5
votes
What is the optimal filter?
The question is rather vague or inaccurate really; or I haven't understood it well.
I would start with saying "there is no such a thing as a 'best' filter (for all use)". A filter is optimal only ...
- 5,587
5
votes
How to derive the "well-known" solution to Unconstrained Array Gain?
A common way is to make use of the Schwarz inequality. First note that:
$$\frac{|w^Hd|^2}{w^HQw} = \frac{|w^HQ^{1/2}Q^{-1/2}d|^2}{w^HQw}$$
Using the Schwarz inequality on the numerator: $$\frac{|w^HQ^{...
- 1,664
3
votes
How to derive the "well-known" solution to Unconstrained Array Gain?
You can solve such a problem using the method of Lagrange multipliers. First note that maximizing the expression in your question is equivalent to minimizing the inverse function:
$$\min_{\mathbf{w}}\...
- 89k
3
votes
Accepted
Constrained Least Squares Filter Design
This can be defined as a non-linear optimization problem and can be solved by
Levenberg-Marquardt method or Trust Region method. MATLAB provides lsqnonlin to solve ...
- 3,183
2
votes
Constrained Least Squares Filter Design
Remark: The answer deals with the Non Negative Least Squares variant the OP asked for.
This is an interesting question I'd like to try solving it without any Toolbox based functions in MATLAB.
First ...
- 19.3k
2
votes
How to derive the "well-known" solution to Unconstrained Array Gain?
First, a sketch of the solution for the maximum SINR beamformer problem
$$
\text{max}_{\mathbf{w}} \frac{|\mathbf{w}^H\mathbf{d}|^2}{\mathbf{w}^H\mathbf{Q}\mathbf{w}}
$$
Start with writing down a ...
- 1,694
2
votes
Accepted
Complex Least Squares Approximation
Even though I think there is a lot of valuable general information in Stanley Pawlukiewicz's answer and in Royi's answer, I think that some specific questions have not been answered, at least as far ...
- 89k
2
votes
Optimal receiver for a binary system employing waveform $s=A\cos^{2}(\pi f_c t \pm \alpha)$
Contrary to what Marcus Muller says, the signal set is a BPSK signal set with a DC offset. The BPSK signal set is not an antipodal BPSK signal set unless $\alpha = \pi/4$; in which case the two ...
- 20.2k
2
votes
Optimal receiver for a binary system employing waveform $s=A\cos^{2}(\pi f_c t \pm \alpha)$
Awesome, your transformation into a sum of a constant and a cosine is wonderful!
Since both signals are cosines shifted from the origin by the same $\frac A2$, that doesn't chang their euclidean ...
- 29.6k
1
vote
Accepted
Minimizing Time Sidelobes with Pulse Compression
You could define a weight sequence $w(n)$ that weighs the error differently for different time indices $n$:
$$\epsilon^2=\sum_{n=0}^{N+K-2} w(n)\big|d(n)-z(n)\big|^2\tag{1}$$
Samples that are too ...
- 89k
1
vote
Complex Least Squares Approximation
Let's assume we have $ m $ points in the Fourier Domain (Discrete Fourier Domain) for our reference frequency response. We'll designate their values by $ y \in \mathbb{C}^{m} $.
Also we need a filter ...
- 19.3k
1
vote
Accepted
Extended Kalman Filter in mechanics, electronics and hydraulics?
It's the application which dictates whether you should use a linearized, extended, unscented Kalman filter or you can go easy with the simpler linear Kalman filter.
In other words, the extended ...
- 28k
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