Questions tagged [math]
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0
votes
2answers
39 views
Resolution of Discrete Fourier Transform is 1/T - Mathematical proof?
In many articles I see that the frequency resolution of the Discrete Fourier Transform (DFT) equals Fs/N where Fs is the sampling rate and N is the total number of samples. Fs/N is equivalent to 1/T ...
0
votes
0answers
29 views
Power, energy and amplitude spectrum from fft in MATLAB - scaling
I've looked at previous answers and none of the answers seem to agree (perhaps I'm missing something obvious):
I would like to know what the correct scaling to use is for
amplitude,
energy and
power ...
0
votes
0answers
16 views
How to find the filter coefficient of matched filter to detect the pulse of a signal?
I would to understand how to find the coefficient of a matched filter.
Let's consider we have a signal and the signal is buried in the following noise. we want to find the coefficient of a matched ...
0
votes
0answers
9 views
How to match resolution of two or more signals
I have received the following 4 time-series signals and wanted to do some data analytics using these. Although the sampling is the same for all of these signals, 2 samples per second, problem is that ...
0
votes
0answers
36 views
On $H_\infty$ norm for transfer function
For a given scenario in the context of control system, I'm trying to investigate how the $H_\infty$ norm can be calculated for a transfer function as follows:
$$G(s)= \frac{w_n^2}{s^2 +2\zeta w_ns +...
0
votes
2answers
79 views
Differential Equations applications in Signal Processing [closed]
Does differential and partial differential equations subject has applications in Signal Processing? I have basic DSP background and know Matrix algebra and Fourier transforms very well, but I am not ...
0
votes
1answer
61 views
How to find the impulse response of filter that outputs the autocorrelation of the input?
I am trying to solve question 9.61 in the attached image.
I guess, by evaluating the autocorrelation for a specific instance, say $ t = 0 $ , that the impulse response, $h(t)$, should be : $$ h (t)...
1
vote
2answers
65 views
Check whether a given equation is linear
$ a = (x, y) \in \mathbb{Z}^2 $ is given as a pixel. My equation in which $g$ is image(matrix) is defined as,
$f(x, y) = 56g(x,y)+93g(x−1,y)+92g(x+1, y)−57g(x, y−1)+555g(x, y+1) $
How can we know ...
0
votes
1answer
43 views
how to set Equalizer's coefficient using generalized eigenvector.
In that paper https://jwcn-eurasipjournals.springeropen.com/articles/10.1186/1687-1499-2012-72 .. Equation 3,
$J(w) = w^HBw/w^HCw$ ..
$B$ and $C$ are matrices, $w$ the filter coefficients vector ...
0
votes
2answers
576 views
What does it mean for a function to have frequencies?
In a lecture, my professor mentioned that $\cos$ has two frequencies. I see that using the inverse Euler's formula we can express $\cos$ as a some linear combination of complex exponentials, each with ...
1
vote
1answer
37 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
40 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
27 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.
0
votes
1answer
56 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
53 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
49 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
37 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
136 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
97 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
63 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
65 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
59 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
266 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
87 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
109 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
48 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
202 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
45 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
164 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
285 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
761 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
87 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
375 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
32 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 ...
7
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
53 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
134 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
74 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
35 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
82 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
40 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
39 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
100 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,...
8
votes
5answers
766 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
79 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
115 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
52 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....
1
vote
3answers
99 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
80 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
189 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 ...
