, and it might indeed be the case that it is overrepresented in traditional textbooks, probably due to historical reasons.
However, I think that its use can be justified in certain situations. I do not agree that computational complexity is no issue ...
7
The LMS algorithm is based on the idea of gradient descent to search for the optimal (minimum error) condition, with a cost function equal to the mean squared error at the filter output. However, it doesn't actually calculate the gradient directly, as that would require knowing:
$$
E(\mathbf{x}[n]e^*[n])
$$
where $\mathbf{x}[n]$ is the input vector and $e[...
7
Windowed Sinc filters can be adaptively generated on the fly on processors barely powerful enough to run the associated FIR filter. Windowed Sinc filters can be generated in finite bounded time.
The generation of some simple windowed Sinc filters can be completely described (and inspected for malware, etc.) in a few lines of code, versus blind use of some ...
7
Slope from all samples obtained
To summarize the question's problem, you want to calculate the slope based on all samples obtained thus far, and as new samples are obtained, update the slope without going through all the samples again.
On the page you cite is the equation for calculation of the slope $m_n$ that together with $b_n$ minimizes the sum of ...
6
That really depends on context, but generally adaptive implies that the calculations are done on-line / on the fly.
In some applications, the filter is updated for a while, then the adaptation is turned off and the last lot of weights are used.
6
To expand on what Peter K. has said, if the signals being used by the filter are stationary, then the filter weights or coefficients can be determined and the filter operates as it was designed without further updates to the filter weights. However, if the signals change, or become quasi-stationary, the filter will adapt continuously.
6
I'll show here one benefit of a windowed design and a trick to get the same benefit from Parks–McClellan.
For half-band, quarter-band etc. filters windowing retains the time-domain zeros of the scaled sinc function, which is the prototypical ideal low-pass filter. The zeros end up in the coefficients, reducing the computational cost of the filters. For a ...
6
The play similar role in those algorithms - the ability to forget the past and adapt to current reality.
In the LMS, the classic implementation has $ \alpha = 1 $.
Namely the optimal weights at any point are function of all inputs.
The Leakage factor allows to weigh the past differently in a damped manner which over times means the far past has ...
5
While I completely agree with Jason R's answer, I would like to add a few things that I consider important. First of all, it is a misunderstanding to believe that for least squares designs the transition band width is smaller than for equi-ripple designs. The width of the transition band depends on many design parameters but it is independent of the ...
5
Kalman filter is the best linear estimator regardless of stationarity or Gaussianity. Also in the Gaussian case it does not require stationarity (unlike Wiener filter). In the linear Gaussian case Kalman filter is also a MMSE estimator or the conditional mean.
5
I'm unsure if this is the answer you are looking for, but why not save and share $|H_i|^2$ in addition to the least squares estimates?
If you have $w_1^*=\frac 1 {|H_1|^2} H_1^t d_1$, $w_2^*=\frac 1 {|H_2|^2} H_2^t d_2$, and also know $|H_1|^2$ and $|H_2|^2$, you get the total least squares estimate with:
$w^*=\frac 1 {|H_1|^2+|H_2|^2} (H_1^t d_1 + H_2^t ...
5
For anyone who is interested, i coincidentally found a paper describing the method implemented in matlab's filtfilt.m. A link to the paper is attached. At least to my understanding matlab's filtfilt.m doesn't implement the Gustafson algorithm.
Sadovsky, P.; Bartusek, K: Optimisation of the Transient Response of a Digital Filter, Radioengineering Vol. 9, No. ...
5
The Jacobian is not computed numerically but analytically and then just evaluated. The frequency response of the IIR filter is
$$H(e^{j\omega})=\frac{b_0+b_1e^{-j\omega}+\ldots+b_Me^{-jM\omega}}{1+a_1e^{-j\omega}+\ldots+a_Ne^{-jN\omega}}=\frac{B(e^{j\omega})}{A(e^{j\omega})}\tag{1}$$
Now you need the derivative with respect to the filter coefficients:
$$\...
5
Let's solve a more general problem (Least Squares with Linear Equality Constraints):
$$
\begin{alignat*}{3}
\arg \min_{x} & \quad & \frac{1}{2} \left\| A x - b \right\|_{2}^{2} \\
\text{subject to} & \quad & C x = d
\end{alignat*}
$$
The Lagrangian is given by:
$$ L \left( x, \nu \right) = \frac{1}{2} \left\| A x - b \right\|_{2}^{2} + {\...
4
A lot has been said already, allow me to add some comments:
Kalman filters are an application of Bayesian probability theory, which means that "a priori information" or "prior uncertainty" can (and must) be specified. As I understand, this is not the case with traditional least-squares fitting. While observations (data) can be weighted with probabilities in ...
4
The idea is to represent all operation sing Matrices.
Once it is done, it is easy to solve the problems as a Least Squares problems.
The way to represent Convolution Operation using a Matrix is by Toeplitz Matrix.
