The signal-analysis tag has no usage guidance.

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### Finite Parseval Frame

Assume that $G$ is a finite vector space over a finite field with order $|G|$. (For example, $G=Z_p^k$). Assume that $\{f_n\}_n$ is a Parseval frame for $l^2(G)$. Can we say that the sequence $\{f_n\}...

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### Boundary behavior of harmonic function on the square

Is there a constant $C$ such that if $u:[0,1]^2\to \mathbb{R}$ is harmonic with $u\in L^\infty(\partial [0,1]^2)$ (if you prefer you can also assume $\|u\|_\infty = 1$ on the boundary and $u$ smooth ...

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### Finite realization of irrational transfer functions

In the field of digital signal processing, linear time-invariant systems play a distinguished role. These are the systems for which there exists an impulse response, a function $h:\mathbb{Z}\to\...

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### When is a mapping the proximity operator of some convex function?

Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ?
That is, given $p : ...

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### Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question:
What's the probability distribution of a deterministic signal? (functional integrals in probability theory)
Clearly my question looks at the same time fairly ...

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110 views

### Discrete Wavelets

I am looking for research that has been done in Discrete wavelets. Let me be specific as Google doesn't give me what I want when I say "discrete wavelets". I don't want countable basis for $ L^2(\...

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445 views

### Weighted Median Filtering

Let's begin with a little review of unweighted median filtering.
Suppose I have a list of $N$ real-valued numbers, $x=x_1,...,x_N$. Let $m_i$ be the median of $K$ consecutive values: $m_i=$ median$(...

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### In which sense Daubechies wavelets converge to the Shannon wavelet?

My question is about wavelets theory. Consider $\psi_n$ the Daubechies wavelet of order $n \geq 1$; that is, the Daubechies wavelet with $n$ vanishing moments. We also define the Shannon wavelet in ...

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### Why is it important to know if a frame is a Parseval frame?

I understand that a Parseval frame is one in which both upper and lower frame bounds equal 1. What's the main advantage to having this be the case? Or, more specifically, if I'm constructing a frame ...

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233 views

### What kind of role has Functional Analysis played in Signal Processing? [closed]

Does it serve mainly as a narration or is there any substantive consequence which might not be derived without tools of functional analysis?

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396 views

### Correlation between two continuous-time stochastic processes

Consider two continous-time stochastic processes $\{A(t)\}_{t \ge 0}$ and $\{B(t)\}_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For ...

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### Linearizing a multifrequency signal

I have a component of a signal
$$\sin (k\omega_1t + \ell\omega_2t)$$
with wavenumbers $k, \ell \in \mathbb{Z}$, frequencies $\omega_1, \omega_2 \in \mathbb{R^+}$ and time $t \in \mathbb{R^+}$. (This ...

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### Multidimensional Filters

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then $...

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### Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T

Given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ instants
$$
\mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T
$$
at Nyquist ...

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### Wavelet transform stability to deformations

I've come across the following claim in a paper of Mallat:
"High frequency instabilities [of a signal representation] to deformations can be avoided by grouping frequencies into dyadic packets in $\...

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### Analogous filter to Kalman filter that maximized mode instead (as opposed to minimizing variance)

I may have a potential application where maximizing the mode (as opposed to typically minimizing the variance) would be useful for state estimates. The situation may arise from skewed distributions ...

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### Relationship between Fourier series & DFT

Sources like http://www.dsprelated.com/dspbooks/mdft/Relation_DFT_Fourier_Series.html explain the equivalence between FS and DFT.
However, isn't there a flaw? When I integrate over the continuous ...

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391 views

### A palindromic polynomial and its derivative have the same number of zeros outside the unit circle. Reference?

I am trying to find the original reference for a lemma attributed to Cohn (as in Schur-Cohn method):
Let $A(z)$ be a palindromic or skew-palindromic polynomial, and denote its derivative by $A'(z)$...

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### Spatial and temporal covariance matrices

Suppose $(x_i(t))$ is a $n$-dimensional time-series, where $t$ is an integer between $1$ and $T$ (time is discrete) and $i$ an integer between $1$ and $n$, and I assume $n<T$. From this time-series,...

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### Bandwidth approximation for a nonlinear problem

Can anyone please help me with this problem.
I must let you know from the beginning that it's not an easy one.
"Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ ,
...

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108 views

### Signal model classification between two possbile candidates

How to decide the most possible signal model between two model candidates besed on the received signal vector?
Assume the received signal vector is $y$, the possible signal model candidates could be:
...

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590 views

### Signal Processing reference for pure mathematician

Before giving a more detailed question below, the basic one is: can anyone recommend a good signal-processing reference which would be maximally readable by a pure mathematician (who nevertheless ...

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### Worst-case error and Cramer-Rao Lower Bound - is there any mathematical relation between them?

