# Questions tagged [math]

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### 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 ...
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 ...
3k views

### What is the math behind median filter's noise reduction property?

I am interested in understanding the mathematical reason for why does applying a median filter on an image (or signal) result in reduction of noise.
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### 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 ...
3k 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 ...
2k views

### Functional analysis for signal processing engineers

It seems that the most advanced pure mathematics course most EE engineers take is Fourier analysis, and after that it's basically 'applied' courses. There's probably a good reason for this, but I'm ...
3k 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 ...
445 views

### Derivation of fixed-point $\tt atan2$ with self-normalization

I'm trying to understand the maths behind this Fast fixed-point $\tt atan2$ calculation with self-normalization. In particular, equation $(2)$ for theta1 appears to ...
1k 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 ...
770 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 ...
251 views

726 views

### How moving part pixel intensity values of video frames becomes dominant compared to stationary part intensities in reconstructed frames?

Hello everyone i want to do dynamic texture video sementation using the Fourier transform in MATLAB. I am applying 3-D fft on dynamic texture video frames (using matlab function 'fftn') and ...
225 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 ...
144 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 ...
103 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 ...
I'm trying to derive the Chernoff bound $\mathrm{erfc}(x) \le \exp(-x^2)$, by first showing: $$\mathrm{erfc}(x) = \frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\exp\left(-\frac{x^2}{\cos^2\theta}\right)d\... 1answer 44 views ### Splines and Gaussians [closed] Higher-order splines all mostly look like Gaussians. I wonder how higher-order splines (from 3rd order onwards) and Gaussians differ. How can one differentiate them? 1answer 212 views ### SIFT: why it is invariant to affine tranlsation? I am studying SIFT. I am a little confused in some of Lowe's algorithm. In order to have a descriptor invariant to translations like scaling, we must get rid of the scaling factor in LoG (Laplacian ... 0answers 143 views ### Career Question: As a person who likes math is signal processing a career that's worth it? I like high level math and I love deep theory based math. In industry will I use high level math and theory to implement my own algorithms? I love high level math but I do not love probability (... 0answers 163 views ### Laplace Transform of -e^{-at}u(-t) I have found a problem in applying Laplace Transform to -e^{-at}u(-t) I am doing these steps:$$ = - \int_{-\infty}^{+\infty} e^{-at}u(-t) e^{-st}dt = - \int_{-\infty}^{0} e^{-at} e^{-st}dt$$... 0answers 538 views ### Tracking 2D positioning with IMU Sensor I am using a miniature car and I want to estimate the position. We can not use GPS modules and most of the tracking systems that I saw, are using IMU sensor with the GPS module. In our car we are able ... 0answers 167 views ### How much is the chance for a pure mathematician (geometer) to get a postdoc in imaging if he switches fields after Ph.D. and apply for one? [closed] Before I start my question, I should probably mention that this question might not be the right question to ask here, but I tried academiabeta, and stackoverflow, but without getting any to-the-point ... 2answers 303 views ### Closed form expression for \sum_{k=0}^{n} \alpha^{-k}? I have seen this sum a couple of times now in the textbook and in the problems book but I don't know how to solve it. I know this might be more suited for math.stackexchange but since this is a fairly ... 2answers 82 views ### What is this theorem in this formula? I just read a paper yesterday and there is confusion: I want to know what is reconstruction fidelity term and the theorem of changing the formula from upper to lower one. Is this theorm from ... 2answers 66 views ### Check whether a given equation is linear  a = (x, y) \in \mathbb{Z}^2  is given as a pixel. My equation in which g is image(matrix) is defined as, f(x, y) = 56g(x,y)+93g(x−1,y)+92g(x+1, y)−57g(x, y−1)+555g(x, y+1)  How can we know ... 1answer 79 views ### How to prove this norm inequality?$$ \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 $$2answers 100 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? 2answers 87 views ### Showing processes as an Equation I have developed an algorithm that counts the number of times a particular block (within a 2D block) crosses zero. Here's an example: ... 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) ... 1answer 39 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 ... 1answer 48 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, ... 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 ... 1answer 120 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 ... 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, ... 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.... 1answer 2k views ### Amplitude calculation I've got an array of different values representing the y-values in a plot: ... 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 ... 0answers 150 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 ... 2answers 288 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 ... 0answers 94 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 ... 1answer 218 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 (... 0answers 372 views ### How to understand energy function in thin plate splines? guys! I have carried out some experiments using matlab warp tools from here. But the results are not good, so I decide to read its original paper Principal Warps: Thin-Plate Splines and the ... 1answer 341 views ### Modeling Load Modulation I'm trying to create a simple NFC/RFID software simulator and am having a hard time understanding the math fully. I know that NFC tags respond by load modulating the 13.56 MHz wave that the reader is ... 0answers 118 views ### Recovery of Bandlimited Signals after Companding and Low Pass Filter Let \Omega > 0 and denote by \mathcal{B}_\Omega the subspace of L^2(\Bbb R) consisting of signals that are bandlimited to (-\Omega, \Omega). Denote \mathcal{L}_{\Omega} : L^2(\Bbb R) \... 1answer 66 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 ... 2answers 579 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 ... 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 ...
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 $v \sim CN(0,2\sigma^2_v)$ ...