6
votes
Accepted
Derivative of $l_1$ norm
Apart from a sign error, your result looks correct. The term with $(1-a_{1k})$ should have a positive sign. Also note that $\text{sgn}(x)$ as the derivative of $|x|$ is of course only valid for $x\neq ...
5
votes
Accepted
How to prove this norm inequality?
Although the question could belong to SE.math, mastering inequalities for $\ell_p$ norms (for $p\ge 1$) or quasinorms (for $0<p< 1$), and their norm ratios and powers, is quite important in ...
5
votes
Accepted
For Schwarz inequality for 2 signals $s_1(t)$ and $s_2(t)$, equality holds if $s_1(t) = c\cdot s_2(t)$ ($c$ is constant). Does $c$ have to be real?
Schwarz Inequality for continuous-time Complex valued functions is given as follows:
$$\left|\int^{\infty}_{-\infty}f(t)^* \cdot g(t) dt \right|^2 \le \int^{\infty}_{-\infty}\left|f(t)\right|^2dt \...
4
votes
Accepted
Differences Between Two $ {L}_{1} $ Norm Minimization Schemes
The first equation you have is often called the Quadratic Problem, which through the use of Duality can be shown to be equivalent to the Basis Pursuit De-Noising (BPDN) given as:
$$ \arg \min_{\...
3
votes
Product of $L^2$-norms
But is there a better bound for a general filter response?
No. The argument about it $H$ being a minimum phase filter doesn't make a difference. In essence you are asking a math question on some ...
3
votes
Accepted
Is there a ''standard'' or commonly accepted reference for the phase of the most usual signals?
Not really.
There isn't even a standard for a sine waves signal. Two major phases are useful and they have even gotten separate signal names: sine and cosine. One is symmetric $\cos(-x) = \cos(x)$ ...
3
votes
norm not preserved after fft: MATLAB
This is expected. There's one of two things happening:
Depending on the math package, the FFT may be scaled by a factor of $N$ (this is the case in scipy.fft). So if the FFT were perfect then the ...
2
votes
Accepted
What is the meaning of $l_p$ norm in this model for sparse channel estimation?
The symbol ${||h||}_p$ refers to the p-norm of the vector. It is calculated as
$${||h||}_p=\sqrt[p]{\left|h_1\right|^p+\left|h_2\right|^p+...+\left|h_n\right|^p}$$
When they write ${||h||}^p_p$, it ...
2
votes
norm not preserved after fft: MATLAB
That's (again, like in your last question) numerical noise. As long as you don't do arbitrary precision arithmetic you will have to live with it. I wouldn't worry too much about a relative error of $...
2
votes
Accepted
Solving a Weighted Basis Pursuit Denoising Problem (BPDN) with MATLAB / CVX
A MATLAB code which implements the problem as defined and solve it using CVX is given by:
...
2
votes
For Schwarz inequality for 2 signals $s_1(t)$ and $s_2(t)$, equality holds if $s_1(t) = c\cdot s_2(t)$ ($c$ is constant). Does $c$ have to be real?
$c$ need not be real number. For the LHS you would get
$$
|\langle s_1,s_2\rangle| = |\langle s_1,cs_1\rangle |=|c|\,|\langle s_1,s_1\rangle |= |c|\|s_1\|^2
$$
For RHS, $$
\|s_1\| \times \|c s_1\| =\...
2
votes
Accepted
Is Sum of Absolute Value / $ {L}_{1} $ Norm of Differences Convex?
It is easier to prove it by using atoms.
The 1st atom is the Absolute Value function $ \left| \cdot \right| $ which is convex.
Then you have linear operation by the subtraction which is convex (Also ...
1
vote
For Schwarz inequality for 2 signals $s_1(t)$ and $s_2(t)$, equality holds if $s_1(t) = c\cdot s_2(t)$ ($c$ is constant). Does $c$ have to be real?
The answer about $c$ being complex has be given already. I would add information on "How is inequality established for complex signals?" Indeed, the inegality is very general, and works in spaces ...
1
vote
What is the meaning of $l_p$ norm in this model for sparse channel estimation?
To add on @Tendero, the expression $\sum_k x_k^p$ is sometimes called the "power $p$-norm" when $p\ge 1$. Most often, you can see mentions of the "squared $\ell_2$ norm" or "$\ell_2$ norm squared". ...
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