Questions tagged [math]

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12
votes
6answers
21k views

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 ...
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 ...
10
votes
4answers
7k 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 ...
9
votes
3answers
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 ...
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 ...
8
votes
2answers
4k 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.
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 ...
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 ...
5
votes
3answers
577 views

Upper Bound for the DFT (FFT) Coefficients of a Bounded Signal

I have a discrete signal $x[n]$ of length say $1024$, such that : $$-1 \leq x[n] \leq 1, \qquad \forall n$$ and let $y[k]$ be its DFT. Is there an upper bound for $|y[k]|$ ? (i.e. for the DFT of a ...
5
votes
1answer
84 views

Why Does the Median Filter Minimize the Absolute Value Error $L_1$ Cost Function?

I can easily prove that the mean filter minimizes the square error $L_2$ cost function using simple calculus. However, how do you prove that the median filter is optimal with respect the absolute ...
5
votes
1answer
125 views

Why Cramér spectral representation and not DTFT for stochastic process

In a lot of time-series analysis references I find (written by mathematicians or statisticians rather than engineers), I find the following signal decomposition for a stochastic process, termed the &...
5
votes
2answers
178 views

Minimum Mean Square Estimator - Equivalent Expressions to Minimize

Given $ M \in \mathbb{R}^{N \times N} $ which is a Positive Definite Matrix. Let $ \hat{x} $ the MMSE of $ x $ given $ z $, namely $ \hat{x} = \mathbb{E} \left[ x \mid z \right] $. Prove the ...
5
votes
1answer
711 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 ...
5
votes
1answer
507 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 ...
4
votes
1answer
1k 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 ...
4
votes
3answers
149 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 ...
4
votes
2answers
15k 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 ...
4
votes
1answer
904 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 ...
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 ...
3
votes
3answers
998 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 ...
3
votes
2answers
284 views

Deriviation of the "Twiddle Sum" property

I can't seem to understand how to derive the "twiddle sum" property: $$\sum_{n=0}^{N-1}W_{N}^{kn}=N \ \delta[k\bmod N] $$ where $$ W_{N} \triangleq e^{\frac{j 2 \pi }{N}} $$ and $$ \delta[n] \...
3
votes
1answer
881 views

Determine Parameters of gaussian distribution to fit a certain Image-Filter

I'd like to know what the parameters for the two-dimensional gaussian distributions are, which create this (I think) quite common image-processing filter: $$ h_\sigma(x,y)= \left[ \matrix { 1&2&...
3
votes
2answers
456 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 ...
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 ...
3
votes
1answer
77 views

Is it possible to compute the PSD of s(t)^2 given the PSD of s(t)?

Given a well-behaved signal $s(t)$ given by, $s(t) = \sum_{i}^{N} \cos( \omega_it + \phi_i) $ with a large $N$ and a well-known power spectral density (PSD), $\tilde{S_1}(\omega)$. Is it possible to ...
3
votes
1answer
339 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 ...
2
votes
2answers
704 views

DFT of pure sinusoidal wave

I'm writing a program in which you can synthesize waves by adding to a sound's Fourier transform, and then inverse the transform to get the modified sound. In order to do this, I need to know what to ...
2
votes
2answers
476 views

Coordinate system in digital image processing

I am trying to understand image coordinate system where I read that it is reversed. So if $f(x,y)$ is actually $f(y,x)$ where $f(\cdot, \cdot)$ is the image. I am a bit confused and would appreciate ...
2
votes
1answer
100 views

Digital integrator

I have been implementing a control software where I need to calculate a magnetic flux based on the measurement of the phase voltages of a three phase grid (basically three sinewaves) according to the ...
2
votes
1answer
101 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\...
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?
2
votes
0answers
586 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 ...
2
votes
0answers
179 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 (...
2
votes
0answers
345 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: $$ = - \int_{-\infty}^{+\infty} e^{-at}u(-t) e^{-st}dt$$ $$ = - \int_{-\infty}^{0} e^{-at} e^{-st}dt$$...
2
votes
0answers
585 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 ...
2
votes
0answers
173 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 ...
1
vote
2answers
939 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 ...
1
vote
2answers
106 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 ...
1
vote
2answers
158 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 ...
1
vote
1answer
129 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 $$ \|z\|_a \geq \|z\|_b \quad \forall a\leq b $$
1
vote
2answers
46 views

Parseval's Theorem for Time Domain Data with DC

I have read several times that the variance for time domain data with a zero mean is equal to the integral of the power spectral density divided by N. From wikipedia, the discrete form of Parseval's ...
1
vote
2answers
325 views

Problems computing the DFT of finite length sequence

I am having trouble finding the same answer as the solution manual for this sequence. The problem asks to compute the DFT of $$ x[n] = \begin{cases} 1 & \text{for even } n \in \{0\...
1
vote
1answer
231 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 ...
1
vote
2answers
125 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?
1
vote
2answers
90 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: ...
1
vote
3answers
6k 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 ...
1
vote
2answers
55 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
2answers
1k views

The Mathematical Relationship Between Derivative and Convolution

What I'm looking for is the mathematical proof of why we can calculate derivatives as a simple convolution of a mask, and how do we get that mask? I know it has something to do with how the derivative ...
1
vote
1answer
56 views

Calculating Haar features

I am confused on how to solve this problem, does anyone have any ideas? https://www.merl.com/publications/docs/TR2004-043.pdf For a Haar filter $H \in \mathbb{R}^{3 \times 5}$ defined as below: What ...
1
vote
2answers
104 views

Question about delayed sampled sinusoid math expression

I have been studying the digital audio processing by using the book <Designing Audio Effect Plugins in C++>. For analog Sinusoid: Complex Sinusoid = $e^{jωt}$ Delayed Sinusoid = $e^{jω(t−n)} = e^...