Questions tagged [math]
The math tag has no usage guidance.
124
questions
0
votes
2answers
73 views
MLE formulation — confusion regarding the terms in the equation (Part1)
If $v \sim CN(0,2\sigma^2_v)$ is a circularly complex Gaussian random variable which acts as the measurement noise in this model
$$y_n = x_n + v_n \tag{1} $$ where $x \sim CN(0,2\sigma^2)$, then is ...
0
votes
0answers
70 views
What kind of math is introduced in this answer related to DSP?
What kind of math is introduced in this answer:
https://dsp.stackexchange.com/a/40225/30897
It smells like standard linear algebra course (which I haven't studied, yet), but is it only that?
Thanks!
0
votes
1answer
107 views
How does the assumption that symbols are equi-probable hold
If I draw a number uniformly between zero and one, what is the probability that they are equal? Mathematically, it should be zero but I don't recall why? Can somebody please help in explaining why the ...
1
vote
2answers
705 views
Autocorrelation of a Shifted Sequence
Suppose I have a sequence $x[k]$ with $\mathcal{Z}$-transform
$$ X(z) = x_{0} + x_{1}z^{-1} + x_{2}z^{-2} + \ldots + x_{N-1}z^{N-1}$$
I know that for real-valued $x[k]$ the $\mathcal{Z}$-transform ...
-1
votes
1answer
138 views
BIBO Stability of a piecewise function
I have a function $y$ defined as
$$
y(t) =
\left\{\begin{array}
~t~ \mbox{ where} |t| \le 3\\
0~ \mbox{ otherwise}
\end{array}
\right .
$$
With a system defined as
$$G(t) = ty(t), $$
is it BIBO ...
1
vote
1answer
315 views
Drawing the modulus from a Transfer function
As you may guess from my other questions i am a student so pardon me any ignorance and guide me to the truth. Thanks
In this exercise from an exam i was given that transfer function with poles and ...
0
votes
1answer
81 views
How is this was derived? DTFT for symmetric pulse
do not understand how the middle expression was derived from above expression,
after all summation\multiplication i get:
$$\frac{e^{-jwN} + e^{jwN} - e^{-jw(N+1)} + e^{jw(N+1)}}{2 - e^{-jw} - e^{jw}...
3
votes
2answers
401 views
Mathematical relationship between highpass and lowpass filtering
Let $g, h_{HP}, h_{LP}: \mathbb{R} \rightarrow \mathbb{R}$ and $G, H_{HP}, H_{LP}$ denote their continuous Fourier transforms under the Fourier operator $\mathcal{F}$. Let $*$ denote the continuous ...
1
vote
1answer
63 views
Should zero element be considered to calculate mean when smoothing spectrogram by window?
My (mass spectrometry) spectrogram does not measure the intensity constantly. I obtain the spectrogram, for example,
...
0
votes
1answer
257 views
What is stochastic differential equation and its need?
A white noise process can be simulated using the Matlab command randn(). The numbers will be drawn from a Normal distribution of zero mean and variance 1.
Is the ...
1
vote
2answers
687 views
Closed form expression for $\sum_{k=0}^{n} \alpha^{-k}$?
I have seen this sum a couple of times now in the textbook and in the problems book but I don't know how to solve it. I know this might be more suited for math.stackexchange but since this is a fairly ...
3
votes
3answers
949 views
What is the meaning of this notation?
I have seen this notation in the book Schaums-Outline a couple of times now and I can't quite understand it. I'd very much appreciate if someone explained what it means. I am thinking that it means ...
1
vote
0answers
201 views
What is the difference between entropy and entropy rate and which one is important in information theory? [duplicate]
By entropy I understand the uncertainty or randomness. But if the uncertainty is high then what is its implication and advantage in transmitting information or as an information source or is high ...
0
votes
1answer
2k views
Z-transform of an impulse signal in discrete time
am trying to compute the Z-transform of the following signal
\begin{equation*}
x\left[n\right]\:=\:\sum_{k=-\infty \:}^{\infty \:}\:\delta \:\left[n-k\right]
\end{equation*}
so I thought it would be
\...
-1
votes
1answer
36 views
Why GCD(1+D, 1+D^2) = 1+D?
I'm reading Digital Communications, Proakis, and in order to classify an encoder as catastrophic or noncatastrophic I need the GCD of the elements in the generator G(D).
I don't really know what kind ...
8
votes
3answers
2k views
Replacing “e” in Euler's formula with another number
Does Euler's formula remain valid if we use any real number other than the constant $e$? For example replacing $e$ with 5 would make the formula look like this: $5^{it}$.
I tried this idea in ...
1
vote
1answer
54 views
Proving Convergence of a limit related to Cross-Spectrum Estimation
Proving asymptotic covariance of Cross-Spectrum Estimation I came across this mathematical problem:
Consider two constants $M,L$ such that $M\to \infty, L \to \infty \hbox{ and } \frac{L}{M} \to 0$, ...
2
votes
0answers
172 views
Career Question: As a person who likes math is signal processing a career that's worth it?
I like high level math and I love deep theory based math.
In industry will I use high level math and theory to implement my own algorithms?
I love high level math but I do not love probability (...
-1
votes
1answer
80 views
Is it mathematically possible to compute Fourier transform without comparing the signal?
If I'm correct, Fourier transform lets me know the magnitude and phase of any given frequency in a signal.
I have a sampled signal and I want to compute its Fourier transform (any frequency) without ...
0
votes
1answer
41 views
Help with Determinants [closed]
Okay so i understand the first det, which is why i got the correct solution. However for the second det, im not sure how to solve it since because of the different coefficents (0,2,4) and i tried of ...
1
vote
2answers
102 views
What is this theorem in this formula?
I just read a paper yesterday and there is confusion:
I want to know what is reconstruction fidelity term and the theorem of changing the formula from upper to lower one. Is this theorm from ...
-2
votes
0answers
42 views
Directional derivative [closed]
I know that D ā v f = ā f ā
ā v Dvāf=āfā
vā is the directional derivative. My question is why must the vector ā v vā be a unit vector? I am sure there is an obvious answer, but my book doesn't really ...
-1
votes
1answer
42 views
What is a “functional”? [closed]
I have read the wiki page (https://en.wikipedia.org/wiki/Functional_(mathematics) but it is not helping.
I understand what a regular function is
input > do something to that input > output.
but ...
0
votes
2answers
105 views
Help in proper notations and mathematical formulation
For a given series, $S = \{-1,0,-2,1,etc\}$. The number of elements in $S$ is $N = 100$. Each $s_i$ belongs to a alphabet from a finite alphabet set $\mathcal{A} = (a_1=0, a_2=1, a_3=2, a_4=3, a_5=-1,...
11
votes
5answers
957 views
Mathematical question that comes out of using bilinear transform
So this is related to the Cookbook and I tried solving it maybe two decades ago, gave up, and was reminded of the unsolved problem. But it's pretty damn straight forward, but I still got slogged down ...
1
vote
1answer
114 views
How to prove this norm inequality?
$$
\left( \sum |z_i|^a \right)^b \geq \left( \sum |z_i|^b \right)^a, \quad \forall a\in [1,2], b\in [2,+\infty)
$$
Actually this is equivalent to
$$
\|z\|_a \geq \|z\|_b \quad \forall a\leq b
$$
0
votes
1answer
217 views
Laplace transform of averaging operator
I am studying dc-dc converter now. I got a problem with Laplace transform of the averaging operator as in the image below.
Can anyone help me derive the Laplace transform result $G_{av}(s)$ as in the ...
1
vote
1answer
53 views
Notation confusion — What is the correct operator for computation of the log-likelihood expression for complex valued data?
This question is an extension of another question of mine asked earlier here Help in understanding if the maximum likelhood estimation is working properly
In that question the inputs were real valued....
3
votes
3answers
136 views
Weighted Sum of Auto Correlation - Lower Bound
Given a vector $ v $ with elements $ {\left\{ {v}_{n} \right\}}_{n = - \infty}^{\infty} $ and denoting $ {v}_{n}^{\left( k \right)} = {v}_{n - k} $, namely, a shifted vector by $ k $ elements (Mind ...
0
votes
1answer
96 views
Unable to understand how the paper simplifies the covariance matrix - Kalman filter
The paper Convergence Analysis of the Unscented Kalman Filter for Filtering Noisy Chaotic Signals
presents the convergence analysis of Unscented Kalman Filter
download
http://www.eie.polyu.edu.hk/~...
1
vote
2answers
193 views
What does the $n$ do in $1/(j \omega)^n$
I'm currently working through a research paper on beam forming. In this paper a magnitude compensation is introduced to compensate for frequency dependency. Due to other calculations low frequencies ...
3
votes
1answer
146 views
Independence of Noise at Each DFT Output
My math may be a little rusty, so I would like confirmation or correction or my calculations here.
Given white noise samples, $x_i$, which are IID and zero-mean, and variance $\sigma^2_x$. I want to ...
0
votes
1answer
81 views
Hi, If I really want to focus on the mathematics of signal procesing, Should I do a Masters in EE or Math? [closed]
I'm an undergraduate electrical engineering student who has an interest in mathematics of signal processing. I want to be able to go to a graduate school where I can focus on more the mathematics and ...
