Questions on the mathematical aspects of signal processing. Please consider first if your question might be more suitable for http://dsp.stackexchange.com/
3
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203 views
Is there a time-domain proof of Nyquist sampling theorem?
For a continuous-time signal $x(t)$ that is bandlimited (in the baseband) to $[-W,W]$, the standard proof of Nyquist sampling theorem proceeds in the frequency domain by examining the Fourier ...
2
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0answers
21 views
Mathematically inclined books on Signal Processing Theory
First off, i know this may seem off topic but i could not find help in signal processing communities so i was hoping there would be people here who both love mathematics and have interest in signal ...
2
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0answers
127 views
Lloyd-Max Quantizer Problem
Consider the function $z = x^2 + y^2$ , for $0 \leq x \leq 10$ and $0 \leq y \leq 10$. Take a regular discretization with steps $\Delta x = \Delta y = 0.1$ and its histogram as the probability ...
2
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0answers
83 views
MFCC - Why 13 Coefficients
Basically I am trying to computer a MFCC and wondered if you can help.
This is the FFT of 1 of the Frames (After I have multiplied the Hamming Window by the Mel Bank Filters) :
Here is the DCT of ...
2
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0answers
87 views
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 ...
2
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0answers
43 views
Estimating the input to a system from a system state
[ Cross-posted to: http://dsp.stackexchange.com/questions/3098/estimating-the-input-to-a-system-from-a-system-state-using-ekf ]
I have a system for which I have obtained a non-linear time-varying ...
2
votes
0answers
97 views
How to solve the recursive relation in Kalman filter?
I was wondering how to solve the Kalman filter's recursive
equation (also see the appendix at the end of this post) for the estimated state $\hat{\textbf{x}}_{n|n}$ at
time $n$, over discrete times ...
2
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0answers
158 views
An example of a “pathological” power-spectral density function?
Suppose that we are given a wide-sense stationary random process $X$ with autocorrelation function $R_X(t)$. Power spectral density $S_X(f)$ of $X$ is then given by the Fourier transform of $R_X(t)$, ...
2
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68 views
Scale invariance and $1/f^2$ power spectrum
In the paper
Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision
I read ...
2
votes
0answers
124 views
Singular Value Decomposition
I want to decompose an image $A$ using the Discrete Wavelet Transform and then find the singular values, $S$, such that $A=USV$. I will then do the same to another image such that $B=USV$. I will ...
2
votes
0answers
63 views
Means to classify data streams or compare similarity
Last year I converted some Matlab code into c to run on embedded Linux. I'm an engineer and normally shy away from maths, but this got me thinking about different ways to classify data or compare the ...
2
votes
0answers
90 views
Analysing an optics model in discrete and continuous forms
A discrete one-dimensional model of optical imaging looks like this:
$I(r) = \sum_i e_i P(r - r_i)$
Here, the $e_i$ are point light sources at locations $r_i$ in the object and $P$ is a point spread ...
1
vote
0answers
14 views
Checking if (discrete) signal is stationary, BIBO stable and
I have this discrete signal
y[n] = sum (x[k+1]h[k-1]), where k goes from -inf to +inf.
I need to check if this signal is stable, stationary, and if it's invertible, i need to find it's inverse ...
1
vote
0answers
45 views
How can I find the compact trigonometric Fourier series from these signals?
I've been stuck on this for a while, but how exactly would I go about calculating the compact trigonometric Fourier series for both of these signals? I have a general formula down for it, but I just ...
1
vote
0answers
18 views
Discrete time fourier transform of partial sum
I came across the following property of the DTFT:
$ \mathcal{F} \Bigg(\sum_{m=- \infty}^{n}x[m]\Bigg) = \frac{1}{1- e^{-j \omega}} X(e^{-j \omega}) + \pi X(e^{-j0}) \sum_{m= ...
1
vote
0answers
26 views
System stability
I need to do an absolute integral of my impulse response of my LTI system so I can find out if the system is stable or not. The general formula is:
$\int_{-\infty}^{\infty} \! |h(t)| \, \mathrm{d} t$ ...
1
vote
0answers
141 views
Projection Slice theorem getting rid of artifacts?
I have employed the fourier(projection) slice theorem in matlab. I have a 3D image, P(x,y,z) defines their pixel intensities at a given location int he image volume, it is discrete and uniform. I ...
