Questions tagged [math]

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5
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6answers
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What is the first derivative of Dirac delta function?

Could you please help me in a simple way, what is the first derivative of a Dirac delta function? I found this answer: The informal answer is a positive Delta function immediately followed by a ...
1
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1answer
43 views

Finding the system output by convolution

pretty new to this concept, so do bear with me. A linear dynamic system is described by the following differential equation: Transfer function H(s) is calculated to be = I've already found the ...
0
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0answers
35 views

Setting some values in a vector to have specific values after multiplying with such matrix

Assume I have a known matrix $X$ of size, for example, $(16$ x $16)$, and a vector $z$ of size $(1$ x $16)$ where only some equispaced values are known, for example $1:4:16$ or values in locations $(...
0
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1answer
45 views

Solve equations with conjugates multiplications

If I have a two variables $x_1$, $x_2$ and two equations. In the first equation the first variable $x_1$ is multiplied with the conjugate of same number which is multiplied with $x_2$, and in the ...
3
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1answer
937 views

Design $ {L}_{2} $ Norm Optimal Infinite Impulse Response (IIR) Filters

It is widely known that matching a FIR filter of fixed length to a band model is an unconstrained QP-problem. The MATLAB function firls() implements a solution to ...
0
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4answers
61 views

On the simplification using trigonometric functions

Assume I have a matrix $D$ whose its entries are as below : Where $A$ and $B$ can be written using using the trigonometric functions for (1) as: My question, Is it possible to simplify (1) more? ...
0
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2answers
42 views

Q format doubts

Is my understanding correct w.r.t Q format. signed Q15.1 format means, 15 bits for fractional and 1 for integer bits SO max range is - 2^15 / 2^15 to + (2 ^ 15 - 1) / 2^15 signed Q7.0 means, 7 bits ...
0
votes
1answer
134 views

Separation of signals without loss of information about the signals themselves

There is a signal that contains: Rectangular pulses on frequency of 1 kHz Rectangular pulses on frequency of 1.4 kHz Other noises. Required: Separate 1kHz pulses from everything else and measure ...
0
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1answer
49 views

Calculate autocorrelation of a sinus

I am a beginer in signal processing. I am currently studying a signal $s(t) = \cos(h(t))$, and I have to calculate its autocorellation. Since $h(t)$ is periodic of period $T_s$, I calculated the value ...
1
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3answers
79 views

Convolution of two rectangular pulses intuition

From what I understand the convolution of the impulse response of a system with the input to that system gives the output. Now if the impulse response is a rectangular function and the input is also ...
0
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2answers
29 views

Understanding Communication Theory Jargon for Statistical Linear Models

Coming from a statistical background, I understand the following formula: \begin{equation} y = X\beta + \epsilon \end{equation} as a linear regression model, with data input-output pair: $(X,y)$, ...
1
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1answer
445 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
56 views

Absolute value based AM envelope detection viewed in the frequency domain

In my understanding when a passband AM signal is demodulated with an envelope detector taking the absolute value restores the original baseband version of the signal and the low-pass filter removes ...
0
votes
0answers
10 views

active contour without edges

i am new to image segmentation.i am studying chanvese ACWE paper1.This is level set equation of chan vese paper ACWE. Above, μ,ν,λ1,λ2 are real parameters, c1,c2 are constants determined for ...
0
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0answers
13 views

Spread out flux in 2D gaussian

I want to generate a 2D Gaussian model with flux spread out for a given radius. From theory, I found out that the total integrated flux of a Gaussian model. Now I want to image it such that total flux ...
1
vote
1answer
52 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 ...
1
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1answer
268 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
0answers
43 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
540 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 ...
8
votes
4answers
2k views

Books/resources for implementing various mathematical functions in fixed point arithmetic for DSP purposes

I'm looking for books or resources that cover the following in detail: implementing mathematical functions (e.g., logarithm, exponential, sine, cosine, inverse) in fixed point arithmetic for DSP ...
0
votes
1answer
28 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 ...
0
votes
2answers
329 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
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0answers
99 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
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 +...
0
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2answers
357 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
129 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
1answer
92 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 ...
1
vote
2answers
112 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 ...
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 ...
0
votes
1answer
48 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
588 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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2answers
46 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) ...
1
vote
1answer
68 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
10k 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
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.
0
votes
1answer
81 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, ...
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
1answer
60 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
237 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
166 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
71 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
69 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
97 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
547 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
121 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 ...
0
votes
1answer
76 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
366 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 ...
10
votes
5answers
873 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
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 ...