For 1D it is pretty straight forward to do (Just pay attention to boundary).
So let's take the simple model in the comment:
$$ g = f \circ h + ...
4
Here's the way I think about a discrete Wiener Filter
Consider a sequence of observations $\mathbf{y} \in \Re^n $
Form a matrix from the input $\mathbf{x} \in \Re^{n+r-1}$ by shifting columns one sample each:
$$
X= \begin{bmatrix}
x_1 & x_2 & ... & x_r \\
x_2 & x_3 & & x_{r+1} \\
x_3 & x_4 & & x_{r+2} \\
... & &...
4
The question really depends on $ f \left( \cdot \right) $.
Yet in order to show how to use FFT we can even use 1D signals.
Let's rewrite the problem:
$$ \hat{x} = \arg \min_{x} \frac{1}{2} \left\| K x - b \right\|_{2}^{2} + \frac{\lambda}{2} \left\| f \left( x \right) \right\|_{2}^{2} $$
The derivative is given by:
$$ g = {K}^{T} \left( K x - b \right) + ...
4
One way to interpret the Tikhonov Regularization is using the Maximum A Posteriori (MAP) framework.
Lets' say we have a model of the form:
$$ \boldsymbol{y} = H \boldsymbol{x} + \boldsymbol{n} $$
Where $ \boldsymbol{n} \sim N \left( 0, {\sigma}_{n}^{2} \right) $, namely Additive White Gaussian Noise, and the prior knowledge about $ \boldsymbol{x} $ is $ \...
4
When dealing with applying a 2D convolution in frequency domain we have to take into account 2 things:
Extending the kernel to the dimension of the input data.
Dealing with the implicit periodic extension of the frequency domain element wise multiplication.
As you wrote in the comments, one way to do it is simple zero padding and fftshift(). Yet this might ...
3
One main difference is the cost function used in the two design methods:
Equiripple filters seek to minimize the maximum error between the desired filter response and the designed approximation.
Least-squares filters seek to minimize the total squared error betwen the desired filter response and the designed approximation.
These different strategies lead ...
3
If you want to solve for single value of $ \lambda $ in the model:
$$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{1} $$
Then you can use Coordinate Descent method which is the fastest and simplest and doesn't require any matrix inversion.
I have a MATLAB code for in my $ {L}_{1} $ Regularized Least Squares ...
3
If you look at the statement of the discrete-time Kalman filtering problem in Anderson & Moore (RIP), you'll notice something:
The covariances $R_k$ and $Q_k$ are time-varying.
Moreover, later in Chapter 3, they go on to prove the best linear estimator property for the Kalman filter in Theorem 2.1, and the proof does not appear to require the noise to ...
3
I don't understand the subscript $n$ notation, however, in the least squares problem that is given by:
\begin{equation}
{\bf{y}}={\bf{H}}{\theta}+\bf{n},
\end{equation}
where ${\bf{n}}\sim\mathcal{N}(\bf{0}, \sigma^2I_N)$ is a zero mean additive white Gaussian noise and $I_N$ is the $N \times N$ identity matrix, the maximum likelihood and the least squares ...
3
You can always augment the matrices to do so.
Let's assume the first model is given by:
$$ {y}_{1} = {H}_{1} * {\theta}_{1} $$
The second model is given by:
$$ {y}_{2} = {H}_{2} * {\theta}_{2} $$
The third model is given by:
$$ {y}_{3} = {H}_{3} * {\theta}_{3} $$
If we assume the number of parameters of the model are the same, namely $ {\theta}_{1}, {\...
3
It is basically an approach choice.
Inside the math is identical.
Usually, when doing Least Squares curve fitting, you're not looking for the Polynomial coefficients but a scaled version of them.
For instance, if you try to estimate Range, Velocity and Acceleration from Range Measurements using LS you need to scale the Polynomial Coefficients according to ...
3
I suggest this reference regarding the comparison between least-squares and Kalman filters :
Fundamentals of Kalman Filtering: A Practical Approach by P. Zarchan & H. Mussof
Especially Chapter 3 (Recursive Least-Squares Filtering) and Chapter 4 (Polynomial Kalman Filters).
In Chapter 4, the authors show that the discrete (time) n-th order polynomial ...
3
Since this is a linear model if you add noise which isn't centered (Non zero mean noise) your estimation will be good up to a bias term.
The easy way to do so is to remove the bias from $ y $ and solve.
In case you know the mean of the added noise, just remove it from your measurement and your model becomes the classic Linear Least Squares.
If you don't ...
3
Our system has the impulse response (why we changed to imaginary instead of real? :) ):
$$h(t)=\mathbb{Im}(Ae(iwt)e(-bt))=Ae^{-bt}sin(wt)$$
With the following structure (ref.) (discretized):
$$H(s)=\frac{Kw}{(s+b)^2+w^2}$$
Under matlab, you only need to use procest:
procest(data,'P2U');
Which models the same structure under the form:
$$H(s)=\frac{K}{(1+2\...
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