1
vote
0answers
83 views
Worst-case error related to Cramer-Rao bound
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 ...
1
vote
0answers
34 views
every continuous signal being modelled as a function
Can every coninuous signal be modelled as a function, which then can be converted into a series of sine and consine functions with unique frequencies?
And let us say that we have some arbitrary ...
1
vote
0answers
192 views
Fast Walsh–Hadamard transform generalization for non-power-of-two orders?
I have to process vectors through a Hadamard matrix of order N.
If N is a power of 2, I can use the Fast Walsh–Hadamard transform; but if N is not a power of two (for instance, N=12), it is not ...
1
vote
0answers
199 views
How to Find Phase Lead/Lag
I have the transfer function $$ H(s) = \frac{s+1}{0.1s+1} $$
I apply the Bilinear Rule with a sampling time T =.25 sec to the transfer function and get a z-domain representation of
$$H(z) = ...
1
vote
0answers
254 views
How can I use the time-frequency uncertainty principle?
I have a signal composed of the summation of a set of sine waves of different frequencies. The amplitude of these sub-signals can change so many times a second.
I have been told that, if I want to ...
1
vote
0answers
75 views
Signal filtering and moving averages
Background
Given a signal $x_n$ for $n=1,2,\dots$ we can consider its filtered values:
$$y_n = \frac{b(L)}{a(L)}x_n$$
where $a(L)=a_0 + a_1L + a_2L^2 + \cdots + a_nL^n$ (similarly for $b$) and $L$ ...
1
vote
0answers
28 views
Name this concept: Comparing equal sized vectors vs. comparing features
If you obtain a vector by taking $n$ discrete samples over some underlying function, then it's easy to compare that vector with another of the same size. With a bunch of $n$-dimensional vectors, you ...
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0answers
60 views
Solving multiple phase angles for multiple equations
I have several equations and each have their own individual frequencies and amplitudes. I would like to sum the equations together and adjust the individual phases, ...
1
vote
0answers
190 views
FFT signal post processing
This is more a "post a suggestion" topic rather than a question. And thank you if you are willing to read this whole.
I've been studing the code in the Nvidia Cuda SDK regarding how to operate a ...
1
vote
0answers
465 views
Partial differentiation of vector to find Jacobian (extended Kalman filter)
I am working through some coursework on self-tuning control and part of one of the questions requires the use of the extended Kalman filter for joint parameter and state estimation. For completeness, ...
1
vote
0answers
89 views
What is the difference between various kalman filters?
What is the difference between additive and multiplicative kalman filters, as well as some other kinds?
I'm also looking for reference texts and articles that describe the algorithms, so ...
0
votes
0answers
32 views
Find the magnitude and phase spectra of H(s) = 1/s+1
Find the magnitude and phase spectra of
H(s) = 1 / s+1
I have no clue what this is asking. My teacher kind of left me hanging could someone help me get started.
0
votes
0answers
32 views
fourier series coefficients for power signal
Given a signal
$$x(t) = -5 +3\cos(3000\pi t-\pi/6) -3\sin(4500\pi t+\pi/2),$$
a) Determine the Fourier series coefficients $X[k]$ for $x(t)$ (do not do any integrals).
b) Suppose that $x(t)$ is ...
0
votes
0answers
24 views
Calculation of the error function.
I have the next two signals:
$X(t)$ and $G(t)$ and a random process $Y(t)=G(t)X(t)$ where $X(t)$ and $G(t)$ are wide sense stationary with expectation values: $E(X)=0, E(G)=1$.
Now, it's also given ...
0
votes
0answers
18 views
is there a Kalman filter for distribution function?
The standard Kalman filter uses a series of measurements observed over time, to decomposite the signal and noise.
However, when I'm modeling the distribution (pdf or cdf) of a variant, is there a ...
0
votes
0answers
75 views
Frequency estimation by Prony's method
My question is related to Prony's method, from Internet there is direct link for this method, but also i would like to show you method
Now i am a little confused, I have read about Pisarenko ...
0
votes
0answers
21 views
Pre-Emphasis of a signal
I'm trying to describe the process of Pre-Emphasis (of a signal) in my equations, but I don't know whether or not this makes sense.
$Y[n] = X[n] - 0.95 \cdot X[n-1]$
Where Y = pre-emphasis after, X ...
