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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 \...
DSP Rookie's user avatar
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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 ...
Laurent Duval's user avatar
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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_{\...
David's user avatar
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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 ...
Hilmar's user avatar
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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 $\cos(-x) = \cos(x)$ ...
Hilmar's user avatar
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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 ...
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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 $...
Matt L.'s user avatar
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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: ...
Royi's user avatar
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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 ...
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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\| =\...
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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 ...
Royi's user avatar
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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 ...
Laurent Duval's user avatar
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". ...
Laurent Duval's user avatar

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