Questions tagged [math]
The math tag has no usage guidance.
136
questions
1
vote
1answer
105 views
Intuitive understanding of frequency
I know that $f = \frac{\text{cycles}}{\text{unit of time}}$ and I know that period $T = \text{time it takes to complete one cycle}$.
So if we take the definition
$f = \frac{1}{T} = \frac{\text{1 ...
1
vote
2answers
56 views
Rearranging exp terms
I have some problems rearranging exp terms to receive a cos or sin. I guess the solution is silly but I‘m not able to spot my mistake.
I know the solution (since i looked it up with wolfram alpha) ...
0
votes
1answer
29 views
Equation Derivation: Help with intermediate steps
As per the attached image above, I don't know how the equation in the first line transformed to the form in second line.
I tried solving it by multiplication but the result was different.
5
votes
1answer
735 views
Laplacian of Gaussian Approximation and Gaussian Blur as the Solution of Heat Equation
While I was reading SIFT paper(Lowe, 2004), I came across the method that he apply "heat diffusion equation" to Gaussian function to derive that
$$ \frac{∂G}{∂σ}
= σ∇^2G
$$
I searched Wikipedia and ...
0
votes
1answer
118 views
How to measure the distance between two circle-like-lines over 360 degree
I am looking for a slim and with that quickly programmable image processing strategy to measure the distance between two circles over 360°. It is important to notice that the circles are not perfect, ...
1
vote
1answer
201 views
How to prove Line Spectral Pair coefficient properties?
I am trying to understand the definition of LSP coefficients from http://www.dspcsp.com/pdf/lsp.pdf.
We take the polynomial $a(x)$ of degree $M$ from the denominator of the LPC system function, and ...
0
votes
1answer
79 views
Explicit a succession using z inverse transform
Is it possible to explicit $y(n)$ of this mathematical succession in recursive form using z inverse transform?:
$ y(0) = 1 \\ y(n+1) = 2y(n) + 3 $
I can't write ...
0
votes
0answers
39 views
Relationships between $f(t)$ signal and $f(t) \times \sin(\omega t)$ signal
Let's assume that there is a signal $f(t)$, $0\le t \le T$. Then, is there any special meaning of the signal $g(t) =f(t) \times \sin(\omega t)$? Like AM transmission. What is shown/heard when we play ...
1
vote
0answers
423 views
NMSE of complex numbers
Actually I am doing an RF measurement and this includes phase and power. i generate a list of powers ranging for example from (-50 to 40) dBm, while the phase rotate to make it for example spiral ...
0
votes
2answers
223 views
MLE parameter estimation -- confusion regarding some terms in the pdf of complex normal r.v (Part 2)
This question is based on the application of the pdf which was an earlier question of mine asked here Confusion regarding pdf of circularly symmetric complex gaussian rv
If $v \sim CN(0,2\sigma^2_v)$ ...
0
votes
2answers
78 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
71 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
122 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
935 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
166 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
394 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
99 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
463 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
81 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
290 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
966 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
1k 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
305 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
3k 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
42 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
55 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
182 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
53 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
107 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 ⃗ v f = ∇ f ⋅ ⃗ v Dv→f=∇f⋅v→ is the directional derivative. My question is why must the vector ⃗ v v→ be a unit vector? I am sure there is an obvious answer, but my book doesn't really ...
-1
votes
1answer
44 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
107 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
1k 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
137 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
2answers
272 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
54 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....
5
votes
3answers
153 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
101 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
194 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
147 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
86 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
6k 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
382 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
973 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
104 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
279 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 ...