I would like to understand the relation (if any) between the Cramer-Rao Lower Bound of estimation theory and the following simple definition of "reconstruction accuracy" which doesn't use any ...

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### Understanding Discrete Cosine Transformation

I'm currently working on some software and a key component is 2D DCT. But my question is more general, as I'm trying to understand the DCT in general, let's say from engineers point of view.
For start,...

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253 views

### Most orthogonal lattice basis

Let $n \in \mathbf{N}$ be a natural number and $v_1,\cdots,v_n$ a set of basis vectors in $\mathbb{R}^n$. How does one find the matrix $g \in \mathbf{GL}_n(\mathbb{Z})$ orthogonalizing these best ...

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### Hammerstein integral equation with inverse of the solution

In signal processing theory I found this integral equation that I recognized to be of Hammerstein type:
$$u(t)-\int_{0}^{1}d\phi cos(\omega t+\phi)\frac{1}{u(\phi)}=0$$
Unfortunately the solution ...

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392 views

### Estimation of Temporal Correlation of Signal

I have a signal and i'd like to estimate its temporal correlation.
My limited understanding is i should compute the PSD by estimation using a parametric model such as AR.
However, i'm not quite ...

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### How Does My Radio Work?

Bear with me for a moment while I invoke the real world; the main question at the end is purely mathematical.
I live in an area with $n$ AM radio stations and $m$ FM radio stations.
AM station ...

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### Convolutive noise removal

I have the time domain signal
$$
u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t)
$$
where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is ...

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### Stochastic process with Bessel function autocorrelation. (Rayleigh (Jakes) fading for radiowave propagation)

Have the following stochastic process $f(t)$ been studied in mathematics ?
It is stationary, Gaussian, $f(t)-$complex independent Gaussians $N(0,1)$.
The autocorrelation is given by the
zero-order ...

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### Signal Analysis/Processing Textbook

Can anybody recommend me a decent Signal Analysis/Processing textbook. If possible one that deals a little with MATLAB. I have an little knowledge of Real Analysis and fourier transforms. Wavelets i ...

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### Generalized fourier transform and convolution?

Let $a(t)$ and $b(t)$ be two equal length sequences indexed by time index $t$.
We know that $a(t) * b(t)$ corresponds to $A(\omega) \odot B(\omega)$ in the frequency domain where $A(\omega)$ and $B(\...

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### Digital Filters [closed]

Can somebody help with the constructing filter by amplitude and phase spectrum? Is it possible?
I try to google it, but unsuccessufully.
I have some thoughts about solving it by system of linear ...

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169 views

### Levelset of band limited function

In a practical application problem I encountered such a question: Given a subset of a N*N Cartesian grid, how to determine if it is a sublevel-set of a band-limited (discrete) function? Here band-...

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### Estimating the derivative of a noisy, non-uniformly sampled function

I have some trading data in the form of (exchange rate, volume, time) tuples. I'm trying to estimate the rate of change of the exchange rate. Of course the trade data is non-uniformly sampled.
Also, ...

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### Estimate on zero-crossings of band passing signals

I'm recently encountering such a problem, which I think is very intuitive but I have no background knowledge on this field:
Given a signal with certain frequency distribution, e.g. we know that the ...

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### Rotationally-Invariant 2D Discrete Transforms

I'm interested in 2D discrete transforms (such as discrete wavelet transforms, Curvelets, Ridgelets, Beamlets etc.) that operate on a discrete unit disk and:
Are invariant to rotations only
Output a ...

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763 views

### Interpolating Wavelet Coefficients

Hi! I was instructed via reddit that this place would be the best place to post this question. Fingers cross you can help...
Ive been writing some code to get rid of noise "spikes" in a signal. I'm ...

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2k views

### Complete formulas book for Communication System engineer

I'm looking for a formulas book.
I'm currently student in Communication Systems and we have several courses involving mainly complex analysis, fourier analysis, signal processing, information theory ...

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### Are there interesting problems involving arbitrarily long time series of small matrices?

Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, ...

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### Periodicity of data

I have some real data (data packets in a router). When I plot it I can see there is a clear periodicity on the dataset (24hours+-).
But how can I discover the periodicity of the data without being by ...

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### How would you compute that “average” ?

I created a DJ-ing application that allows you to mix your MP3s with a real turntable.
So I generated an audio timecode to burn on a CD, left channel is the absolute position, right channel is a ...

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### What is a rigorous statement for “linear time-invariant systems can be represented as convolutions”?

In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of ...

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### Decomposing a 1-d signal into arbitary basis functions

Hi all,
The short-time fourier transform decomposes a signal window into a sin/cosine series.
How would one approximate a signal in the same way, but using a set of arbitrary basis functions instead ...