2
votes
1answer
51 views
Splines and Gaussians [closed]
Higher-order splines all mostly look like Gaussians. I wonder how higher-order splines (from 3rd order onwards) and Gaussians differ. How can one differentiate them?
8
votes
1answer
5k views
Phase correlation vs. normalized cross-correlation
I asked this over at Mathematics Stack Exchange, but since this sort of lies on the border of the questions normally asked over there and the questions you see over here I'll ask it here as well. (As ...
-1
votes
1answer
348 views
FFT: Linear to Logarithmic (TM4C, CMSIS & NeoPixels/ws2812)
I'm trying to perform an FFT using the CMSIS libraries.
My FFT_Output[] provides 1024 bins of data as expected. The issue is that it's linearly spaced and I want to change it to a logarithmic scale ...
4
votes
2answers
2k views
State Estimation by steady state Kalman Filter
I am working with a discret Kalman filter on a System
$x_{k+1}=A_k x_k+B_k u_k+\omega_k$
$y_k=C_k x_k+\upsilon_k$
$E[\omega_k\omega_k^T]=Q$
$E[\upsilon_k \upsilon_k^T]=R$
I have estimated the ...
0
votes
2answers
838 views
Polar form of the Fourier transform of $\sin(t)$
I came across the Fourier transform of sin(t).
It ends up being a purely imaginary (dirac delta) impulse pair. But when considering the frequency domain representation of a signal, we consider the ...
2
votes
1answer
95 views
Derive chernoff bound for $\mathrm{erfc}(x)$
I'm trying to derive the Chernoff bound $\mathrm{erfc}(x) \le \exp(-x^2)$, by first showing:
$$\mathrm{erfc}(x) = \frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\exp\left(-\frac{x^2}{\cos^2\theta}\right)d\...
0
votes
1answer
247 views
Shape of Kaiser windowing function FFT response
I'm trying to determine the shape of the FFT response of a signal which had a kaiser windowing function applied to it. What I mean by this is if the shape is a normal bell curve or a Lorentzian ...
0
votes
2answers
1k views
Convolution by IIR filter, a case where circular convolution is allowed?
The question is, how to attain the frequency domain representation of any signal (which is finite, non-unit length) convolved with an IIR filter.
The correct answer in my opinion is that there is in ...
1
vote
2answers
119 views
How to make the $\ell_2$ norm of all columns and rows of an $n \times n$ matrix equal to $\sqrt{n}$?
I have an $n \times n$ matrix and I would like its columns and rows to have $\ell_2$ norm equal to $\sqrt{n}$. Is this possible?
0
votes
1answer
102 views
Advantages of Splines in Compressed sensing
I am working in compressed sensing, basically in ECG reconstruction
Normally we were using biorthogonal wavelets for reconstruction.
Someone advised me to use exponential spline basis instead of ...
2
votes
0answers
318 views
Laplace Transform of $-e^{-at}u(-t)$
I have found a problem in applying Laplace Transform to $-e^{-at}u(-t)$
I am doing these steps:
$$ = - \int_{-\infty}^{+\infty} e^{-at}u(-t) e^{-st}dt$$
$$ = - \int_{-\infty}^{0} e^{-at} e^{-st}dt$$...
1
vote
1answer
315 views
How to efficiently calculate points of intersection of a straight line and a contour?
Ex.
I have an image, let's say 100x100 pixels with some shape on it. And a set of straight lines that are passing through origin located at some point of image, for example origin located at point (...
9
votes
4answers
6k views
Looking for an arcsin algorithm
Does anyone have a simple algorithm for computing a reasonably accurate arcsine? By "simple" I mean some sort of polynomial that requires <= 5 multiplies per output sample. And by "reasonably ...
-1
votes
3answers
143 views
How to interpret the notation of a transfer function
Probably a stupid question, but I'm slowly making my way through Julius Smiths' filter book. The maths is slightly defeating me at points.
This function $H(e^{j\omega t})$, why is it a function of ...
2
votes
1answer
296 views
SIFT: why it is invariant to affine tranlsation?
I am studying SIFT. I am a little confused in some of Lowe's algorithm.
In order to have a descriptor invariant to translations like scaling, we must get rid of the scaling factor in LoG (Laplacian ...
1
vote
2answers
12k views
How to Calculate Gaussian Kernel for a Small Support Size?
How to calculate the values of Gaussian kernel? I think I understand the principle of it weighting the center pixel as the means, and those around it according to the $\sigma$ but what would each ...
0
votes
1answer
63 views
Getting a part of the invert of Si function
I'm working on an algorithmus that sould need the inverse of the SI function.
since this last once isn't oviouse to calculate, I'm trying to get only the parobla part from the SI function, to be ...
