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25 views

Dummy DSP student request for IDTFT Calculation and Integration Process

We have one DSP course in biomedical engineering, so i have been for 10 years away from mathematical calculation and working in this years by programing by c and ..., so one of our exercise is to ...
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0answers
41 views

Solving equation with convolution

I have the measured signal $y(t)$ that can be modeled in the frequency domain as $Y(f)$: $$Y(f) = X(f)\cdot A(f) - [X(f)\cdot B(f)] \ast C(f)$$ where $\ast$ is the convolution. I know $A(f)$, $B(f)$,...
2
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0answers
132 views

Derivation of ZOH Discretization

I'm trying to understand the derivation of the zero order hold discretization method, and I have a couple of questions about some of the steps. I think I understand the first part, this is just the ...
0
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1answer
20 views

Why is the anisotropic diffusion equivalent to the heat equation when using a constant diffusion coefficient?

The anisotropic diffusion is defined as $$\frac{\partial I}{\partial t} = \mathrm{div} \left( c(x,y,t) \nabla I \right)= \nabla c \cdot \nabla I + c(x,y,t) \Delta I$$ where $\operatorname{div}$ is ...
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2answers
105 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 ...
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0answers
60 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 ...
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0answers
37 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
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0answers
16 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
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0answers
38 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 +...
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2answers
159 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
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1answer
95 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)...
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2answers
101 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 ...
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1answer
46 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
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2answers
583 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
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1answer
48 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
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2answers
41 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
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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.
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1answer
70 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
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1answer
56 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
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1answer
57 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 ...
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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 ...
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0answers
192 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
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2answers
131 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 $v \sim CN(0,2\sigma^2_v)$ ...
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2answers
65 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 ...
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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
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1answer
76 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
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2answers
394 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
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1answer
108 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
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1answer
197 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
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1answer
63 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
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2answers
324 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
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1answer
57 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
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1answer
204 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
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2answers
408 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
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3answers
844 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
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0answers
116 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
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1answer
740 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
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1answer
33 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
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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
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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
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0answers
152 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
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1answer
78 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
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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
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2answers
85 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 ...
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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
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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
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2answers
104 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,...
9
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5answers
796 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
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1answer
80 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
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1answer
147 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 ...