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0answers
88 views
Looking for all possible combinations: Can we find combinations of “triplets with accompanying pitch” in any arbitrary SOAE spectrum?
By the cochlea: Differentiating and squaring of tone combinations, is one of the main characteristics of the "Heerens / de Ru" auditory model Applying physics makes auditory sense
based on the correct ...
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0answers
53 views
Two variable PCA using gradients
Let $x$ and $y$ be two random variables. Using principal component analysis (PCA), I can find a linear projection making the two variables uncorrelated. PCA solves this problem through an eigenvalue ...
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0answers
67 views
How to make normalized cross correlation robust to small changes in uniform regions
the problem is described below:
Given 2 sets of data: A= { 91 87 85 85 84 90 85 83 86 86 90 86 84 89 93 87 89 91 95 97 91 92 97 101 101 },
B = {133 130 129 131 133 136 131 131 135 135 133 133 133 ...
0
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0answers
179 views
Experiment with a Fourier series with missing fundamental. Mathematical explanation?
On this math.stackexchange
on url Can the differentiating and squaring process in the cochlea explain a reported dichotic stimulation experiment?
I asked the question: Can the differentiating and ...
0
votes
0answers
53 views
Decorrelation KL transform property
Definition: The KL transform of vetor $\boldsymbol{u}$ is defined as
$$\boldsymbol{v}=\Phi^{*T}\boldsymbol{u},$$
where the columns of matrix $\Phi$, $\Phi_k$, are the eigenvectors of the ...
0
votes
0answers
61 views
Explain the existence of limit in Persistence Excitation — mostly zero and non-existent?
Definitions
Persistence Excitation on page 121 here or shortly here and here. A signal is PE if this limit exists
$$r_u(\tau)=\lim_{N\rightarrow\infty}\frac 1 N \sum_{t=1}^{N} ...
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0answers
76 views
What is the purpose and usage of convolution?
I am curious of what the purpose and usage of convolution are. Why is convolution created? In layman's term (and in mathematical term), what defines convolution?
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0answers
81 views
Autocorrelation function of a vector
I know this could be a stupid question but I just want to confirm. Let say I have a signal $X$ of dimensionality $K\times N$. Can we calculate the autocorrelation function of $X$? In my opinion, the ...
0
votes
0answers
37 views
Vestigial Filter, find modulated signal?
I have been stuck on this question for a while now. It has to do with vestigial sideband.
I wasn't sure if I should be dividing $H(\omega)$ graph values by 2 because only the positive side of the ...
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votes
0answers
29 views
Prove that sinc function is desired interpolator when given condition holds.
The process $X_t : t \in \mathbb{R} $ is bandlimited with $S_X(\omega) = 0$ for $|{\omega}| > \omega_c$. Show that if:
$X_t = \sum_{n=-\infty}^{\infty} X_{nT}p(t-nT) $ $(m.s.)$
where ...
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0answers
58 views
Wavelet Transforms
I don't know as much as I would like to about Fourier analysis and I know almost nothing about wavelets. So just have a few conceptual questions to determine whether I should pursue their study or ...
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0answers
26 views
Sparsity vs Rank in Compressed Sensing
Is there a big conceptual difference between the Sparsity and Rank of a matrix? Is Sparsity the term used when referring to a signal and Rank used when referring to a matrix?
I am thinking that a ...
0
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0answers
94 views
How to find Impulse response and Time invariance of a given system?
How can I find the if the system is time invariant? This is the question regarding signals and systems.
And how can I find the impulse response for the given signals?
x1[n]={-1,2,1} <----> ...
0
votes
0answers
129 views
Fourier Series Coefficients for Signals
The question is:
We specify the fourier series coefficients of a continuous-time signal that is periodic with period 4. Determine the signal x(t).
$a_k=\begin{cases}
0, & k=0\\
...
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0answers
47 views
Fourier Series for Signals
So my question requires the picture of a graph so here it is. I'm trying to do part $(a)$ and I have worked out all the way up to this part:
$C_k=\frac{1}{2} \int_T \! te^\frac{-j2\pi kt}{T} \, ...
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0answers
63 views
convolution related question: shifting?
i was wondering, for convolution, when we do the graph shifting for h(t-tou) we flip the graph on the y axis and then if t = 0.5, then shouldn't we shift the graph left by 1/2? In the examples I